override public Matrix4x4 GetMetric(Vector4 stpiw, Vector4 pstpiw)
{
Vector4 origin = transform.position;
//We assume all input space-time-position-in-world vectors are Cartesian.
//The Schwarzschild metric is most naturally expressed in spherical coordinates.
//So, let's just convert to spherical to get the conformal factor:
Vector4 cartesianPos = stpiw - origin;
Vector4 sphericalPos = cartesianPos.Cartesian4ToSpherical4();
//Assume that spherical transform of input are Lemaître coordinates, since they "co-move" with the gravitational pull of the black hole:
double rho = cartesianPos.magnitude;
double tau = cartesianPos.w;
//Convert to usual Schwarzschild solution r:
double r = Math.Pow(Math.Pow(3.0 / 2.0 * (rho - SRelativityUtil.c * tau), 2) * radius, 1.0 / 3.0);
//At the center of the coordinate system is a singularity, at the Schwarzschild radius is an event horizon,
// so we need to cut-off the interior metric at some point, for numerical sanity
if (r <= radiusCutoff)
{
Matrix4x4 minkowski = Matrix4x4.identity;
minkowski.m33 = SRelativityUtil.cSqrd;
minkowski.m00 = -1;
minkowski.m11 = -1;
minkowski.m22 = -1;
return(minkowski);
}
else
{
double schwarzFac = 1 - radius / r;
double playerR = ((Vector3)(pstpiw - origin)).magnitude;
double playerSchwarzFac = 1;
if (playerR > radiusCutoff)
{
playerSchwarzFac = 1 - radius / playerR;
}
double sqrtRDivRs = Math.Sqrt(r / radius);
double denomFac = (sqrtRDivRs - 1.0 / sqrtRDivRs) * (sqrtRDivRs - 1.0 / sqrtRDivRs);
//Here's the value of the conformal factor at this distance in spherical coordinates with the orig at zero:
Matrix4x4 sphericalMetric = Matrix4x4.zero;
//(For the metric, rather than the conformal factor, the time coordinate would have its sign flipped relative to the spatial components,
// either positive space and negative time, or negative time and positive space.)
sphericalMetric[3, 3] = (float)(SRelativityUtil.cSqrd * playerSchwarzFac * (r - radius) / (radius * denomFac));
sphericalMetric[0, 0] = (float)(-schwarzFac / (denomFac * playerSchwarzFac));
sphericalMetric[1, 1] = (float)(-r * r);
sphericalMetric[2, 2] = (float)(-r * r * Mathf.Pow(Mathf.Sin(sphericalPos.y), 2));
//A particular useful "tensor" (which we can think of loosely here as "just a matrix") called the "Jacobian"
// lets us convert the "metric tensor" (and other tensors) between coordinate systems, like from spherical back to Cartesian:
Matrix4x4 jacobian = Matrix4x4.identity;
double x = cartesianPos.x;
double y = cartesianPos.y;
double z = cartesianPos.z;
rho = Math.Sqrt(x * x + y * y + z * z);
double sqrtXSqrYSqr = Math.Sqrt(x * x + y * y);
if (rho != 0 && sqrtXSqrYSqr != 0)
{
// This is the Jacobian from spherical to Cartesian coordinates:
jacobian.m00 = (float)(x / rho);
jacobian.m01 = (float)(y / rho);
jacobian.m02 = (float)(z / rho);
jacobian.m10 = (float)(x * z / (rho * rho * sqrtXSqrYSqr));
jacobian.m11 = (float)(y * z / (rho * rho * sqrtXSqrYSqr));
jacobian.m20 = (float)(-y / (x * x + y * y));
jacobian.m21 = (float)(x / (x * x + y * y));
jacobian.m12 = (float)(-sqrtXSqrYSqr / (rho * rho));
jacobian.m22 = 0;
jacobian.m33 = 1;
//To convert the coordinate system of the metric (or the "conformal factor," in this case,) we multiply this way by the Jacobian and its transpose.
//(*IMPORTANT NOTE: I'm assuming this "conformal factor" transforms like a true tensor, which not all matrices are. I need to do more research to confirm that
// it transforms the same way as the metric, but given that the conformal factor maps from Minkowski to another metric, I think this is a safe bet.)
return(jacobian.transpose * sphericalMetric * jacobian);
}
else
{
sphericalMetric.m22 = sphericalMetric.m00;
sphericalMetric.m00 = 1;
sphericalMetric.m11 = 1;
return(sphericalMetric);
}
}
}
