Wednesday, May 20, 2015

Killing Vectors and Hawking Radiation


We start out with the interval (see equations (6) to (23) in Relativistic Doppler Effect ),

(1) ds2 = -dt2 + dx2 + dy2 + dz2

We define the metric as the coefficient of each of the terms in the above:

(2) η00 = -1, η11 = 1,η22 = 1,η33 = 1,and ηij = 0 for i≠j

We can rewrite equation (1) in the general form,

(3) ds2 = ηαβdxα dxβ

The proper time τ is,

(4) dτ2 = - ds2

This yields,

(5) dτ = dt/γ

(6) where γ = (1 - v2)

We measure the velocity with respect to the proper time τ, not the ordinary time t.

(7) uβ =dxβ/dτ

This gives the important result,

(8) u2 = uu = -1

We define a 4-vector momentum as,

(9) pβ =(p0,pi) = (p0,p)

This gives the following:

(10)p2 = muβmuβ = m2u2 = - m2


(11) E2 = m2 + (p)2.

Putting c into the equation,

(11) E2 = m2c4 + p2c2.

Euler-Lagrange Equations for a free particle in motion

Consider two timelike separated points A and B, and all the timelike worldlines. In fig 1, two such lines are illustrated - a straight line path and a nearby path.

By the variational principle, the world line of a free particle between two timelike separated points extremizes the proper time between them. To see this, each curve will have a value in terms of the proper time,

(12) τAB = ∫AB

Using equations (1) and (4),

(13) τAB = ∫AB {dt2 - dx2 - dy2 - dz2}½

We parametrize this equation by choosing σ such that at point A, σ = 0, and at B, σ =1

(14) τAB = ∫01 dσ {(dt/dσ)2 - (dx/dσ)2 - (dy/dσ)2 - (dz/dσ)2}½

This has the same form as the action of equation (1) in The Essential Quantum Field Theory , repeated below

(15) S = ∫ dt L

By making the correspondence:
the action S → τAB,
the time t → σ
and the Lagrangian L → {(dt/dσ)2 - (dx/dσ)2 - (dy/dσ)2 - (dz/dσ)2}½

We can rewrite the Lagrangian L in terms of the general form (equations 3 and 4),

(16) L = { - ηαβ(dxα/dσ) (dxβ/dσ) }½

Also, another form of the Lagrangian is,

(17) L = dτ/dσ

The corresponding Euler-Lagrange equation ( see paragraph below equation 1 in The Essential Quantum Field Theory )


Consider a particle freely moving along the x-axis ( x1 = x, x2 = y =0, x3 = z = 0)

(19)Equation (18) becomes (see appendix A),

(20) Using equation (17), substitute for L in the above, we get,

(21) Now multiply both sides by dσ/dτ, we get,

In case you haven't recognized, this is the equation of a straight line. Integrate once,

(22) dx/dτ = c

Integrate a second time,

(23) x = cτ + d

Hence for the extremal proper time, the world line for a particle freely moving from point A to point B is a straight line path (fig 1).

Killing Vectors

Generally speaking, conservation laws are connected to symmetries. For instance, if there is a symmetry under displacement in time, energy is conserved; under displacement in space, momentum is conserved; under rotations, angular momentum is conserved. However, in GR, the metric is often time dependent, angle dependent, position dependent, etc. So how does one tell if there is a symmetry? One clue is if the metric is independent of one of its coordinates. For instance, say the metric is independent of x1. That means, it transforms as,

(24) x1 → x1 + const.

leaving the metric unchanged

The vector ξ with components,

(25) ξα = (0,1,0,0)

lies along the direction the metric doesn't change. This is a Killing vector (in honor of Wilhelm Killing, German mathematician 1847-1923). A Killing vector is a general way of characterizing a symmetry in any coordinate system. For a freely moving particle, one can show,

(26) ξu = constant, (see appendix B)

(27) Also,ξp = constant, where p is the particle momentum.

Schwarzschild Geometry

In GR, the Minkowsky metric ηαβ is replaced by a more general metric gαβ so that equation (3) now reads as,

(28) ds2 = gαβdxα dxβ

Specifically in a Schwarzschild geometry, the metric reads as, (G=c=1)

(29) g00 = -(1 - 2M/r), g11 = (1 - 2M/r)-1, g22 = r2,g33 = r2sin2θ,and gij = 0 for i≠j

For our purposes, we note that the metric is time-independent, and therefore there is a Killing vector, which has the components,

(30)ξα = (1,0,0,0)

Hawking Radiation

Fig 2 shows a rest-mass zero particle-antiparticle pair which has been created by vacuum fluctuations in such a way that the two particles were created on opposite sides of the horizon of a black hole. The components ξp and ξp' must be equal and opposite so that ξ(p +p') = 0, (value of the vacuum). The particle ( ξp > 0) can propagate and can be seen as radiation by an observer at infinity. This also means that the antiparticle ( ξp' < 0) will be absorbed by the black hole, thus decreasing its mass in the process. This is the basis of Hawking's claim that black holes radiate, and in time, will evaporate.

Appendix A

(A3) Equation (18) now reads as,

First calculate,

Putting it altogether, equation (A3) becomes,


Appendix B

let α =1, Equation (A3) becomes,


(B2) from (A2),

(B3) LHS of (B1) ,

(B4) therefore,

(B5) Now consider,

Using equations (A1) and (17)

Note that we can write,

(B6) η = ηαβ ξα

Substituting in the above,


(B8) From (B4), we get ,

ξu = constant

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