Killing Fields

Let κ^a be a smooth field on our background spacetime (M, g_ab). κ^a is said to be a Killing field if its associated local flow maps Γ_s are all isometries or, equivalently, if £_κ g_ab = 0. The latter condition can also be expressed as ∇(_aκ_b) = 0.

Any number of standard symmetry conditions—local versions of them, at least can be cast as claims about the existence of Killing fields. Local, because killing fields need not be complete, and their associated flow maps need not be defined globally.

(M, g_ab) is stationary if it has a Killing field that is everywhere timelike.

(M, g_ab) is static if it has a Killing field that is everywhere timelike and locally hypersurface orthogonal.

(M, g_ab) is homogeneous if its Killing fields, at every point of M, span the tangent space.

In a stationary spacetime there is, at least locally, a “timelike flow” that preserves all spacetime distances. But the flow can exhibit rotation. Think of a whirlpool. It is the latter possibility that is ruled out when one passes to a static spacetime. For example, Gödel spacetime, is stationary but not static.

Let κ^a be a Killing field in an arbitrary spacetime (M, g_ab) (not necessarily Minkowski spacetime), and let γ : I → M be a smooth, future-directed, timelike curve, with unit tangent field ξ^a. We take its image to represent the worldline of a point particle with mass m > 0. Consider the quantity J = (P^aκ_a), where P^a = mξ^a is the four-momentum of the particle. It certainly need not be constant on γ[I]. But it will be if γ is a geodesic. For in that case, ξⁿ∇_nξ^a = 0 and hence

ξⁿ∇_nJ = m(κ_a ξⁿ∇_nξ^a + ξⁿξ^a ∇_nκ_a) = mξⁿξ^a ∇(_nκ_a) = 0

Thus, J is constant along the worldlines of free particles of positive mass. We refer to J as the conserved quantity associated with κ^a. If κ^a is timelike, we call J the energy of the particle (associated with κ^a). If it is spacelike, and if its associated flow maps resemble translations, we call J the linear momentum of the particle (associated with κ^a). Finally, if κ^a is spacelike, and if its associated flow maps resemble rotations, then we call J the angular momentum of the particle (associated with κ^a).

It is useful to keep in mind a certain picture that helps one “see” why the angular momentum of free particles (to take that example) is conserved. It involves an analogue of angular momentum in Euclidean plane geometry. Figure below shows a rotational Killing field κ^a in the Euclidean plane, the image of a geodesic (i.e., a line) L, and the tangent field ξ^a to the geodesic. Consider the quantity J = ξ^aκ_a, i.e., the inner product of ξ^a with κ^a – along L, and we can better visualize the assertion.

Figure: κ^a is a rotational Killing field. (It is everywhere orthogonal to a circle radius, and is proportional to it in length.) ξ^a is a tangent vector field of constant length on the line L. The inner product between them is constant. (Equivalently, the length of the projection of κ^a onto the line is constant.)

Let us temporarily drop indices and write κ·ξ as one would in ordinary Euclidean vector calculus (rather than ξ^aκ_a). Let p be the point on L that is closest to the center point where κ vanishes. At that point, κ is parallel to ξ. As one moves away from p along L, in either direction, the length ∥κ∥ of κ grows, but the angle ∠(κ,ξ) between the vectors increases as well. It should seem at least plausible from the picture that the length of the projection of κ onto the line is constant and, hence, that the inner product κ·ξ = cos(∠(κ , ξ )) ∥κ ∥ ∥ξ ∥ is constant.

That is how to think about the conservation of angular momentum for free particles in relativity theory. It does not matter that in the latter context we are dealing with a Lorentzian metric and allowing for curvature. The claim is still that a certain inner product of vector fields remains constant along a geodesic, and we can still think of that constancy as arising from a compensatory balance of two factors.

Let us now turn to the second type of conserved quantity, the one that is an attribute of extended bodies. Let κ^a be an arbitrary Killing field, and let T_ab be the energy-momentum field associated with some matter field. Assume it satisfies the conservation condition (∇_aT^ab = 0). Then (T^abκ_b) is divergence free:

∇_a(T^abκ_b) = κ_b∇_aT^ab + T^ab∇_aκ_b = T^ab∇(_aκ_b) = 0

(The second equality follows from the conservation condition and the symmetry of T^ab; the third follows from the fact that κ^a is a Killing field.) It is natural, then, to apply Stokes’s theorem to the vector field (T^abκ_b). Consider a bounded system with aggregate energy-momentum field T_ab in an otherwise empty universe. Then there exists a (possibly huge) timelike world tube such that T_ab vanishes outside the tube (and vanishes on its boundary).

