Weak-Field General Relativity Compared with Electrodynamics

Transcription

1 Weak-Field General Relativity Compared with Electrodynamics Wendeline B. Everett Oberlin College Department of Physics and Astronomy Advisor: Dan Styer May 15, 27

2 Contents 1 Introduction and Overview 3 2 Introduction to Tensors, Spacetime, and Relativity Vectors, Dual Vectors, and Tensors Tensor Manipulation Maxwell s Equations in Tensor Form 8 4 Introduction to General Relativity and the Einstein Equation Vectors, Dual Vectors, and Tensors in Curved Spacetime The Metric Covariant Derivatives and the Metric Connection Riemann, Ricci, and Einstein Tensors Minimal Coupling Principle and Geodesics Einstein s Equation Linearized Gravity Writing the Metric using a Perturbation Gauge Transformation Solving Einstein s Equation for Weak Fields Weak-Field Metric Analysis Verification of Einstein s Equation Calculating Christoffel Symbols Riemann Tensor Symmetries Calculating Riemann, Ricci, and Einstein Tensors Geodesic Equation for a Weak-Field Metric 37 8 Conclusions 38 1

3 9 Appendix A: Stress-Energy Tensor 41 1 Appendix B: Electromagnetism with Differential Forms Acknowledgments 47 2

4 1 Introduction and Overview We can draw many parallels between electromagnetism and general relativity, such as, for example, the fact that both electrostatic and gravitostatic forces fall off as 1/r 2. These parallels become especially clear when we compare the two in tensor notation. However, while Maxwell s equations in electromagnetism have a relatively straight-forward, intuitive understanding in terms of electric and magnetic field components, a similar intuitive understanding of general relativity is not so easily forthcoming. In contrast to electromagnetism, the equations of general relativity are non-linear and involve many more independent degrees of freedom. These factors complicate analysis. Furthermore, general relativity is generally studied in the language of geometry. While other forces of nature, such as electromagnetism and the short-range nuclear forces, are typically thought of as fields on spacetime, gravitational fields are usually analyzed as the curvature of spacetime itself. However, can we analyze general relativity in the familiar field-theoretic manner and from this gain a more intuitive understanding through parallels with these other theories? For electromagnetism the sources are charges and currents (the four-current), and these are related to the fields using Maxwell s equations. For gravity, the sources are energies and momenta (the stress-energy tensor), which are related to the curvature of spacetime using the Einstein equation. To see the parallels between the two theories emerge, we need to write the corresponding equations in the same form, namely, tensor formulation. Maxwell s equations can be written in tensor form (in flat space) as µ F µν = J ν (1) µ G µν =. (2) where F µν is the field tensor given by F µν = E 1 /c E 2 /c E 3 /c E 1 /c B 3 B 2 E 2 /c B 3 B 1 E 3 /c B 2 B 1 (3) and G µν is the dual field tensor given by G µν = B 1 /c B 2 /c B 3 /c B 1 /c E 3 E 2 B 2 /c E 3 E 1 B 3 /c E 2 E 1. (4) [If we assume we already know the structure of the field and dual field tensors (the positions of the electric and magnetic components in the tensors), then writing Maxwell s equations in tensorial form is straightforward. If we want to delve deeper into the structure of these equations, we turn to the mechanics 3

5 of differential forms which deals with covariant, totally antisymmetric tensors which the field and dual field tensors are, and this is treated in Appendix B of the paper.] The analogous equation in general relativity is the Einstein equation: G µν = R µν 1 2 Rg µν = 8πGT µν (5) where R µν is the Ricci tensor, R the Ricci scalar, which are both formed from contractions of the Riemann tensor, which, given in terms of the the Christoffel symbols Γ, is R ρ σµν = µ Γ ρ νσ ν Γ ρ µσ + Γ ρ µλ Γλ νσ Γ ρ νλ Γλ µσ (6) and the Christoffel symbols are formed from the metric g µν (which characterizes the underlying geometry of a space) by Γ σ µν = 1 2 gσρ ( µ g νρ + ν g ρµ ρ g µν ). (7) Essentially, the Christoffel symbols allow a conversion from structures in flat, Minkowski spacetime of special relativity to curved spacetime of general relativity. They are necessary because the ordinary partial derivative in an arbitrary spacetime is no longer linear (tensorial), and therefore the Christoffel symbols serve as a correction factor to make tensorial derivatives in curved spacetime possible (the covariant derivative). In Einstein s equation, T µν is the stress-energy tensor, also referred to as the energy-momentum tensor, which is the generalization of the mass density from Newtonian gravity, and G is Newton s gravitational constant. (An analysis that parallels the structure of electromagnetism for the stress-energy tensor is given in Appendix A.) What we d like to do is to look at the structure of Einstein equation in terms of the gravitational fields rather than the curvature of the underlying spacetime. However, it turns out that Einstein s equation in its full form is a bit unwieldy (ten nonlinear partial differential equations). Therefore, we turn to the limit in which Einstein s equation becomes linear which does have a general solution. In the linearized limit, we can write the metric as a sum of the Minkowski (flat) metric and a small perturbation term which we use only to the first order because we restrict ourselves to only weak gravitational fields. g µν = η µν + h µν (8) The weak-field limit can be used to find the weak-field metric given by ds 2 = (1 + 2Φ)dt 2 + (1 2Φ)(dx 2 + dy 2 + dz 2 ) = diag( 2Φ, 2Φ, 2Φ, 2Φ) (9) where the gravitational potential, Φ, obeys the Newtonian Poisson equation. For our analysis, we use the weak-field metric but we consider that the potential can vary with time, instead of the usual Newtonian gravitational potential which is a function of only the space coordinates. Therefore by comparison of eqns. 8 and 9 we see that we can write our perturbation h µν as h µν = 2Φ(x, x 1, x 2, x 3 )δ µν. (1) 4

