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derivatives 0:9,.|5 (with respect to some specified time coordinate) to be the “momenta”, which together specify the state. (There will also be coordinates and momenta for the matter fields, which we will not consider explicitly.) In fact the equations G',, = 87GT),, do involve second derivatives of the metric with respect to time (since the connection involves first derivatives of the metric and the Einstein tensor involves first derivatives of the connection), so we seem to be on the right track. However, the Bianchi identity tells us that V,G*” = 0. We can rewrite this equation as

Figure 47 derivatives 0:9,.|5 (with respect to some specified time coordinate) to be the “momenta”, which together specify the state. (There will also be coordinates and momenta for the matter fields, which we will not consider explicitly.) In fact the equations G',, = 87GT),, do involve second derivatives of the metric with respect to time (since the connection involves first derivatives of the metric and the Einstein tensor involves first derivatives of the connection), so we seem to be on the right track. However, the Bianchi identity tells us that V,G*” = 0. We can rewrite this equation as

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Consider a garden-variety 2-dimensional plane. It is typically convenient to label the points on such a plane by introducing coordinates, for example by defining orthogonal x and y axes and projecting each point onto these axes in the usual way. However, it is clear that most of the interesting geometrical facts about the plane are independent of our choice of coordinates. As a simple example, we can consider the distance between two points, given
Consider a garden-variety 2-dimensional plane. It is typically convenient to label the points on such a plane by introducing coordinates, for example by defining orthogonal x and y axes and projecting each point onto these axes in the usual way. However, it is clear that most of the interesting geometrical facts about the plane are independent of our choice of coordinates. As a simple example, we can consider the distance between two points, given
We therefore say that the distance is invariant under such changes of coordinates. This is why it is useful to think of the plane as 2-dimensional: although we use two distinct numbers to label each point, the numbers are not the essence of the geometry, since we can rotate axes into each other while leaving distances and so forth unchanged. In Newtonian physics this is not the case with space and time; there is no useful notion of rotating space and time into each other. Rather, the notion of “all of space at a single moment in time” has a meaning independent of coordinates.
We therefore say that the distance is invariant under such changes of coordinates. This is why it is useful to think of the plane as 2-dimensional: although we use two distinct numbers to label each point, the numbers are not the essence of the geometry, since we can rotate axes into each other while leaving distances and so forth unchanged. In Newtonian physics this is not the case with space and time; there is no useful notion of rotating space and time into each other. Rather, the notion of “all of space at a single moment in time” has a meaning independent of coordinates.
traight lines moving at the speed of light is called a light cone; this entire set is invarian inder Lorentz transformations. Light cones are naturally divided into future and past; th et of all points inside the future and past light cones of a point p are called timelik separated from p, while those outside the light cones are spacelike separated and thos m the cones are lightlike or null separated from p. Referring back to (1.3), we see that th nterval between timelike separated points is negative, between spacelike separated points i ositive, and between null separated points is zero. (The interval is defined to be s?, not th quare root of this quantity.) Notice the distinction between this situation and that in th Newtonian world; here, it is impossible to say (in a coordinate-independent way) whether : oint that is spacelike separated from p is in the future of p, the past of p, or “at the sam ”% ime”.
traight lines moving at the speed of light is called a light cone; this entire set is invarian inder Lorentz transformations. Light cones are naturally divided into future and past; th et of all points inside the future and past light cones of a point p are called timelik separated from p, while those outside the light cones are spacelike separated and thos m the cones are lightlike or null separated from p. Referring back to (1.3), we see that th nterval between timelike separated points is negative, between spacelike separated points i ositive, and between null separated points is zero. (The interval is defined to be s?, not th quare root of this quantity.) Notice the distinction between this situation and that in th Newtonian world; here, it is impossible to say (in a coordinate-independent way) whether : oint that is spacelike separated from p is in the future of p, the past of p, or “at the sam ”% ime”.
Later we will relate the tangent space at each point to things we can construct from the spacetime itself. For right now, just think of T;, as an abstract vector space for each point in spacetime. A (real) vector space is a collection of objects (“vectors”) which, roughly speaking, can be added together and multiplied by real numbers in a linear way. Thus, for any two vectors V and W and real numbers a and b, we have
Later we will relate the tangent space at each point to things we can construct from the spacetime itself. For right now, just think of T;, as an abstract vector space for each point in spacetime. A (real) vector space is a collection of objects (“vectors”) which, roughly speaking, can be added together and multiplied by real numbers in a linear way. Thus, for any two vectors V and W and real numbers a and b, we have
Another familiar example occurs in quantum mechanics, where vectors in the Hilbert space are represented by kets, |W). In this case the dual space is the space of bras, (¢|, and the action gives the number (¢|w). (This is a complex number in quantum mechanics, but the idea is precisely the same.) —_ - _ an
Another familiar example occurs in quantum mechanics, where vectors in the Hilbert space are represented by kets, |W). In this case the dual space is the space of bras, (¢|, and the action gives the number (¢|w). (This is a complex number in quantum mechanics, but the idea is precisely the same.) —_ - _ an
for different cases. For spacelike paths we define the path length
for different cases. For spacelike paths we define the path length
We will now approach the rigorous definition of this simple idea, which requires a number of preliminary definitions. Many of them are pretty clear anyway, but it’s nice to be complete. With all of these examples, the notion of a manifold may seem vacuous; what isn’t a manifold? There are plenty of things which are not manifolds, because somewhere they do not look locally like R”. Examples include a one-dimensional line running into a two- dimensional plane, and two cones stuck together at their vertices. (A single cone is okay; you can imagine smoothing out the vertex.)
We will now approach the rigorous definition of this simple idea, which requires a number of preliminary definitions. Many of them are pretty clear anyway, but it’s nice to be complete. With all of these examples, the notion of a manifold may seem vacuous; what isn’t a manifold? There are plenty of things which are not manifolds, because somewhere they do not look locally like R”. Examples include a one-dimensional line running into a two- dimensional plane, and two cones stuck together at their vertices. (A single cone is okay; you can imagine smoothing out the vertex.)
A map ¢ is called one-to-one (or “injective”) if each element of N has at most one element of MZ mapped into it, and onto (or “surjective”) if each element of NV has at least one element of M mapped into it. (If you think about it, a better name for “one-to-one” would be “two-to-two”.) Consider a function ¢: R — R. Then ¢(z) = e” is one-to-one, but not onto; ¢(x) = 2° — z is onto, but not one-to-one; ¢(x) = x° is both; and ¢(x) = 2? is neither.
A map ¢ is called one-to-one (or “injective”) if each element of N has at most one element of MZ mapped into it, and onto (or “surjective”) if each element of NV has at least one element of M mapped into it. (If you think about it, a better name for “one-to-one” would be “two-to-two”.) Consider a function ¢: R — R. Then ¢(z) = e” is one-to-one, but not onto; ¢(x) = 2° — z is onto, but not one-to-one; ¢(x) = x° is both; and ¢(x) = 2? is neither.
The notion of continuity of a map between topological spaces (and thus manifolds) is actually a very subtle one, the precise formulation of which we won’t really need. However the intuitive notions of continuity and differentiability of maps ¢ : R™ — R” between Euclidean spaces are useful. A map from R”™ to R” takes an m-tuple (z',x?,...,2™) to an n-tuple (y', y?,...,y”), and can therefore be thought of as a collection of n functions ¢' of
The notion of continuity of a map between topological spaces (and thus manifolds) is actually a very subtle one, the precise formulation of which we won’t really need. However the intuitive notions of continuity and differentiability of maps ¢ : R™ — R” between Euclidean spaces are useful. A map from R”™ to R” takes an m-tuple (z',x?,...,2™) to an n-tuple (y', y?,...,y”), and can therefore be thought of as a collection of n functions ¢' of
There is nothing illegal or immoral about using this form of the chain rule, but you should be able to visualize the maps that underlie the construction. Recall that when m = n the determinant of the matrix Oy?/Ox* is called the Jacobian of the map, and the map is invertible whenever the Jacobian is nonzero.
There is nothing illegal or immoral about using this form of the chain rule, but you should be able to visualize the maps that underlie the construction. Recall that when m = n the determinant of the matrix Oy?/Ox* is called the Jacobian of the map, and the map is invertible whenever the Jacobian is nonzero.
A chart or coordinate system consists of a subset U of a set M, along with a one-to- one map ¢: U — R”, such that the image ¢(U) is open in R. (Any map is onto its image, so the map ¢: U — @(U) is invertible.) We then can say that U is an open set in M. (We have thus induced a topology on M, although we will not explore this.) C® atlas is an indexed collection of charts {(Ua,¢a)} which satisfies two conditions:
A chart or coordinate system consists of a subset U of a set M, along with a one-to- one map ¢: U — R”, such that the image ¢(U) is open in R. (Any map is onto its image, so the map ¢: U — @(U) is invertible.) We then can say that U is an open set in M. (We have thus induced a topology on M, although we will not explore this.) C® atlas is an indexed collection of charts {(Ua,¢a)} which satisfies two conditions:
A somewhat more complicated example is provided by $?, where once again no single a ERO SRSA ROS MO TE One thing that is nice about our definition is that it does not rely on an embedding of the nanifold in some higher-dimensional Euclidean space. In fact any n-dimensional manifolc an be embedded in R”” (“Whitney’s embedding theorem”), and sometimes we will mak se of this fact (such as in our definition of the sphere above). But it’s important to recognize hat the manifold has an individual existence independent of any embedding. We have nc eason to believe, for example, that four-dimensional spacetime is stuck in some larger space Actually a number of people, string theorists and so forth, believe that our four-dimensiona vorld is part of a ten- or eleven-dimensional spacetime, but as far as GR is concerned the -dimensional view is perfectly adequate.) be er hy ny i ee oe, a, ey ee ee, a: ee a ee: ee i ee: fi
A somewhat more complicated example is provided by $?, where once again no single a ERO SRSA ROS MO TE One thing that is nice about our definition is that it does not rely on an embedding of the nanifold in some higher-dimensional Euclidean space. In fact any n-dimensional manifolc an be embedded in R”” (“Whitney’s embedding theorem”), and sometimes we will mak se of this fact (such as in our definition of the sphere above). But it’s important to recognize hat the manifold has an individual existence independent of any embedding. We have nc eason to believe, for example, that four-dimensional spacetime is stuck in some larger space Actually a number of people, string theorists and so forth, believe that our four-dimensiona vorld is part of a ten- or eleven-dimensional spacetime, but as far as GR is concerned the -dimensional view is perfectly adequate.) be er hy ny i ee oe, a, ey ee ee, a: ee a ee: ee i ee: fi
chart will cover the manifold. A Mercator projection, traditionally used for world maps, misses both the North and South poles (as well as the International Date Line, which involves the same problem with @ that we found for $1.) Let’s take S? to be the set of points in R? defined by (a1)? + (x?)? + (#°)? = 1. We can construct a chart from an open set U;, defined to be the sphere minus the north pole, via “stereographic projection”: the set of points which aren’t included.
chart will cover the manifold. A Mercator projection, traditionally used for world maps, misses both the North and South poles (as well as the International Date Line, which involves the same problem with @ that we found for $1.) Let’s take S? to be the set of points in R? defined by (a1)? + (x?)? + (#°)? = 1. We can construct a chart from an open set U;, defined to be the sphere minus the north pole, via “stereographic projection”: the set of points which aren’t included.