Let S₁ and S₂ be (non-intersecting) spacelike hypersurfaces that cut the tube as in the figure below, and let N be the segment of the tube falling between them (with boundaries included).

Figure: The integrated energy (relative to a background timelike Killing field) over the intersection of the world tube with a spacelike hypersurface is independent of the choice of hypersurface.

By Stokes’s theorem,

∫_S2(T^abκ_b)dS_a – ∫_S1(T^abκ_b)dS_a = ∫_S2∩∂N(T^abκ_b)dS_a – ∫_S1∩∂N(T^abκ_b)dS_a

= ∫_∂N(T^abκ_b)dS_a = ∫_N∇_a(T^abκ_b)dV = 0

Thus, the integral ∫_S(T^abκ_b)dS_a is independent of the choice of spacelike hypersurface S intersecting the world tube, and is, in this sense, a conserved quantity (construed as an attribute of the system confined to the tube). An “early” intersection yields the same value as a “late” one. Again, the character of the background Killing field κ^a determines our description of the conserved quantity in question. If κ^a is timelike, we take ∫_S(T^abκ_b)dS_a to be the aggregate energy of the system (associated with κ^a). And so forth.

Dynamics of Point Particles: Orthogonality and Proportionality

Let γ be a smooth, future-directed, timelike curve with unit tangent field ξ^a in our background spacetime (M, g_ab). We suppose that some massive point particle O has (the image of) this curve as its worldline. Further, let p be a point on the image of γ and let λ^a be a vector at p. Then there is a natural decomposition of λ^a into components proportional to, and orthogonal to, ξ^a:

λ^a = (λ^bξ_b)ξ^a + (λ^a −(λ^bξ_b)ξ^a) —– (1)

Here, the first part of the sum is proportional to ξ^a, whereas the second one is orthogonal to ξ^a.

These are standardly interpreted, respectively, as the “temporal” and “spatial” components of λ^a relative to ξ^a (or relative to O). In particular, the three-dimensional vector space of vectors at p orthogonal to ξ^a is interpreted as the “infinitesimal” simultaneity slice of O at p. If we introduce the tangent and orthogonal projection operators

k_ab = ξ_a ξ_b —– (2)

h_ab = g_ab − ξ_a ξ_b —– (3)

then the decomposition can be expressed in the form

λ^a = k^a_b λ^b + h^a_b λ^b —– (4)

We can think of k_ab and h_ab as the relative temporal and spatial metrics determined by ξ^a. They are symmetric and satisfy

k^a_bk^b_c = k^a_c —– (5)

h^a_bh^b_c = h^a_c —– (6)

Many standard textbook assertions concerning the kinematics and dynamics of point particles can be recovered using these decomposition formulas. For example, suppose that the worldline of a second particle O′ also passes through p and that its four-velocity at p is ξ′^a. (Since ξ^a and ξ′^a are both future-directed, they are co-oriented; i.e., ξ^a ξ′^a > 0.) We compute the speed of O′ as determined by O. To do so, we take the spatial magnitude of ξ′^a relative to O and divide by its temporal magnitude relative to O:

v = speed of O′ relative to O = ∥h^a_b ξ′^b∥ / ∥k^a_b ξ′^b∥ —– (7)

For any vector μ^a, ∥μ^a∥ is (μ^aμ_a)^1/2 if μ is causal, and it is (−μ^aμ_a)^1/2 otherwise.

We have from equations 2, 3, 5 and 6

∥k^a_b ξ′^b∥ = (k^a_b ξ′^b k_ac ξ′^c)^1/2 = (k_bc ξ′^bξ′^c)^1/2 = (ξ′^bξ_b)

and

∥h^a_b ξ′^b∥ = (−h^a_b ξ′^b h_ac ξ′^c)^1/2 = (−h_bc ξ′^bξ′^c)^1/2 = ((ξ′^bξ_b)² − 1)^1/2

so

v = ((ξ’^bξ_b)² − 1)^1/2 / (ξ′^bξ_b) < 1 —– (8)

Thus, as measured by O, no massive particle can ever attain the maximal speed 1. We note that equation (8) implies that

(ξ′^bξ_b) = 1/√(1 – v²) —– (9)

It is a basic fact of relativistic life that there is associated with every point particle, at every event on its worldline, a four-momentum (or energy-momentum) vector P^a that is tangent to its worldline there. The length ∥P^a∥ of this vector is what we would otherwise call the mass (or inertial mass or rest mass) of the particle. So, in particular, if P^a is timelike, we can write it in the form P^a =mξ^a, where m = ∥P^a∥ > 0 and ξ^a is the four-velocity of the particle. No such decomposition is possible when P^a is null and m = ∥P^a∥ = 0.

