It is best to abandon the use of the geodesic hypothesis entirely. The Einstein-Hilbert action is the fundamental formula.
The basic idea in the Einstein-Hilbert action: we can reduce the contribution of the mass M in the integral by slowing down time in the metric g, but we must pay the price of increasing Ricci scalar curvature R. The mass M is considered potential energy in the action, as well as the Ricci scalar curvature R.
Moving a test mass m inside or close to the Sun
Let us calculate an example with the Sun,
M = 2 * 10³⁰ kg,
and m = 1 kg. In our blog post on November 10, 2021 we calculated that if the 1 kg test mass moves at a velocity which is comparable to the escape velocity of the star, then the mass in the star has enough time to adjust to the changed metric.
The Schwarzschild radius of a 1 kg mass is
r_s = 2 GM / c²
= 1.3 * 10⁻²⁷ m.
Let us assume that the Sun has enough time to adjust. If we move the mass m = 1 kg some 1 million kilometers, then we may estimate that some 10³⁰ kg of the mass of the Sun moves ~ 10⁻²⁷ m. The contribution to the inertia of the test mass m is something like 10⁻⁶.
In the Komar mass formula, we integrate
ρ + 3 p
over the volume, where ρ is the energy density. Inside the Sun,
ρ ~ 10²⁰ J/m³,
p ~ 10¹⁶ Pa.
We see that the contribution of pressure to "gravity" is ~ 3 * 10⁻⁴.
The gravity potential of the Sun at the surface of the Sun per kilogram of mass is
1/2 v²,
where v is the escape velocity
v = 600 km/s.
The energy of mass per kilogram is
c²,
where c is the speed of light. The ratio
1/2 v² / c² ~ 2 * 10⁻⁶.
At the center the potential is 3/2 times the potential at the surface. We conclude that the gravity potential contributes 3 * 10⁻⁶ to the inertia.
The conclusion: at the center
1. gravity adds 3 * 10⁻⁶,
2. mass flows add 10⁻⁶,
3. gravity caused by the pressure adds 9 * 10⁻¹⁰
to the inertia of a test mass.
Suppose that the Sun does not have time to adjust to the movement of the 1 kg test mass. If we move the test mass a distance of 1 million kilometers outside the Sun, then some pressure energy moves ~ 1 million km. How large is this energy?
The 1 kg mass stretches the radial metric by a ratio ~ 10⁻³⁶ inside the Sun. The volume change is
~ 10⁻³⁶ * 10²⁷ m³
= 10⁻⁹ m³.
The pressure energy is
~ 10¹⁶ Pa * 10⁻⁹ m³
= 10⁷ J.
A reasonable part of the pressure energy moves ~ 1 million kilometers.
The mass-energy of 1 kg is m c² = 10¹⁷ J.
We see that the flow of pressure energy contributes some
~ 10⁻¹¹
to the inertia of the test mass. If the Sun has time to adjust, then mass flows contribute much more, ~ 10⁻⁷.
What is the magnitude of the ordinary tidal effect on the Sun? How much does that add to the inertia of the 1 kg test mass? The shape of the Sun becomes elongated in the field of that 1 kg. The elongation is something like
m / M ~ 10⁻³¹.
That corresponds to a volume 10⁻¹³ m³, or 10⁻¹⁰ kg. We conclude that the ordinary tidal effect adds
10⁻¹⁰
to the inertia of the test mass.
Is the Einstein-Hilbert action aware of the extra inertia?
Is the Einstein-Hilbert action aware of these various amounts of extra inertia?
■■■■■
• ---------------------------- ■■■ M
■■■■■
m lever
Let us consider a general lagrangian and a mechanical system where moving a test mass m also moves a very large mass M slightly through a lever. Does the lagrangian understand that the inertia of m is greatly increased by the lever?
The action does not depend on time, and is invariant under spatial translations. By Noether's theorem it conserves energy and momentum. The action certainly understand the mechanics of the lever system, and understands that the inertia of the test mass m is effectively larger.