Just thinking of MM and N as sets, we cannot nonchalantly differentiate the map f, since we don’t know what such an operation means. But the coordinate charts allow us to construct the map (wo fod@') : R™ — R”. (Feel free to insert the words “where the maps are defined” wherever appropriate, here and later on.) This is just a map between Euclidean spaces, and all of the concepts of advanced calculus apply. For example f, thought of as an N-valued function on M, can be differentiated to obtain Of /Ox", where the x" represent R”™. The point is that this notation is a shortcut, and what is really going on is
Just thinking of MM and N as sets, we cannot nonchalantly differentiate the map f, since we don’t know what such an operation means. But the coordinate charts allow us to construct the map (wo fod@') : R™ — R”. (Feel free to insert the words “where the maps are defined” wherever appropriate, here and later on.) This is just a map between Euclidean spaces, and all of the concepts of advanced calculus apply. For example f, thought of as an N-valued function on M, can be differentiated to obtain Of /Ox", where the x" represent R”™. The point is that this notation is a shortcut, and what is really going on is
Is it the vector space that we would like to identify with the tangent space? The easiest way to become convinced is to find a basis for the space. Consider again a coordinate chart with coordinates x". Then there is an obvious set of n directional derivatives at p, namely the partial derivatives O,, at p. We are now going to claim that the partial derivative operators {0,} at p form a basis fo: the tangent space T,,. (It follows immediately that J), is n-dimensional, since that is the number of basis vectors.) To see this we will show that any directional derivative can be decomposed into a sum of real numbers times partial derivatives. This is in fact just the familiar expression for the components of a tangent vector, but it’s nice to see it from the big-machinery approach. Consider an n-manifold WM, a coordinate chart ¢: M — R”, ¢ curve y: R— M, and a function f : M — R. This leads to the following tangle of maps:
Is it the vector space that we would like to identify with the tangent space? The easiest way to become convinced is to find a basis for the space. Consider again a coordinate chart with coordinates x". Then there is an obvious set of n directional derivatives at p, namely the partial derivatives O,, at p. We are now going to claim that the partial derivative operators {0,} at p form a basis fo: the tangent space T,,. (It follows immediately that J), is n-dimensional, since that is the number of basis vectors.) To see this we will show that any directional derivative can be decomposed into a sum of real numbers times partial derivatives. This is in fact just the familiar expression for the components of a tangent vector, but it’s nice to see it from the big-machinery approach. Consider an n-manifold WM, a coordinate chart ¢: M — R”, ¢ curve y: R— M, and a function f : M — R. This leads to the following tangle of maps:
If \ is the parameter along y, we want to expand the vector/operator a in terms of the
If \ is the parameter along y, we want to expand the vector/operator a in terms of the
to define our basis. Therefore a change of coordinates induces a change of basis: Since the basis vectors are usually not written explicitly, the rule (2.13) for transforming components is what we call the “vector transformation law.” We notice that it is com- patible with the transformation of vector components in special relativity under Lorentz transformations, V“’ = A“ uV", since a Lorentz transformation is a special kind of coordi- nate transformation, with «!” = AM" yt", But (2.13) is much more general, as it encompasses the behavior of vectors under arbitrary changes of coordinates (and therefore bases), not just linear transformations. As usual, we are trying to emphasize a somewhat subtle ontological distinction — tensor components do not change when we change coordinates, they change when we change the basis in the tangent space, but we have decided to use the coordinates to define our basis. Therefore a change of coordinates induces a change of basis: It’s tempting to think, “why shouldn’t the function f itself be considered the one-form, and df /dX its action?” The point is that a one-form, like a vector, exists only at the point it is defined, and does not depend on information at other points on M. If you know a function in some neighborhood of a point you can take its derivative, but not just from knowing its value at the point; the gradient, on the other hand, encodes precisely the information
to define our basis. Therefore a change of coordinates induces a change of basis: Since the basis vectors are usually not written explicitly, the rule (2.13) for transforming components is what we call the “vector transformation law.” We notice that it is com- patible with the transformation of vector components in special relativity under Lorentz transformations, V“’ = A“ uV", since a Lorentz transformation is a special kind of coordi- nate transformation, with «!” = AM" yt", But (2.13) is much more general, as it encompasses the behavior of vectors under arbitrary changes of coordinates (and therefore bases), not just linear transformations. As usual, we are trying to emphasize a somewhat subtle ontological distinction — tensor components do not change when we change coordinates, they change when we change the basis in the tangent space, but we have decided to use the coordinates to define our basis. Therefore a change of coordinates induces a change of basis: It’s tempting to think, “why shouldn’t the function f itself be considered the one-form, and df /dX its action?” The point is that a one-form, like a vector, exists only at the point it is defined, and does not depend on information at other points on M. If you know a function in some neighborhood of a point you can take its derivative, but not just from knowing its value at the point; the gradient, on the other hand, encodes precisely the information
is more subtle than having the metric depend on the coordinates, since in the example above we showed how the metric in flat Euclidean space in spherical coordinates is a function of r and @. Later, we shall see that constancy of the metric components is sufficient for a space to be flat, and in fact there always exists a coordinate system on any flat space in which the metric is constant. But we might not want to work in such a coordinate system, and we might not even know how to find it; therefore we will want a more precise characterization of the curvature, which will be introduced down the road. A ....f.] VL... yt gst hl UGK {EL lw ee tC tk CUCL nk gat. Ly a) ne |
is more subtle than having the metric depend on the coordinates, since in the example above we showed how the metric in flat Euclidean space in spherical coordinates is a function of r and @. Later, we shall see that constancy of the metric components is sufficient for a space to be flat, and in fact there always exists a coordinate system on any flat space in which the metric is constant. But we might not want to work in such a coordinate system, and we might not even know how to find it; therefore we will want a more precise characterization of the curvature, which will be introduced down the road. A ....f.] VL... yt gst hl UGK {EL lw ee tC tk CUCL nk gat. Ly a) ne |
The concept of moving a vector along a path, keeping constant all the while, is known as parallel transport. As we shall see, parallel transport is defined whenever we have a Having set up the machinery of connections, the first thing we will do is discuss paralle transport. Recall that in flat space it was unnecessary to be very careful about the fact that vectors were elements of tangent spaces defined at individual points; it is actually very natural to compare vectors at different points (where by “compare” we mean add, subtract take the dot product, etc.). The reason why it is natural is because it makes sense, in flat space, to “move a vector from one point to another while keeping it constant.” Then once we get the vector from one point to another we can do the usual operations allowed in ¢ vector space.
The concept of moving a vector along a path, keeping constant all the while, is known as parallel transport. As we shall see, parallel transport is defined whenever we have a Having set up the machinery of connections, the first thing we will do is discuss paralle transport. Recall that in flat space it was unnecessary to be very careful about the fact that vectors were elements of tangent spaces defined at individual points; it is actually very natural to compare vectors at different points (where by “compare” we mean add, subtract take the dot product, etc.). The reason why it is natural is because it makes sense, in flat space, to “move a vector from one point to another while keeping it constant.” Then once we get the vector from one point to another we can do the usual operations allowed in ¢ vector space.