Suppose a particle O with positive mass has four-velocity ξ^a at a point, and another particle O′ has four-momentum P^a there. The latter can either be a particle with positive mass or mass 0. We can recover the usual expressions for the energy and three-momentum of the second particle relative to O if we decompose P^a in terms of ξ^a. By equations (4) and (2), we have

P^a = (P^bξ_b) ξ^a + h^a_bP^b —– (10)

the first part of the sum is the energy component, while the second is the three-momentum. The energy relative to O is the coefficient in the first term: E = P^bξ_b. If O′ has positive mass and P^a = mξ′^a, this yields, by equation (9),

E = m (ξ′^bξ_b) = m/√(1 − v²) —– (11)

(If we had not chosen units in which c = 1, the numerator in the final expression would have been mc² and the denominator √(1 − (v2/c2)). The three−momentum relative to O is the second term h^a_bP^b in the decomposition of P^a, i.e., the component of P^a orthogonal to ξ^a. It follows from equations (8) and (9) that it has magnitude

p = ∥h^a_b mξ′^b∥ = m((ξ′^bξ_b)² − 1)^1/2 = mv/√(1 − v²) —– (12)

Interpretive principle asserts that the worldlines of free particles with positive mass are the images of timelike geodesics. It can be thought of as a relativistic version of Newton’s first law of motion. Now we consider acceleration and a relativistic version of the second law. Once again, let γ : I → M be a smooth, future-directed, timelike curve with unit tangent field ξ^a. Just as we understand ξ^a to be the four-velocity field of a massive point particle (that has the image of γ as its worldline), so we understand ξⁿ∇_nξ^a – the directional derivative of ξ^a in the direction ξ^a – to be its four-acceleration field (or just acceleration) field). The four-acceleration vector at any point is orthogonal to ξ^a. (This is, since ξ^a(ξⁿ∇_nξ_a) = 1/2 ξⁿ∇_n(ξ^aξ_a) = 1/2 ξⁿ∇_n (1) = 0). The magnitude ∥ξⁿ∇_nξ^a∥ of the four-acceleration vector at a point is just what we would otherwise describe as the curvature of γ there. It is a measure of the rate at which γ “changes direction.” (And γ is a geodesic precisely if its curvature vanishes everywhere).

The notion of spacetime acceleration requires attention. Consider an example. Suppose you decide to end it all and jump off the tower. What would your acceleration history be like during your final moments? One is accustomed in such cases to think in terms of acceleration relative to the earth. So one would say that you undergo acceleration between the time of your jump and your calamitous arrival. But on the present account, that description has things backwards. Between jump and arrival, you are not accelerating. You are in a state of free fall and moving (approximately) along a spacetime geodesic. But before the jump, and after the arrival, you are accelerating. The floor of the observation deck, and then later the sidewalk, push you away from a geodesic path. The all-important idea here is that we are incorporating the “gravitational field” into the geometric structure of spacetime, and particles traverse geodesics iff they are acted on by no forces “except gravity.”

The acceleration of our massive point particle – i.e., its deviation from a geodesic trajectory – is determined by the forces acting on it (other than “gravity”). If it has mass m, and if the vector field F^a on I represents the vector sum of the various (non-gravitational) forces acting on it, then the particle’s four-acceleration ξⁿ∇_nξ^a satisfies

F^a = mξⁿ∇_nξ^a —– (13)

This is Newton’s second law of motion. Consider an example. Electromagnetic fields are represented by smooth, anti-symmetric fields F_ab. If a particle with mass m > 0, charge q, and four-velocity field ξ^a is present, the force exerted by the field on the particle at a point is given by qF^a_bξ^b. If we use this expression for the left side of equation (13), we arrive at the Lorentz law of motion for charged particles in the presence of an electromagnetic field:

qF^a_bξ^b = mξ^b∇_bξ^a —– (14)

This equation makes geometric sense. The acceleration field on the right is orthogonal to ξ^a. But so is the force field on the left, since ξ_a(F^a_bξ^b) = ξ^aξ^bF_ab = ξ^aξ^bF_(ab), and F_(ab) = 0 by the anti-symmetry of F_ab.