6 Since we re interested in the gravitational fields more than the potentials we define which for the three space coordinates is just the usual gravitational field. f µ (x, x 1, x 2, x 3 ) = Φ x µ (11) Allowing the potentials to vary with time yet still satisfying the Einstein equation, we arrive at the restriction that the zero component of the field f must be constant for all space and time. We use linearized versions of the equations for the Christoffel symbols, Riemann tensor, Ricci tensor and scalar, and correspondingly, Einstein tensor, to see how these quantities are formulated in terms the fields, f, rather than in terms of the metric. Doing this for the Christoffel symbols is quite straight-forward and the result is given in section 6.2. The Riemann tensor is subject to a number of symmetries so that we (fortunately) don t have to calculate all 256 components. After showing that the number of independent components is only 2, we calculate them in the array in section 6.3. The Ricci tensor and scalar and the Einstein tensor components are calculated on the following pages using contractions of the Riemann tensor. In essence what these calculations perform is a rewriting of the curvature structures of general relativity that are usually formulated in terms of geometry (the metric) in terms of fields. What we find from these calculations is that in contrast to several texts which state that the Riemann tensor in gravity is analogous to the field tensor in electromagnetism, our calculations show that this is not the case. Rather, we find that it is the Christoffel symbols (which contain the fields f µ themselves as components) that are the analogous structure. To get a sense of how the fields result in the motions of test particles and what paths they follow, we look at the geodesic equation, which describes the motion of particles under the influence of no forces (beyond gravity). The results we find are given by eqns. 23 and 24 for the space and time components, respectively. Since forming a clear qualitative understanding of these results is not easily forthcoming, we look at the Raychaudhuri equation which relates geometrically the Ricci tensor to the paths followed by a group of nearby geodesics. The Newtonian limit of this equation makes sense in terms of fields, but the analysis in a less strict limit is still under current work. Furthermore, we d like to use the structures of general relativity now written in terms of fields to assess to what extent gravity mirrors any of the hierarchy of electric and magnetic fields (the fact that in the static situation electric fields dominate magnetic fields and the interrelated relationship of electric and magnetic fields). Specifically, we know a changing electric field generates a magnetic field. We d like to see whether perhaps a changing gravitational field generates any higher structures in gravity, and this analysis is still under current study. Linearized general relativity has been an established area of study for almost 1 years and the parallels with electromagnetism are clearly apparent. Therefore, we suspect that all the results obtained here have been independently discovered and published previously. The object of this project has not been to uncover 5

7 virgin territory, but to look at familiar results in new combinations from new angles. Therefore the goal of the project is not a unique result, and as of yet, we don t report any distinctly new findings. 2 Introduction to Tensors, Spacetime, and Relativity General relativity is a powerful tool to deal with any arbitrary spacetime, but we can introduce useful concepts in a comparatively simple case of general relativity, that of special relativity, where there is no gravity and spacetime is flat. The spacetime of special relativity is known as Minkowski space. Particles in such a space travel along curves through spacetime referred to as worldlines. In general, we can represent the position of a particle by four components, three for space and one for time, and we can write the spacetime distance between two events, the spacetime interval, as ( s) 2 = (c t) 2 + ( x) 2 + ( y) 2 + ( z) 2 (12) where x, y, z are the usual space components and t is time. In contrast to Newtonian mechanics, which provides provides a clear distinction of space from time and correspondingly the notion of simultaneity of events in time, no such distinction is possible in Special Relativity and the universal notion of simultaneity is lost. Using index notation with the Einstein summation convention we can write the position of a particle as x µ where µ runs from to 3 with the zeroth component standing for the time component, ct. We can then write the infinitesimal spacetime interval as ds 2 = η µν dx µ dx ν (13) where η µν is the 4 4 matrix, the metric for Minkowski space, with values 1 η µν = 1 1. (14) 1 Since we re interested in specific situations in some specific coordinate system, we d also like to know how to go about relating measurements in one reference frame to those made in another, i.e. how to relate the position of a particle in one frame expressed by x µ to its position in some other frame, x µ. Specifically, we re interested in frames of reference in which one moves with some fixed velocity relative to the other, since those separated by some fixed distance are relatively trivial and those separated by some fixed (or changing) acceleration are the subject of general, not special, relativity, or at least they must be treated in the limit in which special relativity can approximately hold true, namely, very short periods of time. What we re looking 6

8 for are transformations that leave the interval between events constant. We can consider transformations between frames of reference separated by a constant velocity (also called boosts) by multiplying the position in one frame by a matrix x µ = Λ µ νx ν. (15) (Recall that in index notation, an upper index that is repeated as a lower index in the same equation indicates a sum over that index.) Possibilities for Λ µ ν are limited by the restriction that the spacetime interval should be the same in both reference frames. Thus we write ds 2 = η µν dx µ dx ν = η µ ν dxµ dx ν = η µ ν Λµ µdx µ Λ ν νdx ν (16) so η µν = Λ µ µλ ν νη µ ν. (17) For the interval to be invariant, we must find transformation matrices such that the components of η µν are the same as those of η µ ν. Such transformations are known as Lorentz transformations. 2.1 Vectors, Dual Vectors, and Tensors The concept of vectors is familiar but takes on a different cast in relativity. Vectors exist at a single point in spacetime and can be thought of as tangent vectors at that point, thus the collection of all vectors at a given point, the tangent space, all lie on some subspace which is tangent at that particular point. If we choose to look at a given situation in some frame of reference, we can define a set of basis vectors, ê (µ), and we can then write some vector A as a linear combination of basis vectors, A = A µ ê (µ). Since it is Lorentz transformations that leave a given interval unchanged upon transformation, it is these transformations that we use to transform the components of a vector, since the vector itself must remain invariant in different frames of reference. A dual vector space is the space of all linear maps from a given vector space to the real numbers. We can express a dual vector in terms of the dual basis vectors for a certain frame of reference as ω = ω µ ˆθ(µ). Similarly, vectors themselves are maps from dual vectors to the real numbers. One particularly useful example of a vector is the vector tangent to some curve. If we parametrize the path through spacetime as x µ (λ) then we can write the tangent vector to that path with components V µ = dxµ dλ. (18) As a generalization of the notions of vectors and dual vectors, a tensor of rank (k, l) is a multilinear map from a set of vectors and dual vectors to the reals. Therefore a (, ) type tensor is a scalar, a (1, ) type is a vector, and a (, 1) type is a dual vector. To define a basis for a tensor, we use the tensor product which when used with a type (k, l) tensor and a type (m, n) tensor produces a type (k + m, l + n) tensor. In component notation we write an arbitrary tensor as 7

