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Photon Frequency in a Uniform Gravitational Field

1. Abstract.

In this article f(r), the frequency of a photon emitted upward in a uniform gravitational field, is derived using quasi-classical ideas. The frequency at emission (where r=0) is f0=c/l=6.38e14 Hz (blue light).

2. A Derivation of f(r).

In general the photon is assumed to have a mass of

m=E/c2=hf/c2,           (2.1)

where h is Planck’s constant. Let us assume that the gravitational field acting on the photon is a constant g.

The gravitational force acting on the photon is then

F=-hfg/c2.          (2.3)

Therefore, by Newton 2 the time rate-of-change of the photon’s momentum is:

dP/dt)=-hfg/c2.                   (2.4)

Or, since dt=dr/c in the case of a photon,

dP/dr=-hfg/c3.           (2.5)

But in the case of a photon, we also have

P=hf/c.               (2.6)

Thus

dP/dr=h/c (df/dr).              (2.7)

Equating (2.5) and (2.7), we have

(df/dr)/f=-g/c2.                   (2.8)

Since

(df/dr)/f=d(ln(f))/dr,                  (2.11)

we find

d(ln(f))=-g/c2 dr                  (2.12)

and

ln(f)=-gr/c2 + k1.                  (2.13)

At r=0 (the point at which the photon is  emitted), we have for the constant k1:

k1=ln(f0).            (2.14)

Raising both sides of (2.13) as exponents of e, we find that

f=exp(-gr/c2 + ln(f0)).                  (2.15)

Fig. 1 plots f(r) vs. r. Evidently f(r) linearly decreases with increasing r. Since l=c/f, the wavelength would grow with increasing distance from the point of emission … a behavior known as redshift. Fig.1 plots f v.s r. Appendix 1 provides the Power Basic program used to generate the plot data.

Figure 1 f(r) (Hertz) vs. r (meters)

Appendix 1

#COMPILE EXE

#DIM ALL

'Plot the frequency v.s.r for a photon emitted above the surface of the Earth.

FUNCTION PBMAIN () AS LONG

DIM steps AS LONG

steps=50

DIM g AS DOUBLE     'Gravitational field

g=1e3

DIM c AS DOUBLE     'Speed of light

c=2.99792458e8

DIM f0 AS DOUBLE

f0=6.38e14

DIM k1 AS DOUBLE

k1=LOG(f0)

DIM r(steps) AS DOUBLE         'Distance from Earth center

DIM f(steps) AS DOUBLE    'Frequency

DIM index AS LONG

FOR index=1 TO steps-1

r(index)=index*c/10

f(index)=EXP(-g*r(index)/c^2 + log(f0))

NEXT

OPEN "c:\Users\Marjorie Dixon\Documents\Redshift.dat" FOR OUTPUT AS #1

FOR index=1 TO steps-1

WRITE #1, r(index),f(index)

NEXT

CLOSE #1