connection; the intuitive manipulation of vectors in flat space makes implicit use of the Christoffel connection on this space. The crucial difference between flat and curved spaces is that, in a curved space, the result of parallel transporting a vector from one point to another will depend on the path taken between the points. Without yet assembling the complete mechanism of parallel transport, we can use our intuition about the two-sphere to see that this is the case. Start with a vector on the equator, pointing along a line of constant longitude. Parallel transport it up to the north pole along a line of longitude in the obvious way. Then take the original vector, parallel transport it along the equator by an angle 6, and then move it up to the north pole as before. It is clear that the vector, parallel transported along two paths, arrived at the same destination with two different values (rotated by @). It therefore appears as if there is no natural way to uniquely move a vector from one wngent space to another; we can always parallel transport it, but the result depends on the ath, and there is no natural choice of which path to take. Unlike some of the problems we ave encountered, there is no solution to this one — we simply must learn to live with the ict that two vectors can only be compared in a natural way if they are elements of the same wngent space. For example, two particles passing by each other have a well-defined relative slocity (which cannot be greater than the speed of light). But two particles at different oints on a curved manifold do not have any well-defined notion of relative velocity — the yncept simply makes no sense. Of course, in certain special situations it is still useful to talk 3 if it did make sense, but it is necessary to understand that occasional usefulness is not a ibstitute for rigorous definition. In cosmology, for example, the light from distant galaxies redshifted with respect to the frequencies we would observe from a nearby stationary yurce. Since this phenomenon bears such a close resemblance to the conventional Doppler fect due to relative motion, it is very tempting to say that the galaxies are “receding away om us” at a speed defined by their redshift. At a rigorous level this is nonsense, what Vittgenstein would call a “grammatical mistake” — the galaxies are not receding, since the otion of their velocity with respect to us is not well-defined. What is actually happening that the metric of spacetime between us and the galaxies has changed (the universe has
connection; the intuitive manipulation of vectors in flat space makes implicit use of the Christoffel connection on this space. The crucial difference between flat and curved spaces is that, in a curved space, the result of parallel transporting a vector from one point to another will depend on the path taken between the points. Without yet assembling the complete mechanism of parallel transport, we can use our intuition about the two-sphere to see that this is the case. Start with a vector on the equator, pointing along a line of constant longitude. Parallel transport it up to the north pole along a line of longitude in the obvious way. Then take the original vector, parallel transport it along the equator by an angle 6, and then move it up to the north pole as before. It is clear that the vector, parallel transported along two paths, arrived at the same destination with two different values (rotated by @). It therefore appears as if there is no natural way to uniquely move a vector from one wngent space to another; we can always parallel transport it, but the result depends on the ath, and there is no natural choice of which path to take. Unlike some of the problems we ave encountered, there is no solution to this one — we simply must learn to live with the ict that two vectors can only be compared in a natural way if they are elements of the same wngent space. For example, two particles passing by each other have a well-defined relative slocity (which cannot be greater than the speed of light). But two particles at different oints on a curved manifold do not have any well-defined notion of relative velocity — the yncept simply makes no sense. Of course, in certain special situations it is still useful to talk 3 if it did make sense, but it is necessary to understand that occasional usefulness is not a ibstitute for rigorous definition. In cosmology, for example, the light from distant galaxies redshifted with respect to the frequencies we would observe from a nearby stationary yurce. Since this phenomenon bears such a close resemblance to the conventional Doppler fect due to relative motion, it is very tempting to say that the galaxies are “receding away om us” at a speed defined by their redshift. At a rigorous level this is nonsense, what Vittgenstein would call a “grammatical mistake” — the galaxies are not receding, since the otion of their velocity with respect to us is not well-defined. What is actually happening that the metric of spacetime between us and the galaxies has changed (the universe has
Since 0x7 is assumed to be small, we can expand the square root of the expression in square brackets to find
Since 0x7 is assumed to be small, we can expand the square root of the expression in square brackets to find
AAS we increase the number Of Sharp corners, the null curve comes Closer and Closer to tne timelike curve while still having zero path length. Timelike geodesics cannot therefore be curves of minimum proper time, since they are always infinitesimally close to curves of zero proper time; in fact they maximize the proper time. (This is how you can remember which twin in the twin paradox ages more — the one who stays home is basically on a geodesic, and therefore experiences more proper time.) Of course even this is being a little cavalier; actually every time we say “maximize” or “minimize” we should add the modifier “locally.” It is often the case that between two points on a manifold there is more than one geodesic. For instance, on S$? we can draw a great circle through any two points, and imagine travelling between them either the short way or the long way around. One of these is obviously longer than the other, although both are stationary points of the length functional.
AAS we increase the number Of Sharp corners, the null curve comes Closer and Closer to tne timelike curve while still having zero path length. Timelike geodesics cannot therefore be curves of minimum proper time, since they are always infinitesimally close to curves of zero proper time; in fact they maximize the proper time. (This is how you can remember which twin in the twin paradox ages more — the one who stays home is basically on a geodesic, and therefore experiences more proper time.) Of course even this is being a little cavalier; actually every time we say “maximize” or “minimize” we should add the modifier “locally.” It is often the case that between two points on a manifold there is more than one geodesic. For instance, on S$? we can draw a great circle through any two points, and imagine travelling between them either the short way or the long way around. One of these is obviously longer than the other, although both are stationary points of the length functional.
Sa Ya We won't go into detail about the properties of the exponential map, since in fact w won't be using it much, but it’s important to emphasize that the range of the map is no aecessarily the whole manifold, and the domain is not necessarily the whole tangent space The range can fail to be all of MM simply because there can be two points which are no connected by any geodesic. (In a Euclidean signature metric this is impossible, but not it 1 Lorentzian spacetime.) The domain can fail to be all of T, because a geodesic may rut nto a singularity, which we think of as “the edge of the manifold.” Manifolds which hav such singularities are known as geodesically incomplete. This is not merely a problen or careful mathematicians; in fact the “singularity theorems” of Hawking and Penrose stat chat, for reasonable matter content (no negative energies), spacetimes in general relativit; are almost guaranteed to be geodesically incomplete. As examples, the two most usefu spacetimes in GR — the Schwarzschild solution describing black holes and the Friedmann Robertson-Walker solutions describing homogeneous, isotropic cosmologies — both featur mportant singularities. Dartrine eat WH the macrhinary of narallel tranenart and epnvariant derivatives we are at lac
Sa Ya We won't go into detail about the properties of the exponential map, since in fact w won't be using it much, but it’s important to emphasize that the range of the map is no aecessarily the whole manifold, and the domain is not necessarily the whole tangent space The range can fail to be all of MM simply because there can be two points which are no connected by any geodesic. (In a Euclidean signature metric this is impossible, but not it 1 Lorentzian spacetime.) The domain can fail to be all of T, because a geodesic may rut nto a singularity, which we think of as “the edge of the manifold.” Manifolds which hav such singularities are known as geodesically incomplete. This is not merely a problen or careful mathematicians; in fact the “singularity theorems” of Hawking and Penrose stat chat, for reasonable matter content (no negative energies), spacetimes in general relativit; are almost guaranteed to be geodesically incomplete. As examples, the two most usefu spacetimes in GR — the Schwarzschild solution describing black holes and the Friedmann Robertson-Walker solutions describing homogeneous, isotropic cosmologies — both featur mportant singularities. Dartrine eat WH the macrhinary of narallel tranenart and epnvariant derivatives we are at lac
The (infinitesimal) lengths of the sides of the loop are da and 6b, respectively. Now, we know the action of parallel transport is independent of coordinates, so there should be some tensor which tells us how the vector changes when it comes back to its starting point; it will be a linear transformation on a vector, and therefore involve one upper and one lower index. But it will also depend on the two vectors A and B which define the loop; therefore there should be two additional lower indices to contract with A” and B“. Furthermore, the tensor should be antisymmetric in these two indices, since interchanging the vectors corresponds to traversing the loop in the opposite direction, and should give the inverse of the original answer. (This is consistent with the fact that the transformation should vanish if A and B are the same vector.) We therefore expect that the expression for the change dV” experienced by this vector when parallel transported around the loop should be of the form We have already argued, using the two-sphere as an example, that parallel transport of a vector around a closed loop in a curved space will lead to a transformation of the vector. The resulting transformation depends on the total curvature enclosed by the loop; it would be more useful to have a local description of the curvature at each point, which is what the Riemann tensor is supposed to provide. One conventional way to introduce the Riemann tensor, therefore, is to consider parallel transport around an infinitesimal loop. We are not going to do that here, but take a more direct route. (Most of the presentations in the literature are either sloppy, or correct but very difficult to follow.) Nevertheless, even without working through the details, it is possible to see what form the answer should take. Imagine that we parallel transport a vector V’ around a closed loop defined by two vectors A” and BE:
The (infinitesimal) lengths of the sides of the loop are da and 6b, respectively. Now, we know the action of parallel transport is independent of coordinates, so there should be some tensor which tells us how the vector changes when it comes back to its starting point; it will be a linear transformation on a vector, and therefore involve one upper and one lower index. But it will also depend on the two vectors A and B which define the loop; therefore there should be two additional lower indices to contract with A” and B“. Furthermore, the tensor should be antisymmetric in these two indices, since interchanging the vectors corresponds to traversing the loop in the opposite direction, and should give the inverse of the original answer. (This is consistent with the fact that the transformation should vanish if A and B are the same vector.) We therefore expect that the expression for the change dV” experienced by this vector when parallel transported around the loop should be of the form We have already argued, using the two-sphere as an example, that parallel transport of a vector around a closed loop in a curved space will lead to a transformation of the vector. The resulting transformation depends on the total curvature enclosed by the loop; it would be more useful to have a local description of the curvature at each point, which is what the Riemann tensor is supposed to provide. One conventional way to introduce the Riemann tensor, therefore, is to consider parallel transport around an infinitesimal loop. We are not going to do that here, but take a more direct route. (Most of the presentations in the literature are either sloppy, or correct but very difficult to follow.) Nevertheless, even without working through the details, it is possible to see what form the answer should take. Imagine that we parallel transport a vector V’ around a closed loop defined by two vectors A” and BE:
transporting the tensor first one way and then the other, versus the opposite ordering. The actual computation is very straightforward. Considering a vector field V’, we take Knowing what we do about parallel transport, we could very carefully perform the nec- essary manipulations to see what happens to the vector under this operation, and the result would be a formula for the curvature tensor in terms of the connection coefficients. It is much quicker, however, to consider a related operation, the commutator of two covariant deriva- tives. The relationship between this and parallel transport around a loop should be evident; the covariant derivative of a tensor in a certain direction measures how much the tensor changes relative to what it would have been if it had been parallel transported (since the covariant derivative of a tensor in a direction along which it is parallel transported is zero). The commutator of two covariant derivatives, then, measures the difference between parallel transporting the tensor first one way and then the other, versus the opposite ordering.
transporting the tensor first one way and then the other, versus the opposite ordering. The actual computation is very straightforward. Considering a vector field V’, we take Knowing what we do about parallel transport, we could very carefully perform the nec- essary manipulations to see what happens to the vector under this operation, and the result would be a formula for the curvature tensor in terms of the connection coefficients. It is much quicker, however, to consider a related operation, the commutator of two covariant deriva- tives. The relationship between this and parallel transport around a loop should be evident; the covariant derivative of a tensor in a certain direction measures how much the tensor changes relative to what it would have been if it had been parallel transported (since the covariant derivative of a tensor in a direction along which it is parallel transported is zero). The commutator of two covariant derivatives, then, measures the difference between parallel transporting the tensor first one way and then the other, versus the opposite ordering.
about the curvature is contained in the single component of the Ricci scalar.) Consider a cylinder, R x $*. Although this looks curved from our point of view, it should be clear that we can put a metric on the cylinder whose components are constant in an appropriate coordinate system — simply unroll it and use the induced metric from the plane. In this metric, the cylinder is flat. (There is also nothing to stop us from introducing a different metric in which the cylinder is not flat, but the point we are trying to emphasize is that it can be made flat in some metric.) The same story holds for the torus:
about the curvature is contained in the single component of the Ricci scalar.) Consider a cylinder, R x $*. Although this looks curved from our point of view, it should be clear that we can put a metric on the cylinder whose components are constant in an appropriate coordinate system — simply unroll it and use the induced metric from the plane. In this metric, the cylinder is flat. (There is also nothing to stop us from introducing a different metric in which the cylinder is not flat, but the point we are trying to emphasize is that it can be made flat in some metric.) The same story holds for the torus:
In the metric inherited from this description as part of the flat plane, the cone is flat every- where but at its vertex. This can be seen by considering parallel transport of a vector around various loops; if a loop does not enclose the vertex, there will be no overall transformation, whereas a loop that does enclose the vertex (say, just one time) will lead to a rotation by an angle which is just the deficit angle.