9 T = T µ1 µ k ν1 ν l ê µ1 ê (µk ) ˆθ (ν1) ˆθ (ν l). (19) We can define the transformation of a tensor by generalization from vectors and dual vectors. T µ 1 µ k ν 1 ν l = Λ µ 1 µ1 Λ µ k µk Λ ν1 ν 1 Λ ν l ν T µ1 µ k ν1 ν l l. (2) One very useful tensor is the Kronecker delta, δ µ ρ, a type (1, 1) tensor which relates the metric to the inverse metric, the metric written with upper indices, η µν : η µν η νρ = δ µ ρ. (21) Another useful tensor is the Levi-Civita symbol, usually expressed as a (, 4) tensor: +1 if µνρσ is an even permutation of 123 ɛ µνρσ = 1 if µνρσ is an odd permutation of 123 otherwise. (22) 2.2 Tensor Manipulation There are a few important ways of manipulating tensors to review that will become useful later. First, we can perform a contraction, which turns a type (k, l) tensor into a (k 1, l 1) tensor. This is done through summing over an upper and lower index such as S µρ σ = T µνρ σν. (23) We can change a lower index into an upper one and vice versa by using the metric and inverse metric. Therefore, from a tensor T µν αβ we can form a number of new tensors such as T µνγ β = ηγα T µν αβ (24) T ν γ αβ = η γµ T µν αβ. (25) A tensor is symmetric with respect to a pair of its indices if the tensor remains unchanged under exchange of those indices. Similarly, a tensor is said to be antisymmetric with respect to a pair of its indices if the value changes sign upon exchange of those indices. It is easy to show that symmetry and antisymmetry are properties of the tensor itself, not of its components in a particular frame of reference. 3 Maxwell s Equations in Tensor Form To gain a deeper appreciation for the similarities between electromagnetism and general relativity, we first must examine the form electromagnetism takes when written in tensor notation. The four Maxwell s equations are quite familiar: 8

10 E = 1 ɛ ρ (26) B = (27) E = B (28) t E B = µ J + µ ɛ (29) t where ρ( r, t) is the charge density as a function of space and time and J( r, t) is the current density. The usefulness of Maxwell s equations is that they relate the electric and magnetic fields on one side to their sources, ρ( r, t) and J( r, t), on the other. They show how some orientation of charges in spacetime corresponds with the associated electric and magnetic fields, and we ll see that the Einstein equation in General Relativity performs a very similar function. We can define an important antisymmetric tensor, the electromagnetic field tensor, as E 1 /c E 2 /c E 3 /c F µν = E 1 /c B 3 B 2 E 2 /c B 3 B 1. (3) E 3 /c B 2 B 1 Although we don t discuss the reasons behind its structure here, a brief discussion of them can be found in Appendix B. To begin rewriting Maxwell s equations using the electromagnetic field tensor, first we define the current 4-vector as J µ = (ρ, J x, J y, J z ). We can then write eqns in component notation ɛ ijk j B k E i = J i (31) i E i = J (32) ɛ ijk j E k + B i = (33) i B i =. (34) Here we have used Latin indices i, j, k to refer to only space components, so that they run 1-3, whereas Greek letters will be used for all components, -3, including the time component. Also, the notation µ is a shorthand notation indicating µ = x µ, (35) and ɛ ijk is the 3-dimensional Levi-Civita symbol, so that we write the curl of a vector in terms of the Levi-Civita symbol as ( V) i = ɛ ijk j V k (36) 9

11 for some vector V, since permutations of j and k for a given i will produce the desired alternation of positive and negative derivative terms. Looking at the field tensor written with upper indices (found by F µν = η αµ η βν F αβ ) we can see that we can express the components of E and B as F i = E i (37) F ij = ɛ ijk B k. (38) We can then use this to rewrite the first two Maxwell s equations, eqns. 31 and 32 as j F ij F i = J i (39) i F i = J. (4) Using the antisymmetry of the field tensor, we may rewrite the first equation, eqn. 39, as j F ij + F i = J i (41) and we may combine these first two Maxwell s equations into a single tensor equation: µ F µν = J ν. (42) It can be shown similarly that the second two Maxwell s equations can be written in component tensorial form as [µ F νλ] = (43) where the square brackets indicate an antisymmetrization with respect to the indices µ, ν, λ in which the permutations of indices are added in an alternating sum (terms with an odd number of permutations from the ordering µνλ are added with a minus sign and those with an even number of permutations are added with a plus sign). Using the antisymmetry of the field tensor we can write the antisymmetrization as µ F νλ + ν F λµ + λ F µν =. (44) However, we can also write the second two Maxwell s equations in a slightly different and somewhat more intuitive form using a tensor defined as the dual field tensor G µν to the field tensor: B 1 /c B 2 /c B 3 /c G µν = B 1 /c E 3 E 2 B 2 /c E 3 E 1. (45) B 3 /c E 2 E 1 It turns out that we can relate the field tensor to the dual field tensor by the substitution E/c B and B E/c, and the mechanics of this relation come through operations in the field of differential forms, discussed briefly in Appendix B. 1

