Albert Einstein
In this article we imagine that a photon propagates either (a) “upward”, or (b) “laterally” in an ambient gravitational field that points “downward”(1). We begin by postulating that Newton 2,
F=dP/dt, (1)
is true for every particle in the known Universe. Since P, a particle’s momentum, is:
P=mv (2)
(where m is the particle’s inertial mass), we have by the chain rule:
F=ma + v dm/dt. (3)
In the case of particles whose speed is less than c, m is a function of the constant, mo, and the variable, v:
m =mo(1-v2/c2)-1/2 (4)
=γmo.
Thus, for such particles:
dm/dt=mo(-1/2)(1-v2/c2)-3/2(-2va/c2)] (5)
=γ3mova/c2.
And Eq. 3 becomes:
F = γmoa + vγ3mova/c2. (6)
Now all particles with v<c have nonzero rest masses. But particles with v=c have zero rest masses. And in the latter case the first term in Eq. 6 would be (0/0) and the second term would be (0/03) … both mathematically ambiguous.
However, thanks to the insight of Einstein, light-speed particles (e.g. photons) do have finite masses:
m=hν/c2 (7)
where ν is the frequency of the “carrier” wave.
Since a photon has mass, and since the force felt by an entity in an ambient gravitational field is presumably:
F=mg, (8)
we might wonder “How does a photon behave in such a field?” Presumably, its momentum is:
P=m<c>c (9)
=(h/c2)νc<c>
=(h/c)ν<c>
where <c> is the photon speed’s unit vector. (We note that, although c is a constant, the direction of <c> may vary in time.)
Differentiating Eq. 9, we get
dP/dt=(h/c)[ν(d<c>/dt)+<c>(dν/dt)]. (10)
Thus, by Newton 2,
(hν/c2)g = (h/c)[ν(d<c>/dt) + <c>(dν/dt)], (11)
and
g=dc/dt + c[(dν/dt)/ν]. (12)
If <c> is orthogonal to g, then dν/dt presumably equals zero. In this case Eq. 12 simplifies to:
g= dc/dt (13)
=c(d<c>/dt).
Since c is constant, the direction of <c> is changing … it is curving downward in the present example.
If <c> is antiparallel to g, then dc/dt presumably equals zero. In this case Eq. 12 simplifies to:
g=-c[(dν/dt)/ν]. (14)
We can recast Eq. 14 as
dν/ν=(-g/c) dt(2). (15)
Or, integrating both sides,
ln(ν)=-gt/c+K (17)
where K is a constant. Let us define νo to be the photon’s frequency at time t=0. Then
K=ln(νo) (18)
and
ν=νoe-gt/c). (19)
Assuming g points “downward”, a photon traveling “upward” has a drop in frequency … a phenomenon known as “redshift”.
If the photon’s velocity is normal to the ambient g field, then dν/dt=0 in Eq. 12, and d<c>/dt<0. The trajectory curves downward … a phenomenon known as gravitational lensing.
(1) In gravitomagnetic theory, g would be imaginary and would point “upward”, etc. But the end results are the same as those obtained in the more customary notation used in this article.
(2) Given two parallel vectors A=A<A> and B=B<B>, A/B presumably equals A/B. And given two antiparallel vectors A=A<A> and B=B<B>, A/B presumably equals -A/B.This applies only to parallel and antiparallel vectors.