Notes on Hobson et al., chapter 7

Notes on Hobson et al., chapter 7 In this chapter, we follow Einstein s steps in attempting to include the effects of gravitational forces as the consequences of geodesic motion on a (pseudo- )Riemannian manifold, now that we have learned some of the details of the latter. First and foremost, general relativity must be able to reproduce Newtonian gravity in the appropriate limit. Doing this comprises the first half or so of chapter 7. The latter part of this chapter concerns itself with the three classic tests of intrinsic curvature, and in doing so introduces the Riemann and Ricci tensors, and Ricci scalar. These three classic tests are known as noncommutativity of second covariant derivatives (on vectors and higher tensors), rotation of a vector after parallel transport around a small closed loop, and relative acceleration of neighboring geodesics (or geodesic deviation, for short). The Newtonian Limit Of course, general relativity would not be accepted by any serious physicist if it did not reproduce the well-tested predictions of Newtonian gravity in the appropriate limit! In fact, this is the route Einstein almost certainly followed in trying to relate the admittedly somewhat mathematically abstract formalism of Riemannian geometry to the physical theory of gravity. What is this Newtonian limit? Basically, it is the limit of slow velocities, where slow here means in comparison with the speed of light. So Newtonian gravity should appear in some sort of nonrelativistic limit. As we shall see, the full theory of general relativity is nonlinear in the gravitational field quantities (such as the metric and its derivatives). However, Newtonian gravity is linear: the gravity produced by several objects together equals the sum of the gravities from the individual objects. So we need to linearize things as well to get the appropriate limit. One consequence of this is that we cannot examine the regime of strong gravity in this limit, where the nonlinearities become important. So the Newtonian limit is also the weak-field limit. As with any approximation to a more exact theory, the Newtonian limit is strictly only valid in the regime where it is claimed to be. So if we extrapolate outside this limit, we find some inconsistencies. For instance, we ll see below that the value of γ = dt/dτ gets set to a constant in examining the geodesic equation for timelike geodesics which are nonrelativistic. However, this means that the special-relativistic effect of time dilation due to clock movement is being suppressed in this limit. We may still dot the proper velocity with itself using the full metric; we would find that γ has terms which depend on the velocity, as expected, so it s not really a constant! This just demonstrates that one must not examine the Newtonian limit too carefully, or such inconsistencies will appear. Thus, we ll just be taking this brief look at the Newtonian limit in this chapter, and we ll quickly move on to the full theory. 1

Weak gravity effects on nonrelativistic motion One of the basic assumptions in the Newtonian limit is that the metric does not differ much from the regular metric of special relativity: g µν η µν where η µν = diag(1, 1, 1, 1) in Cartesian coordinates, as we saw in Chapter 5. Of course, if this were an exact equality, we would have geodesic motion equal to that in special relativity, which states that particles move in straight lines with no gravitational effects at all! So the main problem is to find the lowest-order correction to this in some sort of expansion in the strength of the gravitational field, and in the limit of nonrelativistic speeds. Another important assumption is that the correction terms which do appear are slowly varying. Mathematically, this is expressed as g µν t This does not say that the background masses causing the gravitational field cannot move! We all know that when several large masses are in each other s gravitational fields, they must move. Instead, the equation basically asserts that the instantaneous value of the gravitational field is more important in determining the differential equation governing particle motions than the changes in the field are. So, the Newtonian limit excludes situations such as having Jupiter flying close by at nearly the speed of light! The other ingredient is that we are in the nonrelativistic domain. In terms of the four-velocity defined in chapter 5 for particles with mass, this means that u 0 >> max u i, so that the time component, u 0 = d(ct)/dτ, is much greater than each of the spacelike terms u i = dx i /dτ. If we use g µν η µν, this means that γ c γ v 3D, where γ = dt/dτ. (Careful, as the relation dt/dτ = 1/ 1 v 2 /c 2 is now only valid if we are in Minkowski space, with gµν exactly equal to η µν. Otherwise, the full metric line element is necessary to find this relation in the general case.) So we may cancel the γ factors, finding that c v 3D is the nonrelativistic limit, as expected. We expect the Newtonian gravitational force to show up from the geodesic equations in the appropriate limit, so we begin with the general proper-time parameterized geodesic equation: 0 du α dτ + Γα µνu µ u ν = 0 The only places for an effective force to appear is in the connection term, since the first term is the four-acceleration we re trying to find. The nonrelativistic limit says that this will be most important when µ = ν = 0, since then the dominant u 0 term will be multiplying Γ in both slots. We then find the nonrelativistic limit of the geodesic equation to be du α dτ + Γα 00u 0 u 0 0 2