12 Comparing the first two equations of Maxwell s equations with the last two, we see that writing the field components using the dual field vector has done exactly what we would like, namely that it has swapped the electric field terms for the magnetic field terms and vice versa, with the change of some minus signs. Therefore, we can write the second two Maxwell s equations in a very similar form to the first two, namely G µν =. (46) x µ Overall, we have reduced the Maxwell s equations from four to two equations, one for the equations with source charges and currents, and one for equations that are source-free: µ F µν = J ν (47) µ G µν =. (48) 4 Introduction to General Relativity and the Einstein Equation As we move from the relatively straightforward Minkowski spacetime to some arbitrary spacetime that could have curvature, a bit of care is needed in defining what we mean by the structures mentioned in the last section. To describe more complicated and curved spacetimes, we use the concept of manifolds. We are accustomed to working in flat, Euclidean space, R n, the set of n-tuples (x 1,..., x n ), often accompanied by a positive, definite metric that has components δ ij. A (differentiable) manifold corresponds to a space that may globally be very complicated and curved but one which on the local level resembles Euclidean space in the way that functions and coordinates operate (though the metric may not be the same). Therefore, we can think of a differentiable manifold as a space that can be composed of small pieces of R n patched together to form the manifold as a whole. The contrast of global compared to local structure in manifolds is precisely fitting for the structure of general relativity. In essence, general relativity states that the gravitational fields caused by mass-energy is the same as the curving of spacetime, and this concept is embodied by Einstein s Equivalence Principle. In its weak form, the principle states that The motion of freely falling particles are the same in a gravitational field and a uniformly accelerated frame, in a small enough region of space and time. A freely falling particle is one which is influenced by gravity but no other forces, and in the context of general relativity, we define such reference frames as inertial reference frames. The idea behind the weak formulation of the principle is the idea that if one were to perform experiments inside a box it would be impossible to tell whether the box was in a gravitational field or in a frame with uniform acceleration, so long as the box was small enough that the gravitational field was uniform in space (and therefore no inhomogeneities in the field would cause tidal effects) and the experiment was performed for a short enough period of time that any field inhomogeneities would not be evident. Therefore, on any manifold we can always define locally inertial frames of reference, but unlike Minkowski space, on a global scale this is no longer possible. 11

13 4.1 Vectors, Dual Vectors, and Tensors in Curved Spacetime Without embedding our manifold in a higher dimensional space, we lack our previous notion of a vector as tangent to the curves passing through a given point. It is appropriate instead to use the notion of the directional derivative along curves through a specific point. We define a space that is made up of all the smooth functions on a given manifold, and then use the fact that for each curve passing through a desired point we can uniquely assign a directional derivative operator (for a function f we have the assignment f df/dλ at a certain point, where λ is the parameter along the curve), and the space of all of these directional derivatives is equivalent to the notion of a tangent space, the collection of tangent vectors at that point. We can then find a basis for tangent vectors in a given coordinate system as d dλ = dxµ dλ µ, (49) which shows that the partials { µ } are a good set a of basis vectors for the vector space of directional derivatives. We can recognize the components of d/dλ as the same as those defined above for the spacetime of special relativity, eqn. 18, providing further legitimation for using directional derivatives as a replacement for the tangent space of Minkowski space. The only difference is that here, for an arbitrary spacetime, we use the basis vectors ê (µ) = µ. We can find the transformation of basis vectors from one reference frame to another simply by applying the chain rule µ = xµ x µ µ. (5) This relation produces the transformation rule for the components of a tangent vector as V µ = V µ µ (51) xµ µ = V µ (52) x µ so that V µ = xµ x µ V µ. (53) We see that eqn. 53 is compatible with Lorentz transformations in special relativity; indeed, transformations in flat Minkowski space are simply a specific example of a transformation of the form of eqn. 53. Once again we can consider dual vectors to be the linear maps from our newly redefined tangent space to the real numbers. In the same way that the partial derivatives along coordinate axes can be considered as a natural basis for tangent space, the gradients of coordinate functions x µ form a natural basis for dual vectors, expressed as dx µ. We therefore find the transformation properties of dual basis vectors to be dx µ = xµ x µ dxµ (54) and for the components, ω µ = xµ x µ ω µ. (55) 12