We conclude that the Einstein-Hilbert action is aware of the various items of extra inertia which we calculated in the previous section.
The action has to be written very precisely, so that it includes the pressure mass-energy and the kinetic energy of that mass-energy if it is shipped around.
Does a test mass feel immediately the full extra inertia from the flow of pressure energy? Probably not
In our April 20, 2023 blog post we argued that a test charge feels immediately the full extra inertia from its interaction with a static electric field. If that would not be the case, then we should see an oscillation of a static electric field - which has not been observed.
What about a test mass and the flow of pressure energy? We assume that the pressure field is static. Is the test mass aware of the flow of pressure energy which will happen if we move the test mass?
___
/ \
\____/ large mass M with pressure
• ---> test mass m
Suppose that there is pressure in the large mass M, and the system is initially static.
In principle, it is possible that the test mass m would "know" how much pressure energy it will move inside M, if m moves. The system is static. There is plenty of time to communicate information around about the state of the pressure field.
Let us assume that the test mass m feels the inertia of future pressure energy flows immediately. Let us move the test mass m and input some momentum p against the extra inertia.
But then we suddenly remove the pressure inside M through some mechanism. M may be supported by some mechanical structures and we let these structures to come loose.
Now we face a problem: where are we going to put the extra inertia p which we input? There is no pressure energy to move around.
It looks plausible that the test mass m does not feel the inertia of pressure energy flows inside M in advance.
Even less likely is that the test mass would feel in advance the inertia from the ordinary tidal effect, or from the adjustment of the mass M.
Does the test mass m feel the inertia from the direct gravity force of M immediately? Probably yes
In our April 20, 2023 blog post we argued that an electric test charge c does feel the inertia from the Coulomb interaction with a large charge C immediately.
For the analogous problem in gravity we do have empirical data. Photons passing the sun seem to feel the extra inertia from the gravity of the Sun immediately.
Also, if it would take a few milliseconds for atomic clocks to feel the inertia from the gravity of Earth, then someone probably would have noticed a weird behavior of clocks.
(Traditional) general relativity assumes that there is a metric which the large body M creates around itself, and the test mass m must obey this metric: move along a geodesic. That is, in general relativity the test mass m must feel the inertia from the field M immediately. If that were not the case, then the concept of a geodesic would not make sense.
General relativity also assumes that the gravity from the pressure inside M is a part of the metric. Thus, the inertia associated with the pressure is felt immediately by m. This might be wrong.
Let us calculate the presumed inertia from pressure inside Earth. Our test mass is 1 kg.
The ratio of stretching in the Schwarzschild metric at the distance of 10,000 km is 10⁻³⁴. The volume change is
~ 10⁻³⁴ * 10²¹ m³
= 10⁻¹³ m³.
The pressure is ~ 10¹¹ Pa. The pressure energy
~ 0.01 J.
That is only 10⁻¹⁹ of m c² for the 1 kg mass. We conclude that it is impossible to measure the contribution of the pressure to the inertia.
Conclusions
We believe that a test mass m feels the inertia from the direct (newtonian) gravity field of M immediately. There is empirical evidence of this.
We believe that the inertia caused by pressure inside M is not felt immediately. There is a "self-force" from the gravity field of m on m itself, which determines how m behaves. The extra inertia deforms the gravity field of m and the deformation causes a force on m.
The ordinary tidal effect, and mass flows inside M, when M adapts, cause more substantial extra inertia for m. We firmly believe that these are not immediately felt by the test mass m. The extra inertia only unfolds when M changes its shape.
The self-force on an electric charge is a mystery which is well over a century old. In our blog post on October 1, 2021 we wrote about an electron in a periodic motion and the self-force from its own electric field.
For gravity, the self-force might be derivable if we knew the energy of a deformed gravity field. The Schwarzschild metric is seen as the "natural field" of a free test mass. If we deform it in a way or another, we must use energy. Gravitational waves are born from deformations to the field of an accelerating mass.
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