(no sum over 0 here; it s not a dummy index!), or just du α dτ + Γα 00γ 2 c 2 0. Next, we examine this equation for the four possible values of α. Setting α = 0 requires us to find Γ 0 00, which vanishes! Why? Because in our coordinate basis, the formula for the connection symbols is Γ α µν = 1 2 gαβ [ g βν,µ + g µβ,ν g µν,β ] and since the metric is diagonal and all time derivatives vanish, we find that Γ 0 00 = 0 in the Newtonian limit. Plugging this back in gives dγ/dτ 0, so that γ = dt/dτ is a constant. We may then apply the chain rule to convert proper-time derivatives into ordinary time derivatives: d dτ = dt d dτ dt = γ d dt This is true regardless of whether γ changes with τ or not. However, in this case where γ is a constant, we can apply d/dτ to this equation again, finding that second derivatives with respect to τ become second derivatives with respect to t times γ 2, without any derivatives of γ appearing. Thus, spatial indices i for α in the geodesic equation become and we may cancel the γ 2 to give γ 2 d2 x i dt 2 + Γi 00γ 2 c 2 0 d 2 x i dt 2 Γi 00c 2 Again, because we have a diagonal metric and all its time derivatives approximately vanish, we find that the only term contributing to Γ i 00 is the i th derivative of g 00 : Γ i 00 1 2 gii g 00,i with no sum on the fixed spatial index i. This term will vanish unless g 00,i is nonzero, so that term must supply the Newtonian gravitational force. Newtonian gravity is linear in the sources, so we can only keep terms to first order in the gravitational field. So we use η ii = 1 for g ii in calculating Γ i 00 here. Putting it all together then gives d 2 x i dt 2 1 2 g 00,i c 2 The right hand side should be the acceleration due to gravity, of course. The Newtonian force due to gravity is generally mg = Φ, where Φ is the gravitational potential energy, so the right hand side should reduce to (Φ/m)/ x i, 3

giving g 00,i = 2Φ,i /(mc 2 ). Together with the condition that g 00 η 00, the solution is ( g 00 1 + 2Φ ) mc 2 Notation: your book writes Φ for Φ/m here; they are using the gravitational potential energy per mass for Φ, rather than the potential energy itself, which we use here. In fact, this equation is exact when the metric is stationary! The Newtonian approximation is actually in not allowing any of the other components of g µν to vary from η µν. Since gravity is attractive, the gravitational potential energy for a collection of point masses is a negative quantity: Φ = j GM j m r j (1) where the sum runs over all the masses M j which create the gravity which mass m is responding to. We also see from this that Φ/m is independent of the test mass m, as it must be if that combination appears in the expression for the metric. We can apply the metric formula for a clock at rest with a diagonal metric to find that dτ = g 00 dt, The negative value for Φ in this case tells us that 0 < g 00 < 1, so clocks at rest near one of the masses M j will tick more slowly than a clock which is infinitely far away from them all! This is precisely gravitational time dilation, which we presumed must exist when we examined accelerating coordinate systems in special relativity. To escape from the Newtonian limit and find out how to construct the full metric, we need Einstein s equation, which is the subject of the next chapter. First, we ll need to introduce the curvature tensors, as they play starring roles in Einstein s equation. The Riemann curvature tensor Classically, there are two distinct geometric notions of curvature, extrinsic and intrinsic. Extrinsic curvature is a property of how an object (or manifold) is sitting in a higher-dimensional space. For instance, a shoestring could have a very curved appearance as it makes a knot on your shoe, and the surface of a metal cylinder makes for a curved mirror which distorts reflections. Extrinsic curvature of a manifold cannot be measured by a being which lives purely on that manifold, since it depends on the higher dimensions it sits in. On the other hand, intrinsic curvature refers to properties which may always be measured from within the manifold itself. In general relativity, that is the only type of curvature which we are interested in! The reason is that we don t have any way of knowing about what goes on outside of spacetime, if spacetime were to be sitting in some higher-dimensional universe of more than 4 spacetime dimensions. So we ll always mean intrinsic curvature when speaking about curvature. 4