In the metric inherited from this description as part of the flat plane, the cone is flat every- where but at its vertex. This can be seen by considering parallel transport of a vector around various loops; if a loop does not enclose the vertex, there will be no overall transformation, whereas a loop that does enclose the vertex (say, just one time) will lead to a rotation by an angle which is just the deficit angle.
property of Euclidean (flat) geometry is the parallel postulate: initially parallel lines remain parallel forever. Of course in a curved space this is not true; on a sphere, certainly, initially parallel geodesics will eventually cross. We would like to quantify this behavior for an arbitrary curved space. The problem is that the notion of “parallel” does not extend naturally from flat to curved paces. Instead what we will do is to construct a one-parameter family of geodesics, y,(t). ‘hat is, for each s € R, y, is a geodesic parameterized by the affine parameter t. The ollection of these curves defines a smooth two-dimensional surface (embedded in a manifold f of arbitrary dimensionality). The coordinates on this surface may be chosen to be s and provided we have chosen a family of geodesics which do not cross. The entire surface is he set of points x"(s,t) € M. We have two natural vector fields: the tangent vectors to the ppd eed aes
property of Euclidean (flat) geometry is the parallel postulate: initially parallel lines remain parallel forever. Of course in a curved space this is not true; on a sphere, certainly, initially parallel geodesics will eventually cross. We would like to quantify this behavior for an arbitrary curved space. The problem is that the notion of “parallel” does not extend naturally from flat to curved paces. Instead what we will do is to construct a one-parameter family of geodesics, y,(t). ‘hat is, for each s € R, y, is a geodesic parameterized by the affine parameter t. The ollection of these curves defines a smooth two-dimensional surface (embedded in a manifold f of arbitrary dimensionality). The coordinates on this surface may be chosen to be s and provided we have chosen a family of geodesics which do not cross. The entire surface is he set of points x"(s,t) € M. We have two natural vector fields: the tangent vectors to the ppd eed aes
Since S and T are basis vectors adapted to a coordinate system, their commutator van- With this in mind, let’s compute the acceleration:
Since S and T are basis vectors adapted to a coordinate system, their commutator van- With this in mind, let’s compute the acceleration:
In a very big box in a gravitational field, however, the particles will move toward the center of the Earth (for example), which might be a different direction in different regions: The universality of gravitation, as implied by the WEP, can be stated in another, mor« popular, form. Imagine that we consider a physicist in a tightly sealed box, unable tc observe the outside world, who is doing experiments involving the motion of test particles for example to measure the local gravitational field. Of course she would obtain different answers if the box were sitting on the moon or on Jupiter than she would on the Earth But the answers would also be different if the box were accelerating at a constant velocity this would change the acceleration of the freely-falling particles with respect to the box The WEP implies that there is no way to disentangle the effects of a gravitational fielc from those of being in a uniformly accelerating frame, simply by observing the behavior o: freely-falling particles. This follows from the universality of gravitation; it would be possible to distinguish between uniform acceleration and an electromagnetic field, by observing the behavior of particles with different charges. But with gravity it is impossible, since the “charge” is necessarily proportional to the (inertial) mass. To be careful, we should limit our claims about the impossibility of distinguishing gravity
In a very big box in a gravitational field, however, the particles will move toward the center of the Earth (for example), which might be a different direction in different regions: The universality of gravitation, as implied by the WEP, can be stated in another, mor« popular, form. Imagine that we consider a physicist in a tightly sealed box, unable tc observe the outside world, who is doing experiments involving the motion of test particles for example to measure the local gravitational field. Of course she would obtain different answers if the box were sitting on the moon or on Jupiter than she would on the Earth But the answers would also be different if the box were accelerating at a constant velocity this would change the acceleration of the freely-falling particles with respect to the box The WEP implies that there is no way to disentangle the effects of a gravitational fielc from those of being in a uniformly accelerating frame, simply by observing the behavior o: freely-falling particles. This follows from the universality of gravitation; it would be possible to distinguish between uniform acceleration and an electromagnetic field, by observing the behavior of particles with different charges. But with gravity it is impossible, since the “charge” is necessarily proportional to the (inertial) mass. To be careful, we should limit our claims about the impossibility of distinguishing gravity
The solution is to retain the notion of inertial frames, but to discard the hope that they can be uniquely extended throughout space and time. Instead we can define locally inertial frames, those which follow the motion of freely falling particles in small enough regions of spacetime. (Every time we say “small enough regions”, purists should imagine a limiting procedure in which we take the appropriate spacetime volume to zero.) This is the best we can do, but it forces us to give up a good deal. For example, we can no longer speak with confidence about the relative velocity of far away objects, since the inertial reference frames appropriate to those objects are independent of those appropriate to us.
The solution is to retain the notion of inertial frames, but to discard the hope that they can be uniquely extended throughout space and time. Instead we can define locally inertial frames, those which follow the motion of freely falling particles in small enough regions of spacetime. (Every time we say “small enough regions”, purists should imagine a limiting procedure in which we take the appropriate spacetime volume to zero.) This is the best we can do, but it forces us to give up a good deal. For example, we can no longer speak with confidence about the relative velocity of far away objects, since the inertial reference frames appropriate to those objects are independent of those appropriate to us.
clearly this does not carry over to general relativity (outside the weak-field limit). There is a physical reason for this, namely that in GR the gravitational field couples to itself. This can be thought of as a consequence of the equivalence principle — if gravitation did not couple to itself, a “gravitational atom” (two particles bound by their mutual gravitational attraction) would have a different inertial mass (due to the negative binding energy) than gravitational mass. From a particle physics point of view this can be expressed in terms of Feynman diagrams. The electromagnetic interaction between two electrons can be thought of as due to exchange of a virtual photon: But there is no diagram in which two photons exchange another photon between themselves electromagnetism is linear. The gravitational interaction, meanwhile, can be thought o as due to exchange of a virtual graviton (a quantized perturbation of the metric). Th nonlinearity manifests itself as the fact that both electrons and gravitons (and anythin else) can exchange virtual gravitons. and therefore exert a gravitational force:
clearly this does not carry over to general relativity (outside the weak-field limit). There is a physical reason for this, namely that in GR the gravitational field couples to itself. This can be thought of as a consequence of the equivalence principle — if gravitation did not couple to itself, a “gravitational atom” (two particles bound by their mutual gravitational attraction) would have a different inertial mass (due to the negative binding energy) than gravitational mass. From a particle physics point of view this can be expressed in terms of Feynman diagrams. The electromagnetic interaction between two electrons can be thought of as due to exchange of a virtual photon: But there is no diagram in which two photons exchange another photon between themselves electromagnetism is linear. The gravitational interaction, meanwhile, can be thought o as due to exchange of a virtual graviton (a quantized perturbation of the metric). Th nonlinearity manifests itself as the fact that both electrons and gravitons (and anythin else) can exchange virtual gravitons. and therefore exert a gravitational force:
There is nothing profound about this feature of gravity; it is shared by most gauge theories, such as quantum chromodynamics, the theory of the strong interactions. (Electromagnetism is actually the exception; the linearity can be traced to the fact that the relevant gauge group, U(1), is abelian.) But it does represent a departure from the Newtonian theory. (Of course this quantum mechanical language of Feynman diagrams is somewhat inappropriate for GR, which has not [yet] been successfully quantized, but the diagrams are just a convenient shorthand for remembering what interactions exist in the theory.)
There is nothing profound about this feature of gravity; it is shared by most gauge theories, such as quantum chromodynamics, the theory of the strong interactions. (Electromagnetism is actually the exception; the linearity can be traced to the fact that the relevant gauge group, U(1), is abelian.) But it does represent a departure from the Newtonian theory. (Of course this quantum mechanical language of Feynman diagrams is somewhat inappropriate for GR, which has not [yet] been successfully quantized, but the diagrams are just a convenient shorthand for remembering what interactions exist in the theory.)
fe eee Se ee Se eS See Se See eee SA LO Ee eee Se See It is simplest to first consider the problem of evolving matter fields on a fixed background spacetime, rather than the evolution of the metric itself. We therefore consider a spacelike hypersurface & in some manifold M with fixed metric g,,,, and furthermore look at some connected subset S$ in %. Our guiding principle will be that no signals can travel faster than the speed of light; therefore “information” will only flow along timelike or null trajectories (not necessarily geodesics). We define the future domain of dependence of 5, denoted D*(S), as the set of all points p such that every past-moving, timelike or null, inextendible curve through p must intersect S. (“Inextendible” just means that the curve goes on forever, not ending at some finite point.) We interpret this definition in such a way that S itself is a subset of D*(S). (Of course a rigorous formulation does not require additional interpretation over and above the definitions, but we are not being as rigorous as we could be right now.) Similarly, we define the past domain of dependence D~(S) in the same way, but with “past- moving” replaced by “future-moving.” Generally speaking, some points in M will be in one of the domains of dependence, and some will be outside; we define the boundary of D*(S) to be the future Cauchy horizon H*(S), and likewise the boundary of D~(S) to be the past Cauchy horizon H~(S). You can convince yourself that they are both null surfaces.
fe eee Se ee Se eS See Se See eee SA LO Ee eee Se See It is simplest to first consider the problem of evolving matter fields on a fixed background spacetime, rather than the evolution of the metric itself. We therefore consider a spacelike hypersurface & in some manifold M with fixed metric g,,,, and furthermore look at some connected subset S$ in %. Our guiding principle will be that no signals can travel faster than the speed of light; therefore “information” will only flow along timelike or null trajectories (not necessarily geodesics). We define the future domain of dependence of 5, denoted D*(S), as the set of all points p such that every past-moving, timelike or null, inextendible curve through p must intersect S. (“Inextendible” just means that the curve goes on forever, not ending at some finite point.) We interpret this definition in such a way that S itself is a subset of D*(S). (Of course a rigorous formulation does not require additional interpretation over and above the definitions, but we are not being as rigorous as we could be right now.) Similarly, we define the past domain of dependence D~(S) in the same way, but with “past- moving” replaced by “future-moving.” Generally speaking, some points in M will be in one of the domains of dependence, and some will be outside; we define the boundary of D*(S) to be the future Cauchy horizon H*(S), and likewise the boundary of D~(S) to be the past Cauchy horizon H~(S). You can convince yourself that they are both null surfaces.
past Cauchy horizon H~(S). You can convince yourself that they are both null surfaces.
past Cauchy horizon H~(S). You can convince yourself that they are both null surfaces.
