Gravity and the flow of time

c²dt²+ dl²

(2gl/c²)+ dy²+ dz². (3.73) This form of the metric will play a crucial role in our discussion of black holes inChapter 8. [Answer. Use the coordinate transformation 1 + (g ¯x/c²) = (2gl/c²)^1/2.]

Exercise 3.6

Alternative derivation of the Rindler metric There is an alternative way of obtaining the metric inEq. (3.70)which is more direct. Consider an accelerated observer with the tra-jectory h(τ ), f (τ ) and a coordinate velocity u(τ ) ≡ df/dh. At any given instant, there exists a Lorentz frame (t, x) with: (a) the three coordinates axes coinciding with the axes of the accelerating observer and (b) the origin coinciding with the location of the observer.

Show that the Lorentz transformations (with suitable translation of origin) from the global inertial frame coordinates (T, X) to this instantaneously comoving frame is given by (with c = 1)

X = f (τ ) + γ(u) (x + ut) ; T = h(τ ) + γ(u)(t + ux). (3.74) We now define the coordinates for the accelerated observer such that at t = 0 the coordinate labels in the accelerated frame coincide with those in the comoving Lorentz frame. This gives

X = f (τ ) + γ(u)x; T = h(τ ) + γ(u)ux. (3.75) Show that, if we take x and τ as the coordinates in the accelerated frame, then the resulting metric has the form inEq. (3.70).

3.5.2 Gravity and the flow of time

Considering the importance of this result, we shall provide a more physical argu-ment leading to the same conclusion. We know that the annihilation of electrons and positrons can lead to γ-rays and that under suitable conditions one can produce e⁺e⁻ pairs from the radiation. This fact can be used to devise a suitable thought experiment which will prove that gravitational field must necessarily affect the rate of flow of clocks. We will first prove that conservation of energy demands a red-shift of the frequency of photons propagating in a static gravitational field. Using this result and the relation between frequency and clock time, we will demonstrate that the rate of flow of clocks must be affected by the gravitational field.

For the purpose of the argument given below, we may assume that the grav-itational potential near the surface of Earth varies linearly with height; then the potential difference between two points A and B, separated by height L, will be

3.5 The principle of equivalence and the geometrical description of gravity 129

g

x Q

P

X

A B

t

Y tB

t_A

L

Fig. 3.1. The vertical lines at A and B are the world lines of two spatial locations near Earth separated by height L with A being closer to the Earth’s surface. (In this spacetime diagram, the height from Earth’s surface is measured horizontally.) Some simple thought experiments using this arrangement show that gravitational field must affect clock rates.

gL, where g is the acceleration due to gravity. (Figure 3.1shows the situation in a spacetime diagram. The two vertical lines at the points A and B denote the world lines of two spatial locations separated by a height L; the gravitational field is along the negative x-axis in the diagram.) Let us assume that several pairs of sufficiently high energy photons, each of energy ω, were converted into e⁺e⁻ pairs at the point A – which is closer to the surface of the Earth than B. These particles move from point A to point B losing the gravitational potential energy mgL per particle.

At B we annihilate the particles and produce photons which are sent back to A. The sequence of events described above has restored the original configuration. (To be rigorous, one also has to arrange matters such that the momentum balance of the particles is also taken care of. This can be easily done with a suitable mechanism and hence we will not bother about this.⁴) Since the amount of energy available in the form of particles was lower at B than at A (because of the change in the grav-itational potential), it follows that the energy of the photons must change in going from B to A. If this is not the case, we could use the above sequence of events to increase the energy of the system repeatedly. Since the energy of a photon is

proportional to its frequency, the frequencies of photons at A and B must be related (for gL/c²  1) by

ω_A∼= ωB

 1 +gL

c²



. (3.76)

Because the frequency shift derived above is independent of the Planck constant

 (even though the original argument involving pair creation and annihilation was quantum mechanical), it follows that the result should be true even in the clas-sical limit. But in the clasclas-sical limit, one can consider photons as making up an electromagnetic wave with some frequency. This frequency can be determined by counting the number of crests N of a wave train which crosses an observer in a time interval Δt. Let us suppose that such a wave train, made of N crests (and troughs), was emitted from point A towards point B. The head of the wave train travels along some trajectory XY inFig. 3.1. In the absence of a gravitational field, the radia-tion will travel along null lines (at 45^◦ to the axes) but we do not want to make any assumption regarding how gravity affects the trajectory of the rays. Therefore, the curve XY at this stage is just some monotonic continuous curve connecting the points X and Y . Let the tail of the wave train leave the point A after a time interval ΔtAas measured by a local clock located at A. This will travel along the curve P Q. Since the gravitational field is static, nothing changes with time and the curves XY and P Q are locally parallel to each other. It follows from elementary geometry that the time interval between Y and Q measured by a local clock at B is Δt_B = Δt_A. The number of crests in a wave cannot change during the propa-gation either. Since the observers at A and B will attribute to the wave frequencies ω_A = N/Δt_Aand ω_B = N/Δt_B the equality Δt_A = Δt_B leads to ω_A = ω_B which contradicts our previous result inEq. (3.76).

This paradox arises because we had tacitly assumed that the clocks at A and B will run at the same rate as indicated by the coordinate interval Δt at the two loca-tions. But if the line interval in the presence of gravity gets modified to the form in Eq. (3.72), then the clocks at rest (|dx|²= 0) in the gravitational field will run at a rate determined by dτ =

−g00(x)dt which will differ from location to location.

Thus even though the coordinate time differences between the arrival of the head and tail of the wave at A and B are the same, the proper time interval shown by clocks corresponding to ΔtA and ΔtB are different. Hence the frequencies will differ. It is easy to see that if the line interval has the form inEq. (3.72), then the corresponding frequencies do obey the redshift relation inEq. (3.76)to the lowest order in (gL/c²).

All the above features show that the spacetime interval cannot retain its usual special relativistic form in the presence of a gravitational field. A modifica-tion of the metric tensor is absolutely essential if we have to obtain physically

3.5 The principle of equivalence and the geometrical description of gravity 131 reasonable conclusions. For this reason, it is natural to treat the gravitational field as geometrical in origin.

PROJECTS