14 Generalizing to an arbitrary type (k, l) tensor, we find T = T µ1 µ k ν1 ν l µ1 µk dx ν1 dx ν l (56) defined similarly to eqn. 19, and in general, we refer to upper indices as contravariant and lower indices as covariant. By extension we can find the transformation properties of tensors to be similar to eqn. 2. T µ 1 µ k ν 1 ν l = xµ 1 x µ 1 xµ k x µ k x ν 1 x ν 1 xν 1 T µ1 µ k ν1 ν x ν l, (57) The Metric The primary characterizing feature of a manifold is the metric and it is here that we really begin to see the effects of curvature. The metric in general relativity is referred to with the symbol g µν while η µν is reserved only for the metric of Minkowski space. Components of the metric are restricted to be symmetric and if they are also non-degenerate, meaning that the determinant of the tensor does not vanish, then the inverse metric g µν may be defined as g µν g νγ = δ γ µ. (58) Similar to its use in special relativity, the metric in general relativity can be used to raise and lower the indices on tensors. The metric for general relativity embodies many of the concepts we re familiar with in both flat spacetime and in Newtonian gravity. It encompasses many of the innate properties of a given space it represents, such as it allows for the measurement of path length and proper time and therefore a sense of past and future, it determines the shortest spacetime distance between two events and therefore determines the paths that free test particles will travel. Furthermore, it generalizes the Newtonian gravitational potential Φ. The notion of the metric as encompassing the notion of path length becomes clear when we reexamine the definition of the spacetime interval we used for Minkowski space, eqn. 13: ds 2 = η µν dx µ dx ν. We can now recognize that the dx µ are just the dual basis vectors in general relativity, and we can write ds 2 = g µν dx µ dx ν (59) as the interval for any arbitrary spacetime. From this we see that the notion of metric and interval, or distance, become inherently related and interchangeable. The metric also embodies the sense of geometric curvature of a given spacetime. In Minkowski space, we can always make a change of coordinates so that the components of the metric are all constant, because Minkowski space is flat. However, having the components of the metric vary with the coordinates chosen is not sufficient for the metric to describe a space with 13

15 curvature. For example, if we choose to describe flat space using spherical coordinates, we can transform from Minkowski space in rectangular coordinates: ds 2 = (cdt) 2 + dx 2 + dy 2 + dz 2 (6) using the transformations x = r sin θ cos φ (61) y = r sin θ sin φ (62) z = r cos θ (63) so that the interval becomes ds 2 = (cdt) 2 + dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2. (64) Comparing with eqn. 59 we can see that the components of the metric will not all be constant and will depend on the coordinates even though we are still describing flat spacetime. Therefore, although if we re dealing with flat spacetime we can in principle always find the coordinate system in which the metric components are all constant, in practice doing this may be difficult and so we will want to find an additional way of representing the curvature of spacetime besides the metric. The metric also includes within it a sense of the Equivalence Principle. We can write the metric in a particularly useful formulation, known as canonical form, in which the components become g µν = diag( 1, 1,..., 1, +1, +1,..., +1,,,..., ) (65) where diag indicates that the only nonzero components are the diagonal elements, with the values listed. This form looks very familiar since the Minkowski metric is most often cited in canonical form, as above in eqn. 14. It turns out it is always possible to write any arbitrary metric in this form. However, usually this can only be done at a single point on the manifold or in some small neighborhood in which the first derivatives of the metric vanish, though in general the second derivatives do not vanish. The coordinates used to do this are referred to as locally inertial coordinates, incorporating the fact that any spacetime regardless of curvature (so long as it is a differentiable manifold) looks like flat Minkowski space permitted we look in a small enough volume for a short enough duration of time. We know how to relate points that are close enough to each other that we can use special relativistic principles; however, how do we go about relating distinct points that are not so close together? Since we define tensors as maps from vectors and dual vectors to the real numbers, how do we go about relating these maps at different points on a manifold? This issue is of extreme importance, since we d like to be able to take derivatives along curves in the manifold, but the question becomes: with respect to what should we take the derivative? How do we measure the rate of change of something when we don t have a way of relating the same structure at different points along the curve, points which do not lie in the same tangent space? 14

16 As a basis for comparison, we define the concept of parallel transport, which is the idea of transporting a vector along a curve while keeping both its direction and magnitude unchanged. In order to do this, we need a well-defined metric. It turns out that in Minkowski space we can perform such a transformation between two points along any curve through spacetime with the same result, but in curved spacetime the outcome is path-dependent. There is no easy way out of this, and we ll have to cope with the fact that there is no natural choice for how to parallel transport a vector in curved space, and therefore we will always be limited in our attempts to relate quantities from distinctly different points in a manifold. However, if we already have a path in mind, or we look infinitesimally and constrain our interest to the transportation of a vector in a given direction, we can develop the concept of transporting a vector along a path while keeping it as constant as possible. In turn, we use this notion of parallel transport as a reference point for defining the derivative as a deviation away from parallel propagation. The structure underlying the ability to compare nearby points is the concept of the connection, a kind of correction factor for the fact that curved spacetime is not flat, forcing us to find a way to incorporate the effects of curvature into our equations. 4.3 Covariant Derivatives and the Metric Connection We can take an ordinary partial derivative, but it turns out that the partial derivative is not a tensorial operator, in other words, the result of such an operator will not always be a tensor, some of the terms may be nonlinear. However, If our structures are not tensorial then ideas such as that we can look at the same situation from multiple frames of reference and observe the same fundamental structures invariantly are no longer true. On a qualitative level, we can use the concept of parallel propagation to define a rate of change of a vector by comparing to what it would have been if it had just been parallel transported, as mentioned above. On a more quantitative level, clearly, we need to find a replacement for the partial derivative, and this comes in the form of the covariant derivative. We require that the covariant derivative provide the same function in flat space with inertial coordinates as the partial derivative, but that it transforms like a tensor in any arbitrary manifold. We see that in flat spacetime the partial derivative is a map from type (k, l) tensors to type (k, l + 1) tensors using the example: µ S ν γ = T ν µ γ. (66) Therefore we require that the covariant derivative be a map from (k, l) tensors to type (k, l + 1) tensors with the restriction that the map be linear and obey the Leibniz (product) rule. If it obeys the Leibniz rule, then it is always possible to always write the covariant derivative as the partial derivative plus some linear transformation that acts as a correction term to make the result covariant. So for a vector with components µ, we define a set of n matrices (Γ µ ) ρ σ where n is the dimensionality of the underlying spacetime. We refer to Γ ρ µσ as the connection coefficients, and we can now write the covariant derivative of (the components of) a tangent vector as µ V ν = µ V ν + Γ ν µσv σ. (67) 15