In this case © is a nice spacelike surface, but it is clear that D*() ends at the light cone, and we cannot use information on © to predict what happens throughout Minkowski space. Of course, there are other surfaces we could have picked for which the domain of dependence would have been the entire manifold, so this doesn’t worry us too much. PRAIA EN A ROO RERIGLE URENG ORLU LENG: ARMCULLL ENS ENA OR MELE MERLE OWN YY MA UM EARL A somewhat more nontrivial example is known as Misner space. This is a two- imensional spacetime with the topology of R! x $1, and a metric for which the light cones rogressively tilt as you go forward in time. Past a certain point, it is possible to travel on a imelike trajectory which wraps around the S$! and comes back to itself; this is known as a losed timelike curve. If we had specified a surface © to this past of this point, then none f the points in the region containing closed timelike curves are in the domain of dependence f 1, since the closed timelike curves themselves do not intersect =. This is obviously a worse roblem than the previous one, since a well-defined initial value problem does not seem to The usefulness of these definitions should be apparent; if nothing moves faster than light, than signals cannot propagate outside the light cone of any point p. Therefore, if every curve which remains inside this light cone must intersect S, then information specified on S should be sufficient to predict what the situation is at p. (That is, initial data for matter fields given on S can be used to solve for the value of the fields at p.) The set of all points for which we can predict what happens by knowing what happens on S' is simply the union Dt(S)UD~(S). We can easily extend these ideas from the subset S to the entire hypersurface ©. The
In this case © is a nice spacelike surface, but it is clear that D*() ends at the light cone, and we cannot use information on © to predict what happens throughout Minkowski space. Of course, there are other surfaces we could have picked for which the domain of dependence would have been the entire manifold, so this doesn’t worry us too much. PRAIA EN A ROO RERIGLE URENG ORLU LENG: ARMCULLL ENS ENA OR MELE MERLE OWN YY MA UM EARL A somewhat more nontrivial example is known as Misner space. This is a two- imensional spacetime with the topology of R! x $1, and a metric for which the light cones rogressively tilt as you go forward in time. Past a certain point, it is possible to travel on a imelike trajectory which wraps around the S$! and comes back to itself; this is known as a losed timelike curve. If we had specified a surface © to this past of this point, then none f the points in the region containing closed timelike curves are in the domain of dependence f 1, since the closed timelike curves themselves do not intersect =. This is obviously a worse roblem than the previous one, since a well-defined initial value problem does not seem to The usefulness of these definitions should be apparent; if nothing moves faster than light, than signals cannot propagate outside the light cone of any point p. Therefore, if every curve which remains inside this light cone must intersect S, then information specified on S should be sufficient to predict what the situation is at p. (That is, initial data for matter fields given on S can be used to solve for the value of the fields at p.) The set of all points for which we can predict what happens by knowing what happens on S' is simply the union Dt(S)UD~(S). We can easily extend these ideas from the subset S to the entire hypersurface ©. The
nterest, so | won't claim that the issue 1s settled.) A final example is provided by the existence of singularities, points which are not in the nanifold even though they can be reached by travelling along a geodesic for a finite distance. Typically these occur when the curvature becomes infinite at some point; if this happens, he point can no longer be said to be part of the spacetime. Such an occurrence can lead to ‘he emergence of a Cauchy horizon — a point p which is in the future of a singularity cannot ye in the domain of dependence of a hypersurface to the past of the singularity, because here will be curves from p which simply end at the singularity. All of these obstacles can also arise in the initial value problem for GR, when we try to evolve the metric itself from initial data. However, they are of different degrees of trouble-
nterest, so | won't claim that the issue 1s settled.) A final example is provided by the existence of singularities, points which are not in the nanifold even though they can be reached by travelling along a geodesic for a finite distance. Typically these occur when the curvature becomes infinite at some point; if this happens, he point can no longer be said to be part of the spacetime. Such an occurrence can lead to ‘he emergence of a Cauchy horizon — a point p which is in the future of a singularity cannot ye in the domain of dependence of a hypersurface to the past of the singularity, because here will be curves from p which simply end at the singularity. All of these obstacles can also arise in the initial value problem for GR, when we try to evolve the metric itself from initial data. However, they are of different degrees of trouble-
Similarly, if TN is the tangent space at a point q on N, then a one-form w at q (i.e., an element of the cotangent space TN) can be thought of as an operator from TN to R. Since the pushforward ¢* maps T,,M to Typ) N, the pullback ¢, of a one-form can also be thought of as mere composition of maps:
Similarly, if TN is the tangent space at a point q on N, then a one-form w at q (i.e., an element of the cotangent space TN) can be thought of as an operator from TN to R. Since the pushforward ¢* maps T,,M to Typ) N, the pullback ¢, of a one-form can also be thought of as mere composition of maps:
Note that tensors with both upper and lower indices can generally be neither pushed forward nor pulled back. where T4,...a, is a (0,/) tensor on N. We can similarly push forward any (k,0) tensor S#1"# by acting it on pulled-back one-forms:
Note that tensors with both upper and lower indices can generally be neither pushed forward nor pulled back. where T4,...a, is a (0,/) tensor on N. We can similarly push forward any (k,0) tensor S#1"# by acting it on pulled-back one-forms:
Since a diffeomorphism allows us to pull back and push forward arbitrary tensors, it provides another way of comparing tensors at different points on a manifold. Given a diffeo- morphism ¢: M — M and a tensor field T“""* ,,...,,,(@), we can form the difference between the value of the tensor at some point p and ¢,|/T"'""*,,...,(@(p))], its value at ¢(p) pulled back to p. This suggests that we could define another kind of derivative operator on tensor fields, one which categorizes the rate of change of the tensor as it changes under the diffeo- morphism. For that, however, a single discrete diffeomorphism is insufficient; we require a one-parameter family of diffeomorphisms, ¢;. This family can be thought of as a smooth map Rx M — M, such that for each t € R ¢; is a diffeomorphism and @¢, 0 ¢; = 54,4. Note that this last condition implies that ¢p is the identity map. C Nex me ax es woe de ee A ee Be cee ER en oe eee Siege: eos ee De oe BS ee oe oad oa oe oe ee eae efecseees eee lle pea tO | | ae maps”), or we could just as well introduce a diffeomorphism ¢: M — M, after which the coordinates would just be the pullbacks (¢,x7)" : M — R” (“move the points on the man- ifold, and then evaluate the coordinates of the new points”). In this sense, (5.15) really is the tensor transformation law, just thought of from a different point of view.
Since a diffeomorphism allows us to pull back and push forward arbitrary tensors, it provides another way of comparing tensors at different points on a manifold. Given a diffeo- morphism ¢: M — M and a tensor field T“""* ,,...,,,(@), we can form the difference between the value of the tensor at some point p and ¢,|/T"'""*,,...,(@(p))], its value at ¢(p) pulled back to p. This suggests that we could define another kind of derivative operator on tensor fields, one which categorizes the rate of change of the tensor as it changes under the diffeo- morphism. For that, however, a single discrete diffeomorphism is insufficient; we require a one-parameter family of diffeomorphisms, ¢;. This family can be thought of as a smooth map Rx M — M, such that for each t € R ¢; is a diffeomorphism and @¢, 0 ¢; = 54,4. Note that this last condition implies that ¢p is the identity map. C Nex me ax es woe de ee A ee Be cee ER en oe eee Siege: eos ee De oe BS ee oe oad oa oe oe ee eae efecseees eee lle pea tO | | ae maps”), or we could just as well introduce a diffeomorphism ¢: M — M, after which the coordinates would just be the pullbacks (¢,x7)" : M — R” (“move the points on the man- ifold, and then evaluate the coordinates of the new points”). In this sense, (5.15) really is the tensor transformation law, just thought of from a different point of view.
