E = h f.
● /\/\/\/\/\/\/\/\/\/\/\/\/\ photon
M
f f'
o o
/\ /\
observer observer
Let us assume that a photon is emitted in a low gravity potential close to a star M. An observer close to the birthplace of the photon measures a frequency
f
for the photon, using his clock.
The photon must lose some energy when it climbs to outer space, against the gravity potential of M. An observer in outer space measures a frequency
f' < f.
Let us use the standard Schwarzschild coordinates around M. The coordinate frequency of the photon cannot change when it climbs up from the potential of M. Imagine that we have a laser which shoots a beam up from M. We assume that the overall shape of the wave in the beam does not change with coordinate time.
4 waves in 1 coordinate second
● /\/\/\/\ /\/\/\/\
M
o o
/\ /\
observer observer
In one coordinate second, the same number N of waves must pass an observer close to M, as in outer space. The coordinate frequency is the same for both observers.
We conclude that the clock of the observer near M must run slower.
If the clocks of both observers would run at the same rate, then both observers would measure the same frequency f for the photon. Then the photon would not have lost energy as it climbed the potential of M. The "implementation" of the gravity potential would be flawed.
If a hydrogen atom decays close to M, then a single photon is created. What about implementing a gravity potential by deleting some percentage of created photons as they climb up from the potential of M? That will not work, since conservation of energy would be violated.
Massive particles do not require us to change the clock rate near M
The basic property of a potential wall is that a particle m climbing up it must lose energy in some way. Massive particles can shed their kinetic energy. There is no need to assume different clock rates.
But a massless particle, like a photon, can only lose energy by lowering its frequency f. That requires different clock rates near M and in outer space.
Can a classical wave packet shed energy without changing its frequency?
Let us have a classical light wave packet which climbs up the potential of M. A way to reduce the energy of the packet is to lower its frequency.
We can also imagine a process which reduces the amplitude of the wave as it propagates. However, such a process may be complicated to implement.
Why is the radial metric stretched in the Schwarzschild solution?
The stretching probably can be derived from an equivalence principle. But can we find a simpler explanation?
Static electric field? A possible explanation: a static electric field "consists of" virtual particles which only carry momentum, no energy.
Let us have spherically symmetric electric E field around a spherical mass M. To keep the energy of E constant, if the metric of time is g₀, the radial metric must be g₁ = -1 / g₀.
Let us have a mass shell. Inside the shell, the metric of time is squeezed, but the spatial metric is 1. Why is there the energy of the electric field E smaller?
Could it be that the matter in the shell can be seen as "polarized" material which reduces the field energy E? Outside M, it is empty space.
The volume element has the same 4-volume outside M, but inside the mass shell, the volume is smaller.
The mechanism to slow down time is the same as in special relativity? In special relativity we can slow down time using a large velocity. Simultaneously, a ruler contracts in the direction of the movement. Maybe the mechanism which nature uses to change the metric in the Schwarzschild solution is copied from special relativity?
• --> v ●
m r M
Let us have a spherical mass M. Let a test mass m start static at the infinity, and fall toward M. Let v << c be the velocity of m.
The kinetic energy of m is
W = G m M / r,
and
v² = 2 G M / r.
The metric of time of m is slowed down by a factor
sqrt(1 - v² / c²),
and the metric of time for m is
g₀ = -c² * (1 - v² / c²)
= -c² * (1 - 2 G / c² * M / r).
= -c² * (1 - rs / r),
where rs is the Schwarzschild radius of M. The radial metric is
g₁ = 1 + rs / r.
It is the Schwarzschild metric. Is it a coincidence that the metric of a test mass m falling from infinity agrees with the Schwarzschild metric of a static observer?
Comparison to a collapse of a dust ball. Let a uniform dust ball collapse from a very sparse state. Let us assume that the spatial metric in comoving coordinates of dust particles is spatially flat.
Let us have static observers, symmetrically from the center, looking at dust particles flying by. The observers will see the comoving rulers held by dust particles contracted radially. If we convert between the static coordinates and comoving coordinates, do we obtain something like the Schwarzschild metric?
The rulers held the dust particles are contracted by the factor
sqrt(1 - v² / c²) < 1.
These contracted rulers determine a flat spatial metric, by the simultaneousness concept of the dust particles. The static observers have a different view of what is simultaneous.
Let the static observers hold rulers, too. The dust particles see the static rulers length-contracted.
The dust particles figure out that the static observers measure radial distances of static observers by the factor
1 / sqrt(1 - v² / c²) > 1
longer than the distance measured by dust particles. That is, the radial distances measured by static observers are by that factor longer than the flat metric.
We were able to derive the Schwarzschild spatial metric. Our assumption was that the spatial metric in the comoving coordinates of a collapsing dust ball is flat.
The Schwarzschild metric of time follows from the gravity potential.
From the viewpoint of a dust particle, the clock of a nearby static observer runs at a lower rate.
Conclusions
Quantum mechanics seems to require that clocks run slower in a low gravity potential.
We also found a simple way to derive the stretched radial metric of the Schwarzschild solution, if we assume that the spatial metric is flat for a uniform dust ball which initially is very sparse and has a very large (= infinite) radius, and then collapses. The collapse corresponds to a FLRW solution where the curvature parameter k = 0.
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