17 Similarly, we can write the covariant derivative of a dual vector as µ ω ν = µ ω ν Γ σ µνω σ. (68) Because the the connection serves to cancel out the nonlinear part of the ordinary partial derivative, therefore leaving the covariant derivative tensorial, Γ by definition is not itself a tensor. For tensors in general, we can generalize from tangent vectors and dual vectors so that for a type (k, l) tensor, there are k terms of +Γ and l terms of Γ: σ T µ1µ2 µ k ν1ν 2 ν l = σ T µ1µ2 µ k ν1ν 2 ν l +Γ µ1 σλ T λµ2 µ k ν1ν 2 ν l + Γ µ2 σλ T µ1λ µ k ν1ν 2 ν l + Γ λ σν 1 T µ1µ2 µ k λν2 ν l Γ λ σν 2 T µ1µ2 µ k ν1λ ν l. (69) For a given manifold with metric g µν we can define a single unique connection by restricting the connection to be torsion-free, meaning that it is symmetric in its lower indices, and has the property of metric compatibility, meaning that the covariant derivative of the metric with respect to that connection is always zero everywhere, ρ g µν =. We can find that these restrictions produce the existence of a single, unique connection by writing out metric compatibility with all possible permutations of indices: ρ g µν = ρ g µν Γ λ ρµg λν Γ λ ρνg µλ = (7) µ g νρ = µ g νρ Γ λ µνg λρ Γ λ µρg νλ = (71) ν g ρµ = ν g ρµ Γ λ νρg λµ Γ λ νµg ρλ =. (72) If we subtract the second two equations from the first and use the symmetry of the connection we find ρ g µν µ g νρ ν g ρµ + 2Γ λ µνg λρ = (73) We solve for the connection by multiplying by the inverse metric g σρ to find Γ σ µν = 1 2 gσρ ( µ g νρ + ν g ρµ ρ g µν ), (74) which are referred to as the Christoffel connection, and whose components are called the Christoffel symbols. As a correction to the partial derivative to make it account for the differences between flat and curved spacetime, the connection in many ways embodies the amount and nature of the curvature of a particular spacetime, and as we ll see in section 6, it also forms the most important parallel structure to the fields of electromagnetism. 4.4 Riemann, Ricci, and Einstein Tensors Recall that when a vector is parallel transported around a closed loop in curved spacetime the vector will undergo a transformation dependent on the path chosen, a rotation from its initial orientation. The amount 16

18 of this transformation is directly the result of the amount of curvature of the space. Therefore, to define a structure that further embodies the curvature of a space, we can think of propagating a vector V µ around a tiny loop defined by two infinitesimal vectors, A µ and B ν, so that we go first in the direction of A µ, then in the direction of B ν, and then backward along A µ and B ν so that we end up where we started. Since the action of parallel transport is coordinate independent, we should be able to express the change in the vector V µ around the loop using a tensor, which is a linear transformation. Therefore, we can write the expected expression for the change δv ρ experienced by the vector V µ around the loop in the form δv ρ = R ρ σµνv σ A µ B ν (75) where the tensor R ρ σµν is called the Riemann tensor or curvature tensor. If we move around the loop in the opposite direction, corresponding to swapping A µ and B ν in eqn. 75, we should just get the opposite of what we had originally. Thus we know that the Riemann tensor must be antisymmetric in its last two indices. We d like to find an expression for the Riemann tensor in terms of the Christoffel symbols (which are themselves directly related to the metric). To do this, we look at the commutator of two covariant derivatives. Each covariant derivative along a given direction measures the amount that a vector varies compared with if it had been parallel transported, since the covariant derivative of a vector in the direction in which it is parallel transported is zero. Therefore, the commutator of covariant derivatives in two different directions will essentially measure the difference in the resulting transformation of the vector if the vector had been transported first in one direction and then the other or vice versa. If we expand the expression for the covariant derivative using its definition we find [ µ, ν ]V ρ = µ ν V ρ ν µ V ρ (76) = µ ( ν V ρ ) Γ λ µν λ V ρ + Γ ρ µσ ν V σ (same terms with µ and ν swapped) (77) = µ ν V ρ + ( µ Γ ρ νσ)v σ + Γ ρ νσ µ V σ Γ λ µν λ V ρ Γ λ µνγ ρ λσ V ρ +Γ ρ µσ ν V σ + Γ ρ µσγ σ νλv λ (same terms with µ and ν swapped). (78) Expanding, relabeling some dummy indices, and canceling some terms through antisymmetrization, we can rewrite the expression as [ µ, ν ]V ρ = ( µ Γ ρ νσ ν Γ ρ µσ + Γ ρ µλ Γλ νσ Γ ρ νλ Γλ µσ)v σ + (Γ λ νµ Γ λ µν) λ V ρ (79) The last term is the torsion tensor, which will be zero for the Christoffel symbols, since they are always torsion-free. Since the left hand side of the expression is a tensor, then the expression in the parentheses on the right hand side must also be a tensor. Therefore, we can write the value of the Riemann tensor in component form as R ρ σµν = µ Γ ρ νσ ν Γ ρ µσ + Γ ρ µλ Γλ νσ Γ ρ νλ Γλ µσ (8) 17

