In a Previous Lecture, We Introduced Pull-Backs of Differential Forms. We Will Now Put This Into Use and Describe Killing Vectors

6. SYMMETRY In a previous lecture, we introduced pull-backs of differential forms. We will now put this into use and describe Killing vectors. Heuristically, these are infinitesimal diffeomorphisms of the manifold leaving the metric invariant, i.e. infinitesimal isometries. Suppose that ϕ : M N is a diffeomorphism, where M and N are smooth manifolds. We know from Section 3.2.6 that we can pull-back differential forms on N to differential forms→ on M. With vector fields – indeed dual to differential forms – the induced map goes in the other direction: it maps vector fields on M to vector fields on N. The best way to see this is by viewing a vector field on M as a derivation on smooth functions on M and use the definition of pull-back of a smooth function on N. The pushforward vector field ϕ∗v X(N) of v X(M) by the diffeomorphism ϕ : M N is defined by ∈ ∈ ϕ∗v(f) := v(f ϕ)→ (f C (M)). ◦ ∈ ∞ This is intrinsically coordinate independent, but a bit abstract. Let’s derive the corresponding expressions in tensor notation; let {xi} be coordinates in a neighborhood of p on M and {yk} in a neighborhood of ϕ(p) on N. Then, on writing k k i i ϕ∗v =(ϕ∗v) ∂/∂y and v = v ∂/∂x we have k k ∂y i (ϕ∗v) = v . ∂xi ∂yk with the Jacobian matrix ∂xi evaluated at p.

Exercise 6.1. Check this last equality by evaluating ϕ∗v on f C (N). ∈ ∞ Remark. Thus, the pushforward is a map ϕ∗ : X(M) X(N) of sections of the tangent bundles TM and TN, respectively. However, a common understanding → of the pushforward is as a map between tangent spaces: ϕ∗ : TxM Tϕ(x)N for x M. The two are related by ϕ∗ v(x) = ϕ∗v(ϕ(x)) where on the left-hand-side ∈ → ϕ∗ acts by matrix multiplication with the Jacobian matrix. With the deﬁnition of a differential p-form ω on M asa C (M)-multilinear, skew symmetric map ω : X(M)p C (M) we claim that the following∞ relation exists between pullbacks of forms and pushforwards∞ of vector ﬁelds: ∗ → ϕ ω(v1,...,vp) = ω(ϕ∗v1,...,ϕ∗vp). (6.1) This can be seen most easily in tensor notation for which

i1 ip ω(v1,...,vp) = ωi ···i v v . 1 p 1 ··· p Exercise 6.2. Check Eq. (6.1) by choosing local coordinates on M and using tensor notation. More generally – by dropping the condition of skew symmetry on forms – one can deﬁne the pullback of a (0,p)-tensor T by the same formula (6.1), of course re- placing ω by T. In tensor notation, this becomes the familiar tensor transformation law k1 kp ∗ ∂y ∂y (ϕ T)i1···ip = Tk1 ···kp . ∂xi1 ··· ∂xip 44

6.1. Lie derivatives. We already discussed integral curves of a vector field on M. Let us recall that given v X(M) we can construct a parametrized curve on M whose velocity (tangent vector)∈ is equal to v. This amounts to solving the differential equation: dσi(x) t = vi(σ (x)), (6.2) dt t i i and σt(x) is the integral curve of the vector field v starting at x M, i.e. σ0(x) = x. It defines a natural diffeomorphism by following the point x through∈ the flow defined by the integral curve. At ‘time’ t, we arrive (in a smooth manner) at the point σt(x). The flow {σt}t∈R depends smoothly on t and defines a so-called one-parameter subgroup of diffeomorphisms of M with the defining property:

σt(σs(x)) = σs+t(x).

This guarantees that σt is invertible – with inverse given by σ−t – and thus a diffeomorphism.

v σt(x)

FIGURE 7. The ﬂow of the vector ﬁeld v

∗ Exercise 6.3. The diffeomorphism σt : M M induces a map σt : C (M) ∗ ∞ ∗ C (M) on functions (i.e. the pull-back) given by σt(f) = f σt (Note that σt is invertible∞ as well). Show that (6.2) implies that→ ◦ → d σ∗(f) = v(f) (6.3) dt t t=0 for any smooth function f on M. We are interested in how a vector field u changes along the integral curves of another vector field v. We learned from the discussion on connections that com- paring vector fields at different points on a manifold is a subtle business. However, with the notion of pushforward of the previous section, what we can compare is the pushforward of u at σt(x) by the diffeomorphism σ−t and u at the point x (see 45