We then define the Lie derivative of the tensor along the vector field as
We then define the Lie derivative of the tensor along the vector field as
In this language, the issue of gauge invariance is simply the fact that there are a large number of permissible diffeomorphisms between M, and M, (where “permissible” means We can think about this from a highbrow point of view. The notion that the linearized theory can be thought of as one governing the behavior of tensor fields on a flat background can be formalized in terms of a “background spacetime” M,, a “physical spacetime” M,, and a diffeomorphism ¢ : M, — M,. As manifolds M, and M, are “the same” (since they are diffeomorphic), but we imagine that they possess some different tensor fields; on M, we have defined the flat Minkowski metric 7,,,, while on M, we have some metric gag which obeys Einstein’s equations. (We imagine that M, is equipped with coordinates x“ and M, is equipped with coordinates y°, although these will not play a prominent role.) The diffeomorphism @ allows us to move tensors back and forth between the background and physical spacetimes. Since we would like to construct our linearized theory as one taking place on the flat background spacetime, we are interested in the pullback (¢,g9),,. of the physical metric. We can define the perturbation as the difference between the pulled-back physical metric and the flat one:
In this language, the issue of gauge invariance is simply the fact that there are a large number of permissible diffeomorphisms between M, and M, (where “permissible” means We can think about this from a highbrow point of view. The notion that the linearized theory can be thought of as one governing the behavior of tensor fields on a flat background can be formalized in terms of a “background spacetime” M,, a “physical spacetime” M,, and a diffeomorphism ¢ : M, — M,. As manifolds M, and M, are “the same” (since they are diffeomorphic), but we imagine that they possess some different tensor fields; on M, we have defined the flat Minkowski metric 7,,,, while on M, we have some metric gag which obeys Einstein’s equations. (We imagine that M, is equipped with coordinates x“ and M, is equipped with coordinates y°, although these will not play a prominent role.) The diffeomorphism @ allows us to move tensors back and forth between the background and physical spacetimes. Since we would like to construct our linearized theory as one taking place on the flat background spacetime, we are interested in the pullback (¢,g9),,. of the physical metric. We can define the perturbation as the difference between the pulled-back physical metric and the flat one:
Specifically, we can define a family of perturbations parameterized by €
Specifically, we can define a family of perturbations parameterized by €
Using the specific forms (6.30) for the solution and (6.39) for the transformation, we obtain
Using the specific forms (6.30) for the solution and (6.39) for the transformation, we obtain
The effect of a pure Cp wave would be to rotate the particles in a right-handed sense around the ring; they just move in little epicycles.) We can relate the polarization states of classical gravitational waves to the kinds of particles we would expect to find upon quantization. The electromagnetic field has two in- dependent polarization states which are described by vectors in the x-y plane; equivalently. a single polarization mode is invariant under a rotation by 360° in this plane. Upon quan- tization this theory yields the photon, a massless spin-one particle. The neutrino, on the other hand, is also a massless particle, described by a field which picks up a minus sign under rotations by 360°; it is invariant under rotations of 720°, and we say it has spin-5.
The effect of a pure Cp wave would be to rotate the particles in a right-handed sense around the ring; they just move in little epicycles.) We can relate the polarization states of classical gravitational waves to the kinds of particles we would expect to find upon quantization. The electromagnetic field has two in- dependent polarization states which are described by vectors in the x-y plane; equivalently. a single polarization mode is invariant under a rotation by 360° in this plane. Upon quan- tization this theory yields the photon, a massless spin-one particle. The neutrino, on the other hand, is also a massless particle, described by a field which picks up a minus sign under rotations by 360°; it is invariant under rotations of 720°, and we say it has spin-5.
In this case the circle of particles would bounce back and forth in the shape of a “x’
In this case the circle of particles would bounce back and forth in the shape of a “x’
Let us take this general solution and consider the case where the gravitational radiation is emitted by an isolated source, fairly far away, comprised of nonrelativistic matter; these approximations will be made more precise as we go on. First we need to set up some con- ventions for Fourier transforms, which always make life easier when dealing with oscillatory phenomena. Given a function of spacetime ¢(t,x), we are interested in its Fourier transform (and inverse) with respect to time alone, Upon plugging (6.73) into (6.72), we can use the delta function to perform the integral over y°, leaving us with
Let us take this general solution and consider the case where the gravitational radiation is emitted by an isolated source, fairly far away, comprised of nonrelativistic matter; these approximations will be made more precise as we go on. First we need to set up some con- ventions for Fourier transforms, which always make life easier when dealing with oscillatory phenomena. Given a function of spacetime ¢(t,x), we are interested in its Fourier transform (and inverse) with respect to time alone, Upon plugging (6.73) into (6.72), we can use the delta function to perform the integral over y°, leaving us with
We will treat the motion of the stars in the Newtonian approximation, where we can discuss their orbit just as Kepler would have. Circular orbits are most easily characterized by equating the force due to gravity to the outward “centrifugal” force: point where the past light cone of the observer intersects the source. In contrast, the leading contribution to electromagnetic radiation comes from the changing dipole moment of the charge density. The difference can be traced back to the universal nature of gravitation. A changing dipole moment corresponds to motion of the center of density — charge density in the case of electromagnetism, energy density in the case of gravitation. While there is noth- ing to stop the center of charge of an object from oscillating, oscillation of the center of mass of an isolated system violates conservation of momentum. (You can shake a body up and down, but you and the earth shake ever so slightly in the opposite direction to compensate.) The quadrupole moment, which measures the shape of the system, is generally smaller than the dipole moment, and for this reason (as well as the weak coupling of matter to gravity) gravitational radiation is typically much weaker than electromagnetic radiation. Tt ig alwave educational to take a oeneral solution and annly it to 4 anecific cage of
We will treat the motion of the stars in the Newtonian approximation, where we can discuss their orbit just as Kepler would have. Circular orbits are most easily characterized by equating the force due to gravity to the outward “centrifugal” force: point where the past light cone of the observer intersects the source. In contrast, the leading contribution to electromagnetic radiation comes from the changing dipole moment of the charge density. The difference can be traced back to the universal nature of gravitation. A changing dipole moment corresponds to motion of the center of density — charge density in the case of electromagnetism, energy density in the case of gravitation. While there is noth- ing to stop the center of charge of an object from oscillating, oscillation of the center of mass of an isolated system violates conservation of momentum. (You can shake a body up and down, but you and the earth shake ever so slightly in the opposite direction to compensate.) The quadrupole moment, which measures the shape of the system, is generally smaller than the dipole moment, and for this reason (as well as the weak coupling of matter to gravity) gravitational radiation is typically much weaker than electromagnetic radiation. Tt ig alwave educational to take a oeneral solution and annly it to 4 anecific cage of
Of course, this has actually been observed. In 1974 Hulse and Taylor discovered a binary system, PSR1913+16, in which both stars are very small (so classical effects are negligible, or at least under control) and one is a pulsar. The period of the orbit is eight hours, extremely small by astrophysical standards. The fact that one of the stars is a pulsar provides a very accurate clock, with respect to which the change in the period as the system loses energy can be measured. The result is consistent with the prediction of general relativity for energy loss through gravitational radiation. Hulse and Taylor were awarded the Nobel Prize in 1993 for their efforts. For the binary system represented by (6.93), the traceless part of the quadrupole is
Of course, this has actually been observed. In 1974 Hulse and Taylor discovered a binary system, PSR1913+16, in which both stars are very small (so classical effects are negligible, or at least under control) and one is a pulsar. The period of the orbit is eight hours, extremely small by astrophysical standards. The fact that one of the stars is a pulsar provides a very accurate clock, with respect to which the change in the period as the system loses energy can be measured. The result is consistent with the prediction of general relativity for energy loss through gravitational radiation. Hulse and Taylor were awarded the Nobel Prize in 1993 for their efforts. For the binary system represented by (6.93), the traceless part of the quadrupole is
Let’s consider some examples to bring this down to earth. The simplest example is flat three-dimensional Euclidean space. If we pick an origin, then R° is clearly spherically symmetric with respect to rotations around this origin. Under such rotations (7.e., under the flow of the Killing vector fields) points move into each other, but each point stays on an S? at a fixed distance from the origin. It is these spheres which foliate R®. Of course, they don’t really foliate all of the space, since the origin itself just stays put under rotations — it doesn’t move around on some two-sphere. But it should be clear that almost all of the space is properly foliated, and this will turn out to be enough for us.
Let’s consider some examples to bring this down to earth. The simplest example is flat three-dimensional Euclidean space. If we pick an origin, then R° is clearly spherically symmetric with respect to rotations around this origin. Under such rotations (7.e., under the flow of the Killing vector fields) points move into each other, but each point stays on an S? at a fixed distance from the origin. It is these spheres which foliate R®. Of course, they don’t really foliate all of the space, since the origin itself just stays put under rotations — it doesn’t move around on some two-sphere. But it should be clear that almost all of the space is properly foliated, and this will turn out to be enough for us.
In this case the entire manifold can be foliated by two-spheres. This foliated structure suggests that we put coordinates on our manifold in a way whicl is adapted to the foliation. By this we mean that, if we have an n-dimensional manifolx foliated by m-dimensional submanifolds, we can use a set of m coordinate functions u’ o1 the submanifolds and a set of n —m coordinate functions v! to tell us which submanifold w are on. (So 7 runs from 1 to m, while J runs from 1 to n —m.) Then the collection of v’ and u’s coordinatize the entire space. If the submanifolds are maximally symmetric space: (as two-spheres are), then there is the following powerful theorem: it is always possible t« choose the u-coordinates such that the metric on the entire manifold is of the form
In this case the entire manifold can be foliated by two-spheres. This foliated structure suggests that we put coordinates on our manifold in a way whicl is adapted to the foliation. By this we mean that, if we have an n-dimensional manifolx foliated by m-dimensional submanifolds, we can use a set of m coordinate functions u’ o1 the submanifolds and a set of n —m coordinate functions v! to tell us which submanifold w are on. (So 7 runs from 1 to m, while J runs from 1 to n —m.) Then the collection of v’ and u’s coordinatize the entire space. If the submanifolds are maximally symmetric space: (as two-spheres are), then there is the following powerful theorem: it is always possible t« choose the u-coordinates such that the metric on the entire manifold is of the form
symmetric spacetime can be put in the form
symmetric spacetime can be put in the form
is constant. Of course, for a massive particle we typically choose \ = 7, and this relation simply becomes € = —g,,U"U" = +1. For a massless particle we always have e = 0. We will also be concerned with spacelike geodesics (even though they do not correspond to paths of particles), for which we will choose « = —1. Rather than immediately writing out explicit expressions for the four conserved quantitie: associated with Killing vectors, let’s think about what they are telling us. Notice that th symmetries they represent are also present in flat spacetime, where the conserved quantitie: they lead to are very familiar. Invariance under time translations leads to conservation o energy, while invariance under spatial rotations leads to conservation of the three component: of angular momentum. Essentially the same applies to the Schwarzschild metric. We ca1 think of the angular momentum as a three-vector with a magnitude (one component) anc direction (two components). Conservation of the direction of angular momentum mean: that the particle will move in a plane. We can choose this to be the equatorial plane o our coordinate system; if the particle is not in this plane, we can rotate coordinates unti it is. Thus, the two Killing vectors which lead to conservation of the direction of angula mMomentim imply
is constant. Of course, for a massive particle we typically choose \ = 7, and this relation simply becomes € = —g,,U"U" = +1. For a massless particle we always have e = 0. We will also be concerned with spacelike geodesics (even though they do not correspond to paths of particles), for which we will choose « = —1. Rather than immediately writing out explicit expressions for the four conserved quantitie: associated with Killing vectors, let’s think about what they are telling us. Notice that th symmetries they represent are also present in flat spacetime, where the conserved quantitie: they lead to are very familiar. Invariance under time translations leads to conservation o energy, while invariance under spatial rotations leads to conservation of the three component: of angular momentum. Essentially the same applies to the Schwarzschild metric. We ca1 think of the angular momentum as a three-vector with a magnitude (one component) anc direction (two components). Conservation of the direction of angular momentum mean: that the particle will move in a plane. We can choose this to be the equatorial plane o our coordinate system; if the particle is not in this plane, we can rotate coordinates unti it is. Thus, the two Killing vectors which lead to conservation of the direction of angula mMomentim imply
The quantities of interest to us are not only r(A), but also t(A) and @(A). Nevertheless, ve can go a long way toward understanding all of the orbits by understanding their radial yehavior, and it is a great help to reduce this behavior to a problem we know how to solve. A similar analysis of orbits in Newtonian gravity would have produced a similar result; he general equation (7.47) would have been the same, but the effective potential (7.48) would ot have had the last term. (Note that this equation is not a power series in 1/r, it is exact.) n the potential (7.48) the first term is just a constant, the second term corresponds exactly o the Newtonian gravitational potential, and the third term is a contribution from angular nomentum which takes the same form in Newtonian gravity and general relativity. The last erm, the GR contribution, will turn out to make a great deal of difference, especially at mall r.