19 and this form can be shown to agree with the expression given in eqn. 75. From the Riemann curvature tensor we can get a better sense of the relationship between the metric and the curvature of the space it describes. If there exists a coordinate system in which the components of the metric are constant, then the Riemann tensor vanishes, and conversely, if the Riemann tensor vanishes, then we can always construct a coordinate system in which the components of the metric are all constant. In either case, we know for sure we are dealing with flat spacetime. There are some important properties of the Riemann tensor which can aid in calculating its components. We show them using the Riemann tensor with all lowered indices. We have already shown that the tensor is antisymmetric in its last two indices, R ρσµν = R ρσνµ, (81) and it turns out that the tensor is also antisymmetric in its first two indices, R ρσµν = R σρµν. (82) The tensor is symmetric with exchange of the first pair of indices for the second pair, R ρσµν = R νµρσ, (83) and the cyclic sum of permutations in the last three indices vanishes, R ρσµν + R ρµνσ + R ρνσµ =. (84) There are several other structures that are defined using the Riemann tensor which will become useful later in introducing the Einstein equation. The contraction R µν = R λ µλν (85) defines the Ricci tensor R µν, which is symmetric as a result of the symmetries of the Riemann tensor. From the definition of the Riemann tensor in terms of the Christoffel symbols, eqn. 8, and the definition of the Ricci tensor as a contraction of the Riemann tensor, eqn. 85, we can write the Ricci tensor in terms of the Christoffel symbols as R µν = λ Γ λ µν ν Γ λ µλ + Γ λ µνγ σ λσ Γ λ µσγ σ νλ. (86) The trace of the Ricci tensor, the Ricci scalar, is R = R µ µ = g µν R µν. (87) 4.5 Minimal Coupling Principle and Geodesics Now that we have all the structures needed to describe curvature, we start on the real task of general relativity, which is, How is curvature related to mass distribution? We need to know: How does a gravitational 18

20 field affect the behavior of matter? and conversely, How does the distribution of matter determine the gravitational field? For Newtonian gravity, these two questions are answered by a = Φ (88) which relates the acceleration of an object in a gravitational potential, Φ, and 2 Φ = 4πGρ, (89) Poisson s equation, which relates the mass density ρ to the produced gravitational potential, where G is Newton s gravitational constant. For general relativity, the two questions become: How does the curvature of spacetime affect matter in such a way as to manifest gravity? and How does energy-momentum cause the curving of spacetime? The minimal coupling principle provides a straight-forward approach for the generalization of principles from flat to curved spacetime. It instructs that we rewrite a law of physics known to be true in inertial frames in flat spacetime in a coordinate-invariant form (so that the result is a tensorial equation) and then show that the result remains valid in curved spacetime, which we do through comparison at the Newtonian limit. For example, we can look at the movement of a freely-falling particle, which in flat spacetime means that it travels on straight-line paths. Therefore, the second derivative of the position of the particle with respect to the parameter along the path is zero: d 2 x µ =. (9) dλ2 Using the chain rule we can write d 2 x µ dλ 2 = dxν dλ dx µ ν dλ. (91) To move from this equation to one that is tensorial, we simply replace the partial derivative with the covariant derivative to get dx ν dλ dx µ ν dλ = d2 x µ dλ 2 which we recognize as the geodesic equation, d 2 x µ dλ 2 + dx ρ dx σ Γµ ρσ dλ dλ + dx ρ dx σ Γµ ρσ dλ dλ (92) =. (93) We can confirm that the minimal coupling principle works in this case by finding the geodesic equation from a different method, using what we know about parallel transport. To define parallel transport more explicitly, for a curve x σ (λ) in order for an arbitrary tensor T µ1µ2 µ k ν1ν 2 ν l to be parallel transported, the components of the tensor must remain constant along the curve so that d dλ T µ1µ2 µ k ν1ν 2 ν l = dxσ dλ x σ T µ1µ2 µ k ν1ν 2 ν l =. (94) To ensure the result is tensorial, we replace the partial derivative with a covariant one, and define the directional covariant derivative as D dλ = dxσ dλ σ. (95) 19

21 Therefore, for the tensor T to be parallel transported we say that ( ) µ1µ D 2 µ k dλ T = dxσ dλ σt µ1µ2 µ k ν1ν 2 ν l =. (96) ν 1ν 2 ν l With a more concrete structure of parallel transport, we can define a geodesic as a path x µ (λ) along which a tangent vector dx µ /dλ is parallel transported to itself. So long as the connection we use is the Christoffel connection, we can ensure that this definition and the definition of a geodesic as the shortest spacetime distance between two points, the generalization of straight lines in flat spacetime, (found using the principle of least action) are equivalent. Therefore, we can write for a geodesic or equivalently, using the definition of the directional covariant derivative, the same as eqn. 93. d 2 x µ dλ 2 D dx µ dλ dλ = (97) + dx ρ dx σ Γµ ρσ dλ dλ =, (98) 4.6 Einstein s Equation Finally we are at the heart of general relativity: the Einstein equation, which describes how gravitational fields and curvature of spacetime respond to energy and momentum, just as Maxwell s equations describe how electric and magnetic fields respond to charge and current. Just like Newton s second law, which can be demonstrated but not derived from more basic concepts, the Einstein equation is ultimately too fundamental to be derived from first principles, so we ll just state it and then show how it works: R µν 1 2 Rg µν = 8πGT µν (99) where R µν is the Ricci tensor, R the Ricci scalar, T µν the so-called stress-energy tensor, also referred to as the energy-momentum tensor, which is the generalization of the mass density from Newtonian gravity, and G Newton s gravitational constant. Sometimes the left hand side of the equation is written as a single tensor, called the Einstein tensor, for convenience: G µν = R µν 1 2 Rg µν. (1) Since under relativity we think of energy as the same as mass, the stress-energy tensor, T µν, embodies both energy and momentum of a given source distribution. The component of the tensor is the energy density, and the i and i components are the momentum density in the ith direction. The ij components are the stress part of the stress-energy tensor and can be thought of as forces or pressures, for example, for a fluid, the stress tensor becomes the surface pressure of the fluid, the force exerted per unit area. We treat the structure of the stress-energy tensor as a parallel to electromagnetic source vectors in Appendix A. 2