Figure 8). Infinitesimally, this change defines a vector field called the Lie derivative of u along v defined by

d (σ−t)∗u − u Lvu := (σ−t)∗u = lim . dt t 0 t t=0 An equivalent deﬁnition is given→ as a vector at x M by ∈ d L u(x) := (σ− )∗ u(σ (x)) , v dt t t t=0 when (σ−t)∗ is understood as a map from Tσt (x) M to TxM. u

(σ−t)∗ (σ−t)∗u

( ) u σt x

FIGURE 8. The Lie derivative; the dashed vector is the difference ((σ−t)∗u − u)(x)

Exercise 6.4. Use Eq. (6.2) to derive the following expression for the components of the Lie derivative ∂ui ∂vi (L u)i = vj − uj. v ∂xj ∂xj In other words, the Lie derivative on vector ﬁelds is nothing else but the Lie bracket encountered before:

Lvu =[v, u], which also explains the terminology. Consequently, the Lie derivative of a vector field is again a vector field. We conclude from this that in the particular case that v is a coordinate vector field v = ∂k the Lie derivative is just the partial derivative: i i (L∂k u) = u ,k. 6.1.1. Lie derivative of 1-forms. We will now define the Lie derivatives of differential forms. Let us start with 1-forms. In contrast with vector fields, one naturally uses the pullback instead of the pushforward by σt to compare differential forms at different points on the integral curve of a vector field. We define the Lie derivative of a 1-form α along a vector field v d L α := σ∗α. (6.4) v dt t t=0

In tensor notation, we can write

j j (Lvα)i = v αi,j + v ,iαj

This follows easily by taking the derivative with respect to t, putting t = 0 after- k ∗ ∗ ∂σt wards, of (σtα)i := σtα(∂i) = αk(σt(x)) ∂xi where the index k corresponds to the k transformed coordinate system {σt (x)}. As a consequence of this identity, we can prove Cartan’s identity: y y Lvα = v dα + d(v α). which follows directly from the deﬁnitions of the interior product and the exterior derivative.

6.1.2. Lie derivative of covariant tensors. We extend the previous deﬁnition of Lie derivative to contravariant tensors – and in particular p-forms – as follows. For T : X(M)p C (M) we have in terms of the pullback: ∞ → d L T := σ∗T. (6.5) v dt t t=0

Exercise 6.5. Show that for 0-tensors (i.e. smooth functions) f on M, this becomes Lv(f) = v(f).

Of particular interest is the Lie derivative of the metric tensor g on a Riemannian or Lorentzian manifold: d L g(u, v) = g ((σ )∗u, (σ )∗v) (6.6) v dt t t t=0

In tensor notation, this becomes:

k k k (Lvg)ij = v gij,k + v ,igkj + v ,jgik (6.7)

Of course, we are not so happy with this expression since is involves partial derivatives and is therefore not explicitly covariant. However, it turns out that it is equivalent to the following expression:

k k k k k (Lvg)ij = v gij|k + v |igkj + v |jgik = v |igkj + v |jgik (6.8) where the covariant derivatives are with respect to the Levi-Civita connection. Again, observe that if v = ∂k only the partial derivative of the components of the metric appear in the Lie derivative of g.

Exercise 6.6. Check the equivalence of Eq. (6.7) and (6.8).

Remark. In general, one can derive the following formula for the Lie derivative of a (0,p)-tensor T,

j j j (LvT)i ···i = v Ti ···i |j + v Tji ···i + + v Ti ···i j. 1 p 1 p |i1 2 p ··· |ip 1 p−1 47

6.2. Killing vectors. In the previous subsection, we introduced the Lie derivative, which describes the action of a vector field on tensors such as another vector field, one forms or the metric. We now restrict to vector fields which are symmetries of the metric. Suppose that σt solves the differential equation (6.2) for a vector field ξ on a Riemannian or Lorentzian manifold (M, g). Then ξ is called a Killing ∗ vector field if σt is an isometry, i.e. it leaves the metric invariant σtg = g. This is completely equivalent to the condition that the Lie derivative of the metric g along ξ vanishes:

Lξg = 0. (6.9)

6.2.1. Killing’s equation. The last deﬁning equation for a Killing vector ﬁeld can be written in tensor notation using Eq. (6.8). The result is known as Killing’s equation and reads

ξj|i + ξi|j = 2ξ(i|j) = 0. (6.10) where the round bracket denotes the symmetrization of indices.

