**Preliminary**

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

(1) ds

^{2}= -dt

^{2}+ dx

^{2}+ dy

^{2}+ dz

^{2}

We define the

*metric*as the coefficient of each of the terms in the above:

(2) η

_{00}= -1, η

_{11}= 1,η

_{22}= 1,η

_{33}= 1,and η

_{ij}= 0 for i≠j

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

(3) ds

^{2}= η

_{αβ}dx

^{α}dx

^{β}

The

*proper time*τ is,

(4) dτ

^{2}= - ds

^{2}

This yields,

(5) dτ = dt/γ

(6) where γ = (1 - v

^{2})

^{-½}

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

(7) u

^{β}=dx

^{β}/dτ

This gives the important result,

(8) u

^{2}=

**u**•

**u**= -1

We define a 4-vector momentum as,

(9) p

^{β}=(p

^{0},p

^{i}) = (p

^{0},

**p**)

This gives the following:

(10)p

^{2}= mu

^{β}mu

^{β}= m

^{2}u

^{2}= - m

^{2}

And,

(11) E

^{2}= m

^{2}+ (

**p**)

^{2}.

Putting c into the equation,

(11) E

^{2}= m

^{2}c

^{4}+

**p**

^{2}c

^{2}.

**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) τ

_{AB}= ∫

_{A}

^{B}dτ

Using equations (1) and (4),

(13) τ

_{AB}= ∫

_{A}

^{B}{dt

^{2}- dx

^{2}- dy

^{2}- dz

^{2}}

^{½}

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

(14) τ

_{AB}= ∫

_{0}

^{1}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 → τ

_{AB},

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 = { - η

_{αβ}(dx

^{α}/dσ) (dx

^{β}/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 )

(18)

Consider a particle freely moving along the x-axis ( x

^{1}= x, x

^{2}= y =0, x

^{3}= 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 x

^{1}. That means, it transforms as,

(24) x

^{1}→ x

^{1}+ const.

leaving the metric unchanged

The vector

**ξ**with components,

(25) ξ

^{α}= (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)

**ξ**•

**u**= constant, (see appendix B)

(27) Also,

**ξ**•

**p**= constant, where

**p**is the particle momentum.

**Schwarzschild Geometry**

In GR, the Minkowsky metric η

_{αβ}is replaced by a more general metric g

_{αβ}so that equation (3) now reads as,

(28) ds

^{2}= g

_{αβ}dx

^{α}dx

^{β}

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

(29) g

_{00}= -(1 - 2M/r), g

_{11}= (1 - 2M/r)

^{-1}, g

_{22}= r

^{2},g

_{33}= r

^{2}sin

^{2}θ,and g

_{ij}= 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)ξ

^{α}= (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

**ξ**•

**p**and

**ξ**•

**p'**must be equal and opposite so that

**ξ**•

**(p +p')**= 0, (value of the vacuum). The particle (

**ξ**•

**p**> 0) can propagate and can be seen as radiation by an observer at infinity. This also means that the antiparticle (

**ξ**•

**p'**< 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,

(A4)

**Appendix B**

let α =1, Equation (A3) becomes,

(B1)

(B2) from (A2),

(B3) LHS of (B1) ,

(B4) therefore,

(B5) Now consider,

Using equations (A1) and (17)

Note that we can write,

(B6) η

_{1β}= η

_{αβ}ξ

^{α}

Substituting in the above,

(B7)

(B8) From (B4), we get ,

**ξ**•

**u**= constant