Energy of the retarded gravity field in a collapse of a mass shell: the energy is significant
In the case of a collapsing sphere, the retarded field steals energy from the kinetic energy of the collapse.
Suppose that retardation keeps time running at a fraction h / 2 faster at the center of the collapsing shell than it would run if the signal speed were infinite. Then most of the retardation energy is stored in the interaction of the mass near the center, M, and the retarded field. The energy is
~ h / 2 * M c²,
that is, linear in h.
If there is no mass near the center, the energy of the retarded field might be similar to the analogous electric field. Another way to estimate the energy of the gravity field is to calculate the pseudotensor. Generally, the energy density is quadratic in h.
Mark Hindmarsh (2018) gives the following approximate formula for the energy density of gravitational waves:
The dot means the derivative over time. The brackets ⟨ ... ⟩ denote an average over an entire wave. The components hij are differences from the flat Minkowski metric.
Let us calculate a very rough estimate for the collapse into a neutron star. We assume that the mass 3 * 10³⁰ kg forms a thin shell. What is the energy of the retarded gravity field inside the shell?
In the previous blog post we calculated that h₀₀ ≈ 0.13 at the center, that is, time runs 6.4% faster at the center than for a flat metric.
Let us assume that the metric becomes flat when the signal to the center reaches from the radius r = 10 km of the neutron star which just formed. Then
dh₀₀ / dt = h₀₀ c / r.
The energy density of the retarded gravity field is something like
egw = c² / (32 π G) * h₀₀² c² / r²
= c⁴ / (32 π G) * h₀₀² / r²,
and the total energy
E = 1/24 * c⁴ / G * h₀₀² * r
= 1/24 * 8 * 10³³ * 1.5 * 10¹⁰
* 0.13² * 10⁴ J
= 8 * 10⁴⁴ J.
This is interesting. The energy is of the same order of magnitude as we calculated in the previous blog post (13 versus 8) from the interaction of the mass near the center and the retarded gravity field (gravity potential).
Maybe the radial metric g₁₁ is stretched, too? In that case the energy density ggw must be doubled.
The energy of the pure retarded gravity field, without any mass in it, is significant!
What is the energy of the retarded field if we calculate it from the analogous electric field?
In the force formula,
G ~ 1 / (4 π ε₀).
The energy density of a gravity field is then
e = 1 / (8 π G) * Eg²,
where Eg is the strength of the gravity field (force / mass).
If the potential is h₀₀ / 2 * c² over a distance of r = 10 km, the field strength is
Eg = h₀₀ / 2 * c² / r.
The energy density of the retarded gravity field is then
e = c⁴ / (32 π G) * h₀₀² / r²
= egw.
We see that the energy density e agrees with the one which we calculated from the formula for gravitational waves!
An explosion of a mass shell
*** WORK IN PROGRESS ***
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