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Meselwulf
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This short work will help to calculate the varying energy for a non-rotating spherical distribution of mass.
The Energy changing in a Schwartzschild Metric
It is not obvious how to integrate an energy in the Schwartzschild metric unless you derive it correctly. The way this following metric will be presented will be:
\int_{t}^{t'} c^2 d\tau^2 dt = \int_{t}^{t'} (1 - 2\frac{Gm}{\Delta E} \frac{M}{r_s} c^2 dt^{2}) - \frac{dt}{(1-2\frac{Gm}{\Delta E} \frac{M}{r_s})} - r^2 d \phi dt
This will be interpeted as
\int_{t}^{t'} c^2 d\tau^2 dt = \int_{t}^{t'} (1 - 2\frac{Gm}{E - E'} \frac{M}{r_s} c^2 dt^{2}) - \frac{dt}{(1-2\frac{Gm}{E - E'} \frac{M}{r_s})} - r^2 d \phi dt
And this metric is dimensionally-consistent to calculate the energy changes within a metric. Usually, in the spacetime metric, we treat it as a energy efficient fabric. This can be a way to treat a metric with a type of energy variation consistent perhaps with a radiating body.
The Energy changing in a Schwartzschild Metric
It is not obvious how to integrate an energy in the Schwartzschild metric unless you derive it correctly. The way this following metric will be presented will be:
\int_{t}^{t'} c^2 d\tau^2 dt = \int_{t}^{t'} (1 - 2\frac{Gm}{\Delta E} \frac{M}{r_s} c^2 dt^{2}) - \frac{dt}{(1-2\frac{Gm}{\Delta E} \frac{M}{r_s})} - r^2 d \phi dt
This will be interpeted as
\int_{t}^{t'} c^2 d\tau^2 dt = \int_{t}^{t'} (1 - 2\frac{Gm}{E - E'} \frac{M}{r_s} c^2 dt^{2}) - \frac{dt}{(1-2\frac{Gm}{E - E'} \frac{M}{r_s})} - r^2 d \phi dt
And this metric is dimensionally-consistent to calculate the energy changes within a metric. Usually, in the spacetime metric, we treat it as a energy efficient fabric. This can be a way to treat a metric with a type of energy variation consistent perhaps with a radiating body.
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