Example. Consider the 2-dimensional Euclidean plane, so that gij = δij, with i, j = 1,2. Killing’s equation reduces to the differential equations: 1 2 1 2 ∂1ξ = 0, ∂1ξ + ∂2ξ = 0, ∂2ξ = 0 with general solution the vector field 2 1 ξ = a∂1 + b∂2 + c(x ∂1 − x ∂2) Of course, the appearance of the partial derivatives can be understood from the fact that the components of the metric are constant. Thus, the Killing vector fields on the Euclidean plane are the generators of translations and rotation. Example. As a final example, we consider Minkowski space. Instead of an explicit computation using Eq. (6.10), let’s look at the defining equation Lξg = 0 of a Killing vector field. It says that ξ is an infinitesimal generator of a one-parameter family {σt}t∈R of diffeomorphisms respecting the Minkowski metric (i.e. isometries). In other words, the Killing vector fields on Minkowski space are precisely the generators of the Poincarégroup. This group is defined as the group of isometries of Minkowski space which consists of Lorentz transformations and translations. 6.2.2. First integrals of geodesics. Consider a geodesic with parameter s and tangent d vector v = ds . Contracting v with a Killing vector and taking the derivative gives, i d(ξiv ) i j = v ξ | v = 0. (6.11) ds i j i So, ξiv is constant along any geodesic. This can be understood as a special case of Nöther’s theorem: to a symmetry corresponds a conserved quantity, in this case i the function ξiv along the integral curve of v. We will see in a moment that this equation is extremely useful for computing geodesics. Example. Consider n-dimensional Euclidean space with Cartesian coordinates. The metric tensor is constant in these coordinates, therefore all the coordinate vector 48

ﬁelds are Killing vectors. For any geodesic, each of the n components of the tangent vectorx ˙ i are constant. Therefore, the geodesics in Euclidean space are the straight lines in Cartesian coordinates.

i 6.2.3. Killing’s second equation. For any Killing vector ξ , the ﬁrst derivative ξi|j is antisymmetric and can be thought of as a 2-form. Indeed, if we lower the index to 1 make ξ a 1-form, then this derivative is just − 2dξ. Since any second exterior derivative vanishes, 0 = d2ξ. This can be computed in terms of covariant derivatives. It is proportional to ξi|jk antisymmetrized. In other words,

0 = ξi|jk + ξj|ki + ξk|ij.

Now, using the deﬁnition of the Riemann tensor,

l ξi|jk = ξk|ji − ξk|ji = ξlR kji.

Rearranging the indices gives Killing’s second equation

i i l ξ |jk = R jklξ .

The second derivative of ξ at any point is determined from the values of ξ and the curvature at that point. A Killing vector on a connected manifold is completely specified by its value and first derivative at any point. Geometrically, an infinitesimal isometry is completely determined by the translation and rotation it gives at any point.

6.2.4. The d’Alembertian of a Killing vector. If we contract the indices j and k in the last equation, we see that the d’Alembertian of a Killing vector in given simply by contracting with (minus) the Ricci tensor:

i i|k ik l i j ξ ξ k = R klξ = −R ξ . (6.12) ≡ j

6.3. Geodesics in Schwarzschild geometry. We will now demonstrate how Killing vectors – and in particular the first integrals of geodesics – can be used to find the geodesics in a Schwarzschild spacetime. This allows us to derive the three classical predictions of general relativity. Although we will derive the Schwarzschild spacetime as the unique static and spherically symmetric solution to Einstein’s equation in a later lecture, let us define for the moment the Schwarzschild metric by

2m 2m −1 ds2 = 1 − dt2 − 1 − dr2 − r2(dθ2 + sin θdφ). r r

If one wishes to restore the constants G and c (which we have set to 1) to connect to experiment, one replaces m by Gm/c2 in this deﬁnition. As we will see later, this metric is rich in isometries, the group of isometries is R O(3). The inﬁnitesimal versions of these symmetries will be heavily used in computi× ng geodesics to arrive at the three classical tests. 49

6.3.1. Gravitational redshift. Consider first the timelike Killing vector field ξ = ∂/∂t. Suppose that O1 and O2 are two static observers, that is, their velocities u1 and u2 (or tangent vectors to their worldlines) point in the direction of the Killing vector field ξ. Suppose that O1 sends a light signal at P1 to O2 who receives it at P2; it travels on a null geodesic with tangent vector, say, k. Then the frequency of emis- µ µ sion is given by ω1 = kµu1 whereas the frequency measured by O2 is ω2 = kµu2. Now, since u1 and u2 are unit vectors pointing in the direction of ξ, we can write ξµ ξµ uµ = , uµ = . 1 κ 2 κ √ξ ξκ √ξ ξκ P1 P2

