Let us call the "true geometry" of spacetime something which is determined by orbits of an infinitesimal test mass. The test mass only interacts through gravity.
The metric of general relativity is approximately the true geometry in the case of a spherically symmetric mass. However, it does not include tidal effects, which close to the mass can be substantial.
The stress-energy tensor in the Einstein field equations includes essentially the mass-energy density, movement of energy, and pressure, if we ignore shear stresses.
Thus, some effects of pressure do affect the calculated metric. Let us try to analyze what is included.
Adding extra pressure: the Ehlers et al. metric for a spherical mass which is squeezed by a membrane on the surface
Jürgen Ehlers et al. (2005) calculated the metric. How does pressure from the external membrane affect the metric inside the mass?
The metric in the bulk of the sphere is the interior Schwarzschild metric. The spatial metric of the equatorial plane inside the sphere is the surface of a polar region of a 3-dimensional ball whose radius is R.
(Picture by Djexplo https://commons.wikimedia.org/wiki/File:Latitude_and_Longitude_of_the_Earth.svg )
Let
A₁ = the latitude of the "effective" border of the polar region, including the pressure of the membrane;
A₀ = the latitude of the border of the bulk of the sphere.
We can increase the pressure inside the bulk of the sphere either by
1. adding more mass of the density ρ around it, down to the latitude A₁, or
2. increasing the pressure by tightening the membrane on the surface.
Above, a₁ = sin(A₁). Let us define a₀ = sin(A₀).
We are interested in how much slower does time run at the center of the sphere relative to its border, if we increase pressure by squeezing the sphere with the membrane.
The metric shows that the closer we are to the pole, the slower time runs. Furthermore the rate of time depends on the sine of the latitude.
According to Wikipedia,
R = sqr(r₀³ / r_s),
where r₀ is the coordinate radius of the sphere (not proper), and r_s is the Schwarzschild radius of it.
The rate of time at the center is
1/2 (3 a₁ - 1)
while at the border it is
1/2 (3 a₁ - a₀).
Let us calculate a case where the latitudes A₀ and A₁ are almost 90 degrees. The pressure at the center is not very big and we can calculate most things using newtonian gravity.
Let us have a sphere of a radius r₀ and a density ρ. We add more matter, so that the new radius is r₁.
We have
R = sqrt(r₀³ / r_s)
= sqrt( r₀³ c² / (2 G ρ * 4/3 π r₀³) )
= sqrt( 3 c² / (8 π G ρ) ),
a₀ = sin(A₀)
= sin(r₀ / R)
= 1 - b₀
≅ 1 - 1/2 r₀² / R²,
a₁ = 1 - b₁
≅ 1 - 1/2 r₁² / R².
We introduced b₀ and b₁ which are small positive numbers.
The rate of time at r₀ divided by the rate at the center is
T = (1 - 3/2 b₁ + 1/2 b₀) / (1 - 3/2 b₁)
≅ 1 + 1/2 b₀ + 3/4 b₀ b₁,
where we used the fact that b₁ is small.
T ≅ 1 + 1/4 r₀² / R² + 3/16 r₀² r₁² / R⁴.
We assume that r₀ >> 1. We let
r₁ = r₀ + 1.
The increase in T, relative to the setting r₁ = r₀, is approximately
3/8 r₀³ / R⁴
= 8/3 π² r₀³ G² ρ² / c⁴.
The energy of an extra 1 kg test mass placed at the center of the sphere, as seen by an observer at r₀, is reduced by
E = 8/3 π² r₀³ G² ρ² / c²,
because of the slowdown of time. That is, if an observer at the center would send that 1 kg of mass-energy as radiation to the r₀ observer, the redshift would cut off that much energy.
The extra pressure from 1 meter of matter is
p = G ρ * ρ * 4/3 π r₀³ / r₀²
= 4/3 π r₀ G ρ².
Let us put 1 kg of extra matter to the center of the sphere and calculate how much it increases the volume of the sphere. We assume that the 1 kg mass stretches the radial metric according to the Schwarzschild metric of that 1 kg mass. We denote by r_kg the Schwarzschild radius of 1 kg mass:
r₀
V = ∫ 1/2 r_kg / r * 4 π r² dr
0
= 2 π r₀² G / c².
The extra pressure p does the work
E' = p V
= 8/3 π² r₀³ G² ρ² / c².
We see that E' = E. If we increase the pressure by p, we can explain the lower potential E of the 1 kg mass at the center solely by the fact that when the 1 kg is lowered, the volume of the sphere increases by V, and the extra pressure p does the extra work E.
Conjecture. General relativity calculates correctly the direct effect on the metric of time caused by the volume increase around a test mass combined with pressure. The rate of time slows down according to the work done by pressure.
However, it may be that general relativity miscalculates the effect of pressure on the radial metric. General relativity claims that the spatial metric remains the same, even if we increase the pressure. In the interior Schwarzschild metric, R only depends on ρ, not on the pressure.
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