The quantities of interest to us are not only r(A), but also t(A) and @(A). Nevertheless, ve can go a long way toward understanding all of the orbits by understanding their radial yehavior, and it is a great help to reduce this behavior to a problem we know how to solve. A similar analysis of orbits in Newtonian gravity would have produced a similar result; he general equation (7.47) would have been the same, but the effective potential (7.48) would ot have had the last term. (Note that this equation is not a power series in 1/r, it is exact.) n the potential (7.48) the first term is just a constant, the second term corresponds exactly o the Newtonian gravitational potential, and the third term is a contribution from angular nomentum which takes the same form in Newtonian gravity and general relativity. The last erm, the GR contribution, will turn out to make a great deal of difference, especially at mall r.
and of course c = 3.00 x 10° cm/sec. This gives wy = 2.35 x 107“ sec~'. In other words, the major axis of Mercury’s orbit precesses at a rate of 42.9 arcsecs every 100 years. The observed value is 5601 arcsecs/100 yrs. However, much of that is due to the precession of equinoxes in our geocentric coordinate system; 5025 arcsecs/100 yrs, to be precise. The eravitational perturbations of the other planets contribute an additional 532 arcsecs/100 yrs, leaving 43 arcsecs/100 yrs to be explained by GR, which it does quite well. The gravitational redshift, as we have seen, is another effect which is present in the weak field limit, and in fact will be predicted by any theory of gravity which obeys the Principle of Equivalence. However, this only applies to small enough regions of spacetime; over larger distances, the exact amount of redshift will depend on the metric, and thus on the theory under question. It is therefore worth computing the redshift in the Schwarzschild geometry. We consider two observers who are not moving on geodesics, but are stuck at fixed spatial coordinate values (71, 6;, 61) and (r2, 02, 2). According to (7.45), the proper time of observer zi will be related to the coordinate time t by
and of course c = 3.00 x 10° cm/sec. This gives wy = 2.35 x 107“ sec~'. In other words, the major axis of Mercury’s orbit precesses at a rate of 42.9 arcsecs every 100 years. The observed value is 5601 arcsecs/100 yrs. However, much of that is due to the precession of equinoxes in our geocentric coordinate system; 5025 arcsecs/100 yrs, to be precise. The eravitational perturbations of the other planets contribute an additional 532 arcsecs/100 yrs, leaving 43 arcsecs/100 yrs to be explained by GR, which it does quite well. The gravitational redshift, as we have seen, is another effect which is present in the weak field limit, and in fact will be predicted by any theory of gravity which obeys the Principle of Equivalence. However, this only applies to small enough regions of spacetime; over larger distances, the exact amount of redshift will depend on the metric, and thus on the theory under question. It is therefore worth computing the redshift in the Schwarzschild geometry. We consider two observers who are not moving on geodesics, but are stuck at fixed spatial coordinate values (71, 6;, 61) and (r2, 02, 2). According to (7.45), the proper time of observer zi will be related to the coordinate time t by
Let’s consider what we have done. Acting under the suspicion that our coordinates may not have been good for the entire manifold, we have changed from our original coordinate t to the new one w, which has the nice property that if we decrease r along a radial curve null curve % = constant, we go right through the event horizon without any problems. (Indeed, a local observer actually making the trip would not necessarily know when the event horizon had been crossed — the local geometry is no different than anywhere else.) We therefore conclude that our suspicion was correct and our initial coordinate system didn’t do a good job of covering the entire manifold. The region r < 2GM should certainly be included in our spacetime, since physical particles can easily reach there and pass through. However, there is no guarantee that we are finished; perhaps there are other directions in which we can extend our manifold.
Let’s consider what we have done. Acting under the suspicion that our coordinates may not have been good for the entire manifold, we have changed from our original coordinate t to the new one w, which has the nice property that if we decrease r along a radial curve null curve % = constant, we go right through the event horizon without any problems. (Indeed, a local observer actually making the trip would not necessarily know when the event horizon had been crossed — the local geometry is no different than anywhere else.) We therefore conclude that our suspicion was correct and our initial coordinate system didn’t do a good job of covering the entire manifold. The region r < 2GM should certainly be included in our spacetime, since physical particles can easily reach there and pass through. However, there is no guarantee that we are finished; perhaps there are other directions in which we can extend our manifold.
Each point on the diagram is a two-sphere.
Each point on the diagram is a two-sphere.
Now we can draw pictures of each slice, restoring one of the angular coordinates for clarity: event horizon, while the boundary of region II is called the future event horizon. Region IV, meanwhile, cannot be reached from our region I either forward or backward in time (nor can anybody from over there reach us). It is another asymptotically flat region of spacetime, a mirror image of ours. It can be thought of as being connected to region I by a “wormhole,” a neck-like configuration joining two distinct regions. Consider slicing up the Kruskal diagram into spacelike surfaces of constant 2):
Now we can draw pictures of each slice, restoring one of the angular coordinates for clarity: event horizon, while the boundary of region II is called the future event horizon. Region IV, meanwhile, cannot be reached from our region I either forward or backward in time (nor can anybody from over there reach us). It is another asymptotically flat region of spacetime, a mirror image of ours. It can be thought of as being connected to region I by a “wormhole,” a neck-like configuration joining two distinct regions. Consider slicing up the Kruskal diagram into spacelike surfaces of constant 2):
Remember that it is only valid in vacuum, for example outside a star. If the star has a radius larger than 2GM, we need never worry about any event horizons at all. But we believe that there are stars which collapse under their own gravitational pull, shrinking down to below r = 2GM and further into a singularity, resulting in a black hole. There is no need for a white hole, however, because the past of such a spacetime looks nothing like that of the full Schwarzschild solution. Roughly, a Kruskal-like diagram for stellar collapse would look like the following: holes and wormholes. While we are on the subject, we can say something about the formation of astrophysical black holes from massive stars. The life of a star is a constant struggle between the inward pull of gravity and the outward push of pressure. When the star is burning nuclear fuel at its core, the pressure comes from the heat produced by this burning. (We should put “burning” in quotes, since nuclear fusion is unrelated to oxidation.) When the fuel is used up, the temperature declines and the star begins to shrink as gravity starts winning the struggle. Eventually this process is stopped when the electrons are pushed so close together that they resist further compression simply on the basis of the Pauli exclusion principle (no two fermions can be in the same state). The resulting object is called a white dwarf. If the mass is sufficiently high, however, even the electron degeneracy pressure is not enough, and the electrons will combine with the protons in a dramatic phase transition. The result is a neutron star, which consists of almost entirely neutrons (although the insides of neutron stars are not understood terribly well). Since the conditions at the center of a neutron star are very different from those on earth, we do not have a perfect understanding of the equation of state. Nevertheless, we believe that a sufficiently massive neutron star will itself
Remember that it is only valid in vacuum, for example outside a star. If the star has a radius larger than 2GM, we need never worry about any event horizons at all. But we believe that there are stars which collapse under their own gravitational pull, shrinking down to below r = 2GM and further into a singularity, resulting in a black hole. There is no need for a white hole, however, because the past of such a spacetime looks nothing like that of the full Schwarzschild solution. Roughly, a Kruskal-like diagram for stellar collapse would look like the following: holes and wormholes. While we are on the subject, we can say something about the formation of astrophysical black holes from massive stars. The life of a star is a constant struggle between the inward pull of gravity and the outward push of pressure. When the star is burning nuclear fuel at its core, the pressure comes from the heat produced by this burning. (We should put “burning” in quotes, since nuclear fusion is unrelated to oxidation.) When the fuel is used up, the temperature declines and the star begins to shrink as gravity starts winning the struggle. Eventually this process is stopped when the electrons are pushed so close together that they resist further compression simply on the basis of the Pauli exclusion principle (no two fermions can be in the same state). The resulting object is called a white dwarf. If the mass is sufficiently high, however, even the electron degeneracy pressure is not enough, and the electrons will combine with the protons in a dramatic phase transition. The result is a neutron star, which consists of almost entirely neutrons (although the insides of neutron stars are not understood terribly well). Since the conditions at the center of a neutron star are very different from those on earth, we do not have a perfect understanding of the equation of state. Nevertheless, we believe that a sufficiently massive neutron star will itself
The point of the diagram is that, for any given mass M, the star will decrease in radius until it hits the line. White dwarfs are found between points A and B, and neutron stars between points C' and D. Point B is at a height of somewhat less than 1.4 solar masses; the height of D is less certain, but probably less than 2 solar masses. The process of collapse is complicated, and during the evolution the star can lose or gain mass, so the endpoint of any given star is hard to predict. Nevertheless white dwarfs are all over the place, neutron stars are not uncommon, and there are a number of systems which are strongly believed to contain black holes. (Of course, you can’t directly see the black hole. What you can see is radiation from matter accreting onto the hole, which heats up as it gets closer and emits radiation. ) The process is summarized in the following diagram of radius vs. mass
The point of the diagram is that, for any given mass M, the star will decrease in radius until it hits the line. White dwarfs are found between points A and B, and neutron stars between points C' and D. Point B is at a height of somewhat less than 1.4 solar masses; the height of D is less certain, but probably less than 2 solar masses. The process of collapse is complicated, and during the evolution the star can lose or gain mass, so the endpoint of any given star is hard to predict. Nevertheless white dwarfs are all over the place, neutron stars are not uncommon, and there are a number of systems which are strongly believed to contain black holes. (Of course, you can’t directly see the black hole. What you can see is radiation from matter accreting onto the hole, which heats up as it gets closer and emits radiation. ) The process is summarized in the following diagram of radius vs. mass