Substituting this into the formula’s for ω1 and ω2 yields

2m κ 1 − ω √ξ ξ | r2 1 = κ P2 = q . λ ω2 √ξ ξλ|P 2m 1 1 − r q 1 µ Here, we used Eq. (6.11) which states in this case that kµξ is conserved along the null geodesic swept out by the light signal (so that its values at P1 and P2 coincide) and we have inserted the explicit expression for gtt in the Schwarzschild metric.

u P2 1 k

O 2 O1

If the emitter is closer to the center of graviational attraction, i.e. r2 > r1, we have ω2 <ω1, the frequency of light will be decreased (“redshifted”). In the limit case where m is much less that r1 and r2, we ﬁnd that ω − ω Gm Gm 1 2 − + 2 2 , ω1 ≈ c r1 c r2 after restoring the G’s and c’s in this equation. This gravitational redshift has been measured and found to be in agreement with this prediction to within the 1% ex- perimental uncertainty. 50

6.3.2. Starlight deflection. For the bending of light by the gravitational field of the sun, say, we describe the null geodesics in the Schwarzschild metric, making heavy use of the above integrals of motion arising from the Killing vectors. First, we claim that a geodesic C starting in the equatorial plane Σ := {θ = π/2} with initial tangent vector C˙ (0) in TC(0) Σ will remain in Σ. For this, we use the symmetry of the Schwarzschild metric under the diffeomorphism given by ϕ : (t, r, θ, φ) (t, r, θ−π, φ) to establish that geodesics starting in and tangent to the fixed point7 set of ϕ, which in this case is exactly the equatorial plane Σ. Suppose on the contrary→ that C(s) / Σ for some s. Then ϕ(C(s)) = C(s) but the geodesic ϕ C satisfies the same differential∈ equation as C does with6 the same initial con- ◦ ditions since ϕ(C(0)) = C(0) and ϕ(C˙ (0)) = C˙ (0). This contradicts the uniqueness property of geodesics. Since we can always rotate our coordinate system so that C(0) Σ and C˙ (0) TC(0) Σ, we can assume in what follows that θ = π/2 and ∈ ∈ θ˙ = 0.

Corresponding to the Killing vectors ξ = ∂/∂t and ζ = ∂/∂φ there are two conserved quantities: 2m E := ξ kµ = 1 − t,˙ (6.13) µ r µ 2 L := ζµk = r φ.˙ (6.14) We call these quantities energy and angular momentum, respectively. For null geodesics (t(s), r(s),π/2,φ(s)) with tangent k = (t,˙ r,˙ φ˙ ), we have that g(k,˙ k˙ ) = 0, or, explicitly

2m 2m −1 1 − t˙ 2 − 1 − r˙ 2 − r2φ˙ 2 = 0. r r In terms of the energy and angular momentum this reads 1 L2 mL2 1 r˙ 2 + − = E2. (6.15) 2 2r2 r3 2 Thus, we reduced our geodesic equation to the equation of motion of a particle of 1 2 mass 1 and energy 2E on a one-dimensional line in the effective potential V(r) = 1 L2 mL2 2 r2 − r3 . Exercise 6.7. Compute the point of equilibrium of this potential and show that it is unstable. The orbit described by the equilibrium point is called a trapped ray. The potential has a maximum L2/54m2 at this point. Hence, an orbit initally at radius smaller 1 2 2 2 than the equilibrium can escape only if 2E > L /54m . Conversely, an orbit at 1 2 2 2 radius greater than the equilibrium will not get trapped if 2E

From it, we derive ds 1 = . dr L2 2m 1 − r2 1 − r q and by defintion of L we also have dφ/ds = L/r2. Combining these two, we obtain dφ L = . dr 2 L2 2m r 1 − r2 1 − r q We will integrate this equation to obtain the deflection of light by the Schwarzschild geometry. Suppose that the orbit comes in from infinity, that is, r for s − . From the last equation, we conclude that φ must be constant for r , say, 2 2 →∞ → ∞ φ φ0 as s − . If L > 54m , then the orbit will not get trapped and also → ∞ φ φ1 will become constant as s , in which case r . The deflection angle → → ∞ is given by δ = φ1 − φ0 − π. Note that in flat spacetime (i.e. m = 0) we have → →∞ →∞ φ1 − φ0 = π, which explains the π in this formula. The minimum r0 of r – at the perihelion of the orbit – satisfies the equation L2 mL2 − 3 = 1. (6.16) r0 r0 We will use this last equation to evaluate the integral L φ = 2 ∞ dr, Z 2 r0 2 L 2m r 1 − r2 1 − r q at least, up to first order in m. The factor 2 appears because of the symmetry of the orbit with respect to the perihelion at r = r0. 2 Exercise 6.8. (1) Insert the expression for L in terms of r0 obtained from Eq. −1 (6.16) into the above integral, and make the substitutions u = mr0 and v = r0/r to arrive at 1 1 φ = 2 dv. 2 3 Z0 √1 − 2u − v + 2uv (2) Show that the first order contribution in m to φ is given by 1 3 −1 1 − v 2r0 2 3/2 dv. Z0 (1 − v ) dφ Hint: compute dm and evaluate at m = 0. (3) Evaluate this integral to obtain the prediction for the deflection of light: 4m δ . ≈ r0 When we insert the constants G and c this becomes 4Gm δ 2 ≈ c r0 which, for a light ray grazing the surface of the sun is 1,74 seconds of arc. This has been measured to about 1% accuracy. 52