It turns out that curvature is a property of two-dimensional surfaces! No curvature exists for one-dimensional manifolds. An ant who could only move, see, and measure along the long axis of your shoestring would have no way of telling whether the way it sits in three dimensions is curved (or knotted, for that matter). For manifolds of dimensions higher than two, curvature may occur separately in each of the independent two-dimensional slices (which we ll just call surfaces ), in general. A nontechnical way of thinking about curvature of a surface is to examine whether a flat piece of paper may be placed on that surface without tearing or crinkling the piece of paper. So by this definition, the surface of a cylinder has no (intrinsic) curvature, since you can put a flat piece of paper on it without tearing or crinkling (rolling it up to match the cylinder s shape does not require either of these). By the same test, the surface of a sphere really is curved, as is the fender on a bicycle (or aerodynamic car). More technically, the three classic signatures of intrinsic curvature are: failure of covariant derivatives to commute (on components of any tensor field besides a scalar), rotation of a vector after parallel transport around a small closed curve, and relative acceleration of neighboring geodesics. Although the first two of these seem rather different (the first requires a vector or tensor field, and the second requires only a single vector at a point), they are very closely related. In fact, the relation is very similar to that of the two jobs the connection symbols Γ perform: covariant differentiation (of the components of a field) and parallel transport (of a single vector or tensor). Parallel transport is defined to keep things constant, so that the covariant derivative of the vector field formed by parallel transport is zero. (The precise definition of constant is that derivatives of the object vanish.) The third test is also closely related to the first two, for two closely spaced geodesics form a surface (spanned by the direction of the geodesic, and the direction of their separation). Breaking up this surface into tiny squares around which parallel transport is performed on each relates geodesic deviation to the other two tests. Commutation of covariant derivatives Your book defines the Riemann tensor using the commutator of second derivatives on a covector: [ µ ν ν µ ] v α = R β ανµv β which, after a short calculation, gives (this is eq. [7.13] of your text) R β ανµ = ν Γ β αµ µ Γ β αν + Γ σ αµγ β σν Γ σ ανγ β σµ from which we can see that the Riemann tensor is antisymmetric in its last two indices: R β ανµ = R β αµν. (More about this and other symmetries later.) So we may also write the above relation as [ µ ν ν µ ] v α = R β αµνv β 5