22 One concern with the Einstein equation is ensuring that it satisfies conservation of energy, which for general relativity takes the form µ T µν =. (11) In addition to the four symmetries obeyed by the Riemann tensor, it also obeys a principle known as the Bianchi identity: [λ R ρσ]µν = λ R ρσµν + ρ R σλ + σ R λρ =. (12) If we contract twice on the Bianchi identity, we have µ R ρµ = 1 2 ρr. (13) Comparing this with the definition of the Einstein tensor, we can see that the twice-contracted Bianchi identity is equivalent to µ G µν =. (14) Therefore, if we look at the Einstein equation, we can see that energy conservation is always satisfied if we use the Einstein tensor, which is why we use the particular combination of Ricci tensor and scalar found in eqn. 99. Delving deeper into the Einstein equation, we see that the curvature on the left hand side is related to the energy-momentum on the right by a proportionality constant, 8πG, which will be fixed by comparison with Newtonian gravity. Our main concern vis-à-vis the minimal coupling principle is ensuring that Einstein s equation reduces to the Poisson equation of Newtonian gravity, 2 Φ = 4πGρ, which it replaces in moving from special to general relativity. To look at the Einstein equation in the Newtonian limit, we restrict ourselves to only weak gravitational fields, small velocities, and negligible pressures in the stress-energy tensor. We can write the stress-energy tensor for a perfect fluid source of energy-momentum as T µν = (ρ + p)u µ U ν + pg µν (15) where U µ is the four-velocity of the fluid and ρ and p are the rest-frame energy and momentum densities, respectively. Since we re looking in the Newtonian limit, we can neglect the pressure, and therefore, the stress-energy tensor reduces to T µν = ρu µ U ν. (16) We can work in the rest frame of the fluid we re working with so that U µ = (U,,, ), (17) and we normalize the four-velocity using g µν U µ U ν = 1. (18) We consider our weak gravitational field to be represented by a small perturbation away from a flat Minkowski metric, so that we can write g µν = η µν + h µν where h µν is a small perturbation term, and it turns out 21

23 for Newtonian gravity, the component of the perturbation is h = 2Φ where Φ is the Newtonian gravitational potential. We ll deal with where this comes from later when discussing the weak-filed metric and general relativity in the linearized limit. We approximate that U = 1 since the energy density ρ is close to zero for situations close to flat spacetime. Therefore, in the Newtonian limit T = ρ. (19) and all the other components are negligible. If we contract over both sides of the Einstein equation, we find that R = 8πGT, (11) where T is the stress-energy scalar just as R is the Ricci scalar, and we can use this to write an equivalent form of the Einstein equation as ( R µν = 8πG T µν 1 ) 2 T g µν. (111) Therefore using eqn. 19 we find the component of the Ricci tensor to be R = 4πGρ (112) Next we find the relationship between the Ricci tensor components and the perturbation factors, h µν, using the definition of the Ricci tensor in terms of the Christoffel symbols, eqn. 86, and the definition of the Christoffel symbols in terms of the metric. Since in the Newtonian limit we re only concerned to within the first order in the perturbation of the metric, we can neglect the factors of (Γ) 2 in eqn. 86. Thus we calculate R as R = Γλ x λ Γλ λ x (113) ( ) 1 = λ 2 ηλσ ( h σ + h σ σ h ) (114) = 1 2 δij i j h (115) = h (116) where we have neglected the time derivatives of the Christoffel symbols in the first line and those of the metric in the second line, since in the Newtonian limit we consider only static fields. (Note that this will no longer be the case in the linearized limit). Therefore, we can write using eqn h = 8πGρ (117) from which we can see, with h = 2Φ, that Einstein s equation yields precisely the Poisson equation of Newtonian gravity: 2 Φ = 4πGρ. (118) 22

24 5 Linearized Gravity Einstein s equation comprises ten partial differential equations for the ten unknown coefficients of the metric g µν, which, unlike Maxwell s equations, are nonlinear. Although there is some progress to be made with Einstein s equation in its complete form, namely solutions such as flat space and for the spherically symmetric and static Schwarzchild metric (often situations where symmetries can make solving the Einstein equation easier), there is no general solution to the equation. However, for situations that are very close to flat spacetime, we can find a general solution. Working in this limit is quite common, though as computational techniques become more extensive and developed, the need for this limit to simplify calculation is sometimes no longer necessary. The linearized limit is not so strict as the Newtonian limit, in which we are constrained only to weak fields, small velocities, and small pressures in the stress-energy tensor. Here, we are restricted similarly to weak fields, but we allow for the possibility of large velocities and pressures. 5.1 Writing the Metric using a Perturbation The weakness of the gravitational field means that, as before, we can write the metric as a sum of the Minkowski metric plus a perturbation term, so that g µν = η µν + h µν (119) where all of the components of h µν are much less than one, and where η µν is the usual Minkowski metric with components η µν = diag( 1, 1, 1, 1). The inverse metric is g µν = η µν h µν (12) since because h µν is small, we can ignore all factors higher than first order in this perturbation term. First we can calculate the Christoffel symbols for such a metric. Γ ρ µν = 1 2 gρλ ( µ g νλ + ν g λµ λ g µν ) (121) = 1 2 ηρλ ( µ h νλ + ν h λµ λ h µν ) (122) since first order partial derivatives of the Minkowski metric vanish, and we can neglect the factor of h ρλ in the inverse metric since we are only concerned with first order perturbations. In a vacuum, essentially the equivalent of Maxwell s equations without source, the Einstein equation reduces to R µν =. (123) When we substitute our metric, eqn. 119, into this equation, the expansion has two terms, one for the Minkowski metric and one for the perturbation. The first term vanishes, since the Ricci tensor for flat 23