Einstein Field Equation solution calculator.

Accepts stress-energy source terms of matter (density, specific internal energy, pressure) and electromagnetism (electric field, magnetic field).

Converges on solution via...

• 1. linear solver with solution vector corresponding to stress-energy source terms (which is constantly changing per-iteration, since the stress-energy tensor is based on the metric, which is being solved for)
• 2. JFNK solver whose zero is equal to the EFE: $G_{ab} - 8 \pi T_{ab}$

goals:

• 1. (CHECK) verify Newtonian gravity matches Schwarzschild gravity from on to above the surface (I'm not doing this here, but I've done this in my solar system project)
• 2. (CHECK) verify Schwarzschild stellar model gravity due to analytical connection equations is maximal at surface, $\rightarrow 0$ as $r \rightarrow 0$ and $r \rightarrow \infty$
• 3. (CHECK) verify Schwarzschild stellar model gravity due to finite difference matches the analytical connection equation values
• 4. (CHECK) verify, with flat spacetime and stellar stress-energy, that the EFE constraint error is equal to the stellar body stress-energy within the body, and zero elsewhere.
• 5. (CHECK) verify, with Schwarzschild stellar model and stellar stress-energy, that the EFE constraint is near zero everywhere. ... well this works inside the planet and outside the planet, but on the changes in rho this has some numerical errors.
• 6. (CHECK) add time 2nd derivatives to metric prims, from this compute time derivatives of connections, from that the Riemann, and then the Ricci and Einstein
• 6. (sort of) get JFNK to work. My goal is to start at flat space, given Earth's stress-energy source term, and come up with Earth's gravity, as the Schwarzschild analytical answer already gives us. I am getting descent, however it is going really slow, so I'm going to port this project to OpenCL...

Finite Difference Results:

Volumetric graph of the gravitational acceleration using finite differencing for the Schwarzschild-based TOV equation of structure for the Earth's parameters.

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Analytical Calculations:

Schwarzschild-based equations of structure with exhaggerated density.

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Newton, Schwarzschild-based EOS gravity using Earth's parameters, and Kerr-based EOS gravity. Schwarzschild-based EOS is taken from Misner, Thorne, and Wheeler's "Gravitation" box 23.2 on the TOV equations of structure starting from a Schwarzschild metric. Note that the Kerr metric is approximated within the body as the Kerr metric, using for mass all matter contained within the radial coordinate. This is not a correct equation of structure for the Kerr metric, as the correct Kerr EOS has not yet been solved for.

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Differences in between various gravity models, using Earth's parameters.

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Difference between Kerr metric with and without rotation. Note that Kerr without rotation exhibits less gravitational acceleration than Kerr with rotation, while Schwarzschild (which is rotation-less) exhibits more gravitational acceleration than Kerr.

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Kerr metric increase in acceleration due to faster rotation.

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Some thoughts on results ...

Here are the variable ranges, initializing the stress-energy prims with the earth, initializing the metric prims with a flat metric:

after 0 iterations:
#	ix	iy	iz	rho	det-1	alpha-1	gravity	analyticalGravity	EFE_tt(g/cm^3)	EFE_ti	EFE_ij	G_ab
min	0	0	0	0	0	0	0	0.93126130172052	-5.5147098661213	0	0	0
max	15	15	15	4.0950296770075e-24	0	0	0	9.4304273489865	0	0	0	0

after 1 iterations of jfnk
#	ix	iy	iz	rho	det-1	alpha-1	gravity	analyticalGravity	EFE_tt(g/cm^3)	EFE_ti	EFE_ij	G_ab
min	0	0	0	0	-0.57528624373743	-0.2483215332056	-1.2855936147291	0.93126130172052	-2.9155846148211	5.6376265101891e-25	5.7454085369078e-27	6.8400743120253e-27
max	15	15	15	4.0950296770075e-24	-0.57528624373725	-0.24832153320289	1.3218539855677	9.4304273489865	1.220838980645	4.1759391983984e-21	3.1036763335186e-23	3.8474511176799e-23