6.3.3. Perihelion precession. We now consider timelike geodesics describing the tra- jectories of physical test particles. We still have two conserved quantities along the geodesic, namely energy and angular momentum as defined in Eq. (6.13). Recall that timelike geodesics (t(τ), r(τ),π/2,φ(τ)) parametrized by proper time, have unit tangent vector k = (t,˙ r,˙ φ˙ ), so that in terms of energy and angular momentum we have 1 1 m L2 mL2 1 r˙ 2 + − + − = E2. (6.17) 2 2 r 2r2 r3 2 This equation describes the (1-dim) motion of a particle of unit mass and energy 1 2 1 m L2 mL2 2E in the potential V(r) = 2 − r + 2r2 − r3 . Note that the last term is not present in the Newtonian description. The circular orbits in this potential are described by the critical points of V. We compute its first two derivatives m L2 3mL2 2m 3L2 12mL2 V ′(r) = − + , V ′′(r) = − + − , r2 r3 r4 r3 r4 r5 and derive that the circular orbits are at L2 √L4 − 12m2L2 rc = ± . 2m We distinguish three case: L2 < 12m2: there are no circular orbits; just as in the previous section, de- • 1 2 pending on the energy 2E the orbit is either a crash or an escape orbit. 2 2 L = 12m : there is one circular orbit at rc = 6m which is unstable. • L2 > 12m2: there are two circular orbits; in the large L limit, we find that • their radii are given by 3Gm and L2/m.

Let us consider the last case more closely, and write r1 and r2 for the two so- lutions, with r1 < 6m< r2. One can check that r2 corresponds to a stable orbit. 1 2 Orbits with r initially close to r2 and energy 2E slightly larger than V(r2) will have r oscillating between rmin and rmax. If we assume that r2 is large – by taking for in- stance the large L limit – the relativistic correction mL2/r3 to the potential is small and we can consider the orbit as nearly elliptical. Thus, let us consider perturbations around an elliptic orbit with perihelion r = rmin at τ = 0 and φ = 0. The perihelion precession is ψ = φ0 − 2π where φ0 is the angle at which perihelion is reached again at some proper time τ0. As before, we can solve from Eq. (6.17) dτ 1 = dr E2 − 2V(r) p which with dφ/dτ = L/r2 can be combined to give dφ/dr. We can integrate this to obtain rmax L ψ = 2 dr − 2π Z 2 2 rmin r E − 2V(r) p We will not carry out the computation of this integral but sketch how to obtain its ﬁrst order (in m) contribution. 53

2 Observe that E − 2V(r) vanishes for rmin and rmax. This allows to express • 2 2 E and L in terms of r, rmin, rmax. In fact, 1 1 1 1 1 E2 − 2V(r) = f − − r rmin r r rmax and one only has to solve for the function f. One can further express E2 and L2 in terms of the eccentricity e and semimajor axis a of the orbit by writing

rmin = a(1 − e), rmax = a(1 + e). One can express the integral as • v2 1 ψ = 2 dv − 2π Zv1 u(v0 − v)(v − v1)(v2 − v)

by making thep substitutions u = 2m/a, v1 = 1/(1 + e), v2 = 1/(1 − e) and v0 + v1 + v2 = 1/u. dφ Finally, one takes the derivative dm , putting m = 0 afterwards; this leads to • the integral v2 v + v + v 3π 2 1 2 dv = Z 2 v1 (v − v1)(v2 − v) 1 − e This leads to ﬁrstp order in m to a perihelion precession rate of 6πGm ψ ≈ c2a(1 − e2) per revolution. For the orbit of Mercury around the sun, this leads to a perihelion precession of 5 10−7 per revolution. With about 415 revolutions per centry, this gives a perihelion× rate of 43 seconds of arc per century for the orbit of Mercury. This is in excellent agreement with the observed value.