**Gravitational
Red Shift**

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Gravitational red shift is
often interpreted as the result of spacetime curvature. The argument used to
arrive at this conclusion is based on that proposed by Schild [1]. From herein
it will be assumed that the gravitational field is static, otherwise there is no
guarantee that there will be a gravitational red shift in all generality. Marsh
and Nissim-Sabat gave a counter argument to Schild in the American Journal of
Physics, [2] which is discussed below. Schild’s argument is basically the
following; consider two systems **S**_{1}
and **S _{2}**
located in a gravitational field as shown in Figure 1.

Let the gravitational
potential F(**z)**
be defined such that F(0)
= 0. Two pulses of light are emitted from **S**_{1}
towards **S _{2}
**spaced in time by the amount D

Regardless of
how the field acts on the light, the worldlines will be congruent. By this it is
meant that the shape of the curves will be identical but merely translated
upwards on the spacetime diagram. AC = BC as shown in Figure 2. However, it is
argued, since it is known from experiment that D*t*
> D*t*’.
Thus, Schild argues, AC ¹
BC and therefore the diagram is non-Euclidean, curved, spacetime. That the first
derivation of gravitational redshift by Albert Einstein was in flat spacetime
never seemed to bother Schild!

What’s wrong
with Schild’s argument? First one needs to be careful when interpreting the
statement "The frequency of light decreases..." Caution must be
exercised when using "the" when discussing relativity. The frequency
reckoned but which observer? Which clock is used to reckon this change? Secondly
we note that when spacetime diagrams are drawn to illustrate Schild’s
argument, they are always drawn *without* a scale on the time axes. This
lack of a time scale leads one to make easily make an error. Each time axis
represents the worldline of the clock located at the origin of its coordinate
system. The clock in **S**_{1}
records the coordinate time in **S**_{1}
whereas the clock in **S _{2}**
records the coordinate time in

Note that these are really two spacetime diagrams superimposed on top of
each other, the difference between them being the origins are separated by the
distance *h*, the spatial separation of the clock, i.e. **S**** _{2}**
is located at a distance

**Schwarzschild Metric
**

In order to describe * gravitational
redshift *consider a spherical symmetric massive body (e.g.
planet, star, black hole etc.) whose gravitational field is described by the

*ds* is a scalar quantity which represents the invariant spacetime interval
between two events in spacetime. For two events which have a spacelike spacetime
separation the proper time between two events is related to *ds *by *ds *=
*c*^{2}*d*t^{2}.
*t* is known as the * coordinate time*.

Consider two clocks

Let

Define

D*t*_{e}
=
Coordinate time interval between the two signals which are emitted from *C*_{e}

D*t*_{r}
= Coordinate time interval between the two signals which are received at *C*_{r}

It then follows that

Therefore the frequency of
the light**, ****as measured by any
single observer**, does not change as the light moves through the
gravitational field!

According to Eq. (1)
for two closely spaced pulses *dt*_{r}
= *dt*_{e}
. Let *dt*
= *dt*_{r}
= *dt*_{e}
= the
coordinate time interval. The time interval recorded on *C*1
between the emission of two pulses is the *proper time* between the two
pulses and will be denoted by *d**t*_{e}
. The
time interval recorded on *C***2**
between the reception of the same two pulses is the *proper time* between
the two pulses arriving at *C*_{2}
and denoted *d**t*_{r}.
These intervals are related to the proper time intervals through the
metric. For simplicity let *dr = d**q = df
= *0 (i.e. the
clocks are at rest). Substituting
these into the Schwarzschild metric gives

Upon rearranging terms we get

Note that frequencies are
related to time intervals by *f* = 1/*dt*. Define the following
frequencies

*f*
= frequency as measured on far-away clock

*f*_{e
}=
frequency as measured on *C*_{e
}

*f*_{r}
= frequency as measured on *C*_{r}

Then

Equating Eq. 3b by Eq. 3a becomes upon rearranging terms

This is the relation for
gravitational redshift. Eq. 5 shows that when *r*_{e
} ¹
*r*_{r}
then *f*_{e
}¹
*f*_{r}.
Consider now that the light consists of photons. Multiply Eq. (4) through by
Plank's constant *h*, noting that the energy of a photon is *E* = *hf*,
to get

E is then the energy of
photon as measured by a far-away observer. *K*_{e
}_{
}and *K*_{r}
are the energies of the photons when measured by devices collocated with the
clocks at *P*_{e}
and at *P*_{r}_{
}respectively. These energies may be thought of as the kinetic energies of
the photons. Eq. 5 can be rearranged to give

Eq. 6 can be simplified by
using the approximation (1+*d*)^{n}
»
1 + *nd*,

neglecting second order
terms

whew *m* = *K*_{r
}*/c*^{2} = gravitational mass of photon as
measured by observer at r = re. *U* º
-*GMm*/*r* = *Gravitationnal Potential Energy* of photon and
K = *Kinetic Energy* of Photon (energy of photon measured locally). Then

Define *E* = *K*
+ *U* = Kinetic Energy + Potential Energy = total inertial energy.
Therefore

which is simply the
expression for the conservation of energy. Since this holds for any* r*
it follows that *E* = constant along the photon's trajectory. I.e. *The
total energy of a photon moving through a gravitational field is constant.
*

[1] A. Schild, Monist **40**,
20 (1962) **Evidence for Gravitational Theories**, edited by C. Moller
(Academic, New York, 1962); *Conference Internationalesur les theories
relativistes de la gravitation*, edited by L. Infeld (Pergamon, Oxford.
1964).

[2] **Does a gravitational red shift necessarily imply that spacetime is
curved?**, by G.E. Marsh and C. Nissim-Sabat. *Am. J. Phys*. *Vol. 43,
No. 3, *March (1975)

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