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Photon Frequency in an Inverse Square Gravitational Field

1. Abstract.

In this article f(r), the frequency of a photon emitted upward in the Earth’s gravitational field, is derived using quasi-classical ideas. r is the distance from the center of the Earth. A photon is assumed to be emitted from a point r0=c/10 meters from the Earth’s center (well above the surface). The frequency at emission is f0=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. Evidently, if m decreases as r increases, then so does f. The gravitational field acting on the photon is assumed to be

g=-GM/r2           (2.2)

where M is the Earth’s mass and r is the distance from Earth’s center. The gravitational force acting on the photon is thus

F=-hfGM/r2c2.           (2.3)

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

dP/dt)=-hfGM/r2c2.           (2.4)

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

dP/dr=-hfGM/r2c3.            (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=-GM/r2c2.           (2.8)

Or, defining a constant k by

k=-GM/c2,                   (2.9)

we have

(df/dr)/f=k/r2.            (2.10)

Since

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

we find

d(ln(f))=k/r2 dr          (2.12)

and

ln(f)=-k/r + k1.            (2.13)

At r=c/10 (the point at which the photon is  emitted), we have for the constant k1:

k1=ln(f0)+k/c/10.                (2.14)

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

f=exp(-k/r + k1).                  (2.15)

Fig. 1 plots f(r) vs. r. Evidently f(r) does decrease 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 vs. 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 Constamt

G=6.67408e-11

DIM M AS DOUBLE     'Mass of earth

M=5.97219e24

DIM c AS DOUBLE     'Speed of light

c=2.99792458e8

DIM f0 AS DOUBLE

f0=6.38e14

DIM k AS DOUBLE

k=-G*M/c^2

DIM k1 AS DOUBLE

k1=LOG(f0)+k/c/10

DIM r1(steps) AS DOUBLE         'Distance from Earth center

DIM f(steps) AS DOUBLE    'Frequency

DIM index AS LONG

FOR index=1 TO steps-1

r1(index)=index*c/10

f(index)=EXP(-k/r1(index) + k1)

NEXT

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

FOR index=1 TO steps-1

WRITE #1, r1(index),f(index)

NEXT

CLOSE #1