(Note the switching of µ and ν from the previous version, which introduced the minus sign.) A similar calculation on the components of a contravariant vector gives [ µ ν ν µ ] v α = +R α βµνv β with the same placement of µ and ν as the equation immediately above it, but now with a positive sign on the Riemann tensor. Also, note that the roles of the first two indices have switched places, as they must in order to be consistent with the tensor notation (one up and one down index for any summed dummy index), since the Riemann tensor is always contracted with the vector index (up or down). At this point, we can say something about the various indices on the Riemann tensor: the second two indices specify the plane for which curvature is being measured, and the first two indices do the actual shuffling of the vector indices. Here, the plane is really the plane made by the two directions along which covariant derivatives are being performed. Notice that if µ = ν, the Riemann tensor vanishes! This is trivially true since a a a a is clearly zero; you really do need a plane for curvature to be present. What about other tensors? The general rule follows much the same pattern that the rules for including Γ factors in covariant derivatives of tensors do: we get a sum of terms, with one positive Γ factor for each upstairs index of the tensor in turn, and one negative Γ factor for each down index. Similarly, we get a sum of the Riemann tensor with a positive sign contracted with each up index in turn, and a negative sign for each down index of the tensor (to use the µν ordering as was done in the latter equations for vectors above). This leaves no Riemann tensor terms for covariant derivatives of a scalar! This is because ( µ ν ν µ )f = 0 if f is a scalar (without indices attached). Proving this is not as trivial as one might think at first! Although the first covariant derivative of a scalar is just the ordinary derivative (so that you might think we re just restating the commutation of ordinary derivatives), the covariant derivative of a scalar is now a (co)vector! So the second covariant derivatives involve the appearance of some Γ connection factors, and it turns out that the fact that we required Γ α µν = Γ α νµ in coordinate bases provides the necessary cancellation. You are strongly urged to show this yourself using the rules for µ ; it s not difficult! Parallel transport around closed curves The Riemann tensor also appears when checking a vector for rotation after parallel transport around a small closed curve in some plane. This job of the Riemann tensor is often used to illustrate curvature, but it is seldom used in calculations in general relativity. Your book gives the equation for this job as equation (7.21), which is still a bit obscure-looking, so here is a plain explanation in words: If a vector is parallel transported around a small closed curve of area A, the vector is rotated by an amount proportional to the Riemann tensor at the point of origin, times the area A of the curve. 6

This immediately again shows that curvature requires at least two dimensions, or the area of the loop would vanish! Again, we see the roles played by the various indices of the Riemann tensor: the final two indices specify the plane formed by the closed loop, and the first two indices rotate the vector components. (Of course, this again generalizes to higher tensors as a sum of such terms.) Note that only rotations are allowed, and not stretching, because one of the requirements of parallel transport was that it preserves the magnitudes of vectors. Geodesic deviation The third classic test of curvature is the failure of initially parallel neighboring geodesics to remain parallel, also known as geodesic deviation for short. In essence, this is a measure of a counterexample to Euclid s famous fifth postulate of geometry: that two initially parallel lines remain parallel for all time if the lines are straight. If they do, the space is flat (Euclidean), if not, it is curved! The geodesic deviation equation is given by equation (7.24) of your book (and again repackaged as [7.25] and [7.26] together): D 2 ξ a Du 2 + dx c Ra dxd cbd ξb du du = 0. In the equation, ξ represents the vector which points from one geodesic to the next, and D/Du is the operation of covariant differentiation along a curve, defined back in chapter 3 (the tangent vector to the curve which is parameterized by u, contracted with covariant derivative along the curve). Again, we see from the index placement that the tangent to one of the curves itself and the deviation vector ξ go with the last two indices of the Riemann tensor, so they specify the plane of curvature. The other curve s tangent vector (which is equal to the first s, to lowest order) is in the slot where vector components get rotated. Relative rotation of one geodesic s tangent with respect to the other changes their separation as we move along in u, so it shows up as an acceleration of the deviation vector between the two. Since only covariant derivatives make intrinsic geometric sense, the acceleration expression involves only capital Ds (and so they have ordinary d and connection symbols when written out in full). The geodesic deviation equation is often used in general relativity, for it describes tidal forces produced by gravity! For instance, the Earth as a whole moves as though its mass were all concentrated at the center. So only the center of mass actually follows a geodesic as the Earth moves in response to the gravity of the moon and sun. Points nearer to and farther from the moon than the earth s center feel weaker and stronger lunar gravity (respectively) than the center does, so there s a relative tidal force from this. In a similar way, the interior of the Earth feels tidal stresses (except at the center of mass), which cause a slight heating. When tidal forces are stronger (such as is the case on Jupiter s moon Io), the stresses can and do actually trigger volcanism. More generally, tidal forces are the signature of a nonuniform gravitational field. A uniform gravitational field is always equivalent to a uniformly accelerating reference frame, and so becomes Minkowski spacetime in freely falling 7