radius r = u—v. The metric in these coordinates is given by These ranges are as portrayed in the figure, on which each point represents a 2-sphere c with corresponding ranges given by
radius r = u—v. The metric in these coordinates is given by These ranges are as portrayed in the figure, on which each point represents a 2-sphere c with corresponding ranges given by
The Minkowski metric may therefore be thought of as related by a conformal transfor-
The Minkowski metric may therefore be thought of as related by a conformal transfor-
The (u”,v”) part of the metric (that is, at constant angular coordinates) is now conformally related to Minkowski space. In the new coordinates the singularities at r = 0 are straight lines that stretch from timelike infinity in one asymptotic region to timelike infinity in the other. The Penrose diagram for the maximally extended Schwarzschild solution thus looks like this: The only real subtlety about this diagram is the necessity to understand that 7+ and 7 are distinct from r = 0 (there are plenty of timelike paths that do not hit the singularity). Notice also that the structure of conformal infinity is just like that of Minkowski space, consistent with the claim that Schwarzschild is asymptotically flat. Also, the Penrose diagram for < collapsing star that forms a black hole is what you might expect, as shown on the next page Once again the Penrose diagrams for these spacetimes don’t really tell us anything we didn’t already know; their usefulness will become evident when we consider more genera black holes. In principle there could be a wide variety of types of black holes, depending o1 the process by which they were formed. Surprisingly, however, this turns out not to be the case; no matter how a black hole is formed, it settles down (fairly quickly) into a state whicl is characterized only by the mass, charge, and angular momentum. This property, whict must be demonstrated individually for the various types of fields which one might imagin« go into the construction of the hole, is often stated as “black holes have no hair.” You
The (u”,v”) part of the metric (that is, at constant angular coordinates) is now conformally related to Minkowski space. In the new coordinates the singularities at r = 0 are straight lines that stretch from timelike infinity in one asymptotic region to timelike infinity in the other. The Penrose diagram for the maximally extended Schwarzschild solution thus looks like this: The only real subtlety about this diagram is the necessity to understand that 7+ and 7 are distinct from r = 0 (there are plenty of timelike paths that do not hit the singularity). Notice also that the structure of conformal infinity is just like that of Minkowski space, consistent with the claim that Schwarzschild is asymptotically flat. Also, the Penrose diagram for < collapsing star that forms a black hole is what you might expect, as shown on the next page Once again the Penrose diagrams for these spacetimes don’t really tell us anything we didn’t already know; their usefulness will become evident when we consider more genera black holes. In principle there could be a wide variety of types of black holes, depending o1 the process by which they were formed. Surprisingly, however, this turns out not to be the case; no matter how a black hole is formed, it settles down (fairly quickly) into a state whicl is characterized only by the mass, charge, and angular momentum. This property, whict must be demonstrated individually for the various types of fields which one might imagin« go into the construction of the hole, is often stated as “black holes have no hair.” You
impossible to detect, since they average out to give zero total energy (although nobody know: why; that’s the cosmological constant problem). In the presence of an event horizon, though occasionally one member of a virtual pair will fall into the black hole while its partner escape: to infinity. The particle that reaches infinity will have to have a positive energy, but the total energy is conserved; therefore the black hole has to lose mass. (If you like you cai think of the particle that falls in as having a negative mass.) We see the escaping particle as Hawking radiation. It’s not a very big effect, and the temperature goes down as the mas: goes up, so for black holes of mass comparable to the sun it is completely negligible. Still in principle the black hole could lose all of its mass to Hawking radiation, and shrink tc nothing in the process. The relevant Penrose diagram might look like this: On the other hand, it might not. The problem with this diagram is that “information is lost” — if we draw a spacelike surface to the past of the singularity and evolve it into the future, part of it ends up crashing into the singularity and being destroyed. As a result the radiation itself contains less information than the information that was originally in the spacetime. (This is the worse than a lack of hair on the black hole. It’s one thing to think that information has been trapped inside the event horizon, but it is more worrisome to think that it has disappeared entirely.) But such a process violates the conservation of information that is implicit in both general relativity and quantum field theory, the two theories that led to the prediction. This paradox is considered a big deal these days, and there are a number of efforts to understand how the information can somehow be retrieved. A currently popular explanation relies on string theory, and basically says that black holes have a lot of hair, in the form of virtual stringy states living near the event horizon. I hope you will not be disappointed to hear that we won’t look at this very closely; but you should know what the problem is and that it is an area of active research these days.
impossible to detect, since they average out to give zero total energy (although nobody know: why; that’s the cosmological constant problem). In the presence of an event horizon, though occasionally one member of a virtual pair will fall into the black hole while its partner escape: to infinity. The particle that reaches infinity will have to have a positive energy, but the total energy is conserved; therefore the black hole has to lose mass. (If you like you cai think of the particle that falls in as having a negative mass.) We see the escaping particle as Hawking radiation. It’s not a very big effect, and the temperature goes down as the mas: goes up, so for black holes of mass comparable to the sun it is completely negligible. Still in principle the black hole could lose all of its mass to Hawking radiation, and shrink tc nothing in the process. The relevant Penrose diagram might look like this: On the other hand, it might not. The problem with this diagram is that “information is lost” — if we draw a spacelike surface to the past of the singularity and evolve it into the future, part of it ends up crashing into the singularity and being destroyed. As a result the radiation itself contains less information than the information that was originally in the spacetime. (This is the worse than a lack of hair on the black hole. It’s one thing to think that information has been trapped inside the event horizon, but it is more worrisome to think that it has disappeared entirely.) But such a process violates the conservation of information that is implicit in both general relativity and quantum field theory, the two theories that led to the prediction. This paradox is considered a big deal these days, and there are a number of efforts to understand how the information can somehow be retrieved. A currently popular explanation relies on string theory, and basically says that black holes have a lot of hair, in the form of virtual stringy states living near the event horizon. I hope you will not be disappointed to hear that we won’t look at this very closely; but you should know what the problem is and that it is an area of active research these days.
The nakedness of the singularity offends our sense of decency, as well as the cosmic cen- sorship conjecture, which roughly states that the gravitational collapse of physical matter configurations will never produce a naked singularity. (Of course, it’s just a conjecture, and it may not be right; there are some claims from numerical simulations that collapse of spindle- like configurations can lead to naked singularities.) In fact, we should not ever expect to find [he nakedness of the singularity offends our sense of decency, as well as the cosmic cen-
The nakedness of the singularity offends our sense of decency, as well as the cosmic cen- sorship conjecture, which roughly states that the gravitational collapse of physical matter configurations will never produce a naked singularity. (Of course, it’s just a conjecture, and it may not be right; there are some claims from numerical simulations that collapse of spindle- like configurations can lead to naked singularities.) In fact, we should not ever expect to find [he nakedness of the singularity offends our sense of decency, as well as the cosmic cen-
The extremal black holes have A(r) = 0 at a single radius, r = GM. This does represent an event horizon, but the r coordinate is never timelike; it becomes null at r = GM, but is spacelike on either side. The singularity at r = 0 is a timelike line, as in the other cases. So f the external (asymptotically flat) universe, at least as seen from the black hole. But they ee this (infinitely long) history in a finite amount of their proper time — thus, any signal hat gets to them as they approach r_ is infinitely blueshifted. Therefore it is reasonable o believe (although I know of no proof) that any non-spherically symmetric perturbation hat comes into a Reissner-Nordstrgm black hole will violently disturb the geometry we have lescribed. It’s hard to say what the actual geometry will look like, but there is no very ood reason to believe that it must contain an infinite number of asymptotically flat regions onnecting to each other via various wormholes. Nase Three — GM? = p* + q mii: Dey RTO fe (O. 2.2
The extremal black holes have A(r) = 0 at a single radius, r = GM. This does represent an event horizon, but the r coordinate is never timelike; it becomes null at r = GM, but is spacelike on either side. The singularity at r = 0 is a timelike line, as in the other cases. So f the external (asymptotically flat) universe, at least as seen from the black hole. But they ee this (infinitely long) history in a finite amount of their proper time — thus, any signal hat gets to them as they approach r_ is infinitely blueshifted. Therefore it is reasonable o believe (although I know of no proof) that any non-spherically symmetric perturbation hat comes into a Reissner-Nordstrgm black hole will violently disturb the geometry we have lescribed. It’s hard to say what the actual geometry will look like, but there is no very ood reason to believe that it must contain an infinite number of asymptotically flat regions onnecting to each other via various wormholes. Nase Three — GM? = p* + q mii: Dey RTO fe (O. 2.2
In this expression the two vectors / and n are given (with indices raised) by
In this expression the two vectors / and n are given (with indices raised) by
Plugging in the definitions of EF and L, we see that this condition is equivalent to
Plugging in the definitions of EF and L, we see that this condition is equivalent to
in time as well as space. and for closed universes,
in time as well as space. and for closed universes,
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