coordinates. So tidal forces, and the Riemann tensor, roughly measure the size beyond which reference frames cease to have physics which mimics that of special relativity. Notice that the Riemann tensor has dimensions of inverse area, since each covariant derivative of a coordinate with length has dimensions of inverse length. So the size beyond which you will notice tidal forces in a plane is roughly the inverse of the components of the Riemann tensor which correspond to that plane. Symmetries of the Riemann tensor As a tensor with four indices, the Riemann tensor could have as many as n 4 independent components, in n spacetime dimensions (this is 256 for n = 4). However, it possesses a number of important symmetries, which reduce the number of independent components greatly. These symmetries are easiest to state when we examine the tensor in fully covariant (0, 4) form (or fully contravariant form, of course). We ll follow the usual course and refer to the parallel-transport job of the Riemann tensor in analyzing these, as it s easiest to talk about, even though it s the least used of the three jobs the Riemann tensor has. Antisymmetry in last two indices: R abcd = R abdc. As we mentioned before, this is because the final indices label a plane, and you need two different indices to specify a plane! So it s at least off-diagonal in these indices. The fact that we have only antisymmetric and not symmetric off-diagonal components is most easily seen when performing integration of areas, which requires an orientation of each plane. Oriented planes have an intrinsic antisymmetry, just as the area between a curve f(x) and the x axis counts as negative when f lies below the axis in integration. Antisymmetry in first two indices: R abcd = R bacd. Since the first two indices perform the rotation of tensor components, this is a property of rotation matrices! They must be off-diagonal (else they would stretch the x component of a vector pointing purely in the x direction, if for example R xxyt were nonzero, which isn t allowed since parallel transport preserves lengths). Also, symmetric off-diagonal terms perform a shearing and not a rotation of little areas, so only off-diagonal ones remain since parallel transport must also preserve relative angles between two vectors. Symmetry when swapping first two indices with last two: R abcd = R cdab. This says that if the plane of parallel transport is switched with the plane of vector rotation, the total result is identical (which is interesting)! Cyclic antisymmetry of last three indices: R abcd + R acdb + R adbc = 0. This is not an independent symmetry, but may be derived by manipulating the three symmetries already mentioned. Bianchi identity: e R abcd + c R abde + d R abec = 0. This is a (covariant) differential identity, and so is of a bit of different character than the others. The derivatives signify that we re looking at the tensor at slightly different points. In full, it says that if we take any closed 3-dimensional cube and parallel transport a vector around all 6 faces (preserving the outward orientation of each face), then we get zero total rotation! This makes perfect sense when you draw a 8

picture: each edge of a cube gets traversed twice under such a prescription, once in each direction, and so the net rotations all must cancel. The derivative terms arise from moving from one face to its opposite face: instead of examining the Riemann tensor at a point, we must also invoke its covariant derivatives. Symmetries of the tensor in other forms If you are wondering how to use these symmetries in the (1,3) form of the Riemann tensor or other forms, the solution is just to raise and/or lower the same index at the same time in all terms of the equation expressing the symmetry. For example, we may contract the third symmetry with the inverse metric: and so g ae R abcd = g ae R cdab R e bcd = R cd e b and so on. Contractions of the Riemann tensor Since we have four indices, we may contract over any two. This produces either zero (if done over two antisymmetric indices), or what is known as the Ricci tensor, or (after two contractions) the Ricci scalar. These tensors play a primary role in the Einstein equation which is the central equation of state in general relativity, and so we will examine them in the notes to chapter 8, where the Einstein equation is introduced. 9