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Varying coordinate systems in GR has given me a new perspective that may help to resolve a problem that has been nagging at me ever since I began working with GR. In every problem I've ever dealt with, a complex mathematical result describes an impossible scenario, something that cannot occur. Yet in the Schwarzschild coordinate system, we have a coordinate radius beyond which all sorts of complex results arise, below the event horizon. This says nothing should exist within this space, since we cannot have clocks that tick at a complex rate and we cannot have complex lengths. This space below 2m, then, really shouldn't exist imo. Yet using Schwarzschild coordinates, we can still plot coordinate distances between zero and 2 m according to a distant observer, that a freefaller falls into within finite proper time, although we can't describe what happens below 2 m.

I had always thought of Schwarzschild coordinates as the "real" coordinate system, while all others were psuedo-systems for ease of derivation that must be switched back to Schwarzschild to gain the "actual" results. But if Schwarzschild is just as arbitrary a coordinate system as any other, then we can just as easily make a different coordinate choice. Of course, even in deriving Schwarzschild, we must follow certain rules: the locally measured speed of light is always c, SR is valid locally, the equivalence principle holds, the EFE equations are valid, locally measured angular momentum is constant, the locally measured energy of a particle per local time dilation is constant, etc. I will add one more which will limit my choice of coordinate systems even further. It is simply that we will only measure "real" space with our coordinate system.

My choice of coordinate systems, at least the simplest I've found so far, then, is

r1 = r (1 - 2 m / r), r = r1 (1 + 2 m / r1) which transforms the metric to

ds^2 = c^2 dt^2 / (1 + 2 m / r1) - dr1^2 (1 + 2 m / r1) - dθ^2 r1^2 (1 + 2 m / r1)^2

I have now turned the event horizon at r = 2 m in Schwarzschild into a point singularity at r1 = 0. Nothing below r1 = 0 exists within this coordinate system, so complex space is eliminated. Problem solved. This is the way I believe it should be, and if I am allowed to use any coordinate system I choose, this is definitely the one that makes sense to me. If a freefalling observer were to fall all the way to the point singularity in this coordinate system, in a finite time according to his own watch, then since that is also where the point mass must exist, the freefaller will immediately strike the mass rather than falling further into some complex space. If the mass were not struck somehow, and the freefaller could keep going, he should simply pass through the point in real space and begin decelerating as he now travels away from the point while his watch continues to tick at a gravitationally time dilated rate, although to the distant observer, it would take him an infinite amount of time to reach the point singularity.

The transformations are now

dt' = dt / sqrt(1 + 2 m / r1)

dr' = dr sqrt(1 + 2 m / r1)

with tangent distances measured as (1 + 2 m / r1) greater locally. Locally, tangent and radial speeds of particles would be measured as

v_t' = v_t (1 + 2 m / r1)^(3/2)

v_r' = v_r (1 + 2 m / r1)

Of course, there are still plenty of other coordinate systems that create a point singularity, r1 = r sqrt(1 - 2 m / r) for instance. The only other rule that must apply is that r1 approximates r as r goes to infinity. If we want to describe -r1 as the same as r1 but in the opposite direction, so that r1 and -r1 must give the same time dilations and length contractions, then we could use r1 = sqrt(r^2 - (2 m)^2). Even if others might not agree, since there is theoretically no right or wrong in GR coordinate choices, I truly believe this condition of only allowing real space should be included in the coordinate choice. Other conditions might narrow it down further as well. Does anyone have any other conditions for the coordinate choice in mind that might help things make more sense to you?

I had always thought of Schwarzschild coordinates as the "real" coordinate system, while all others were psuedo-systems for ease of derivation that must be switched back to Schwarzschild to gain the "actual" results. But if Schwarzschild is just as arbitrary a coordinate system as any other, then we can just as easily make a different coordinate choice. Of course, even in deriving Schwarzschild, we must follow certain rules: the locally measured speed of light is always c, SR is valid locally, the equivalence principle holds, the EFE equations are valid, locally measured angular momentum is constant, the locally measured energy of a particle per local time dilation is constant, etc. I will add one more which will limit my choice of coordinate systems even further. It is simply that we will only measure "real" space with our coordinate system.

My choice of coordinate systems, at least the simplest I've found so far, then, is

r1 = r (1 - 2 m / r), r = r1 (1 + 2 m / r1) which transforms the metric to

ds^2 = c^2 dt^2 / (1 + 2 m / r1) - dr1^2 (1 + 2 m / r1) - dθ^2 r1^2 (1 + 2 m / r1)^2

I have now turned the event horizon at r = 2 m in Schwarzschild into a point singularity at r1 = 0. Nothing below r1 = 0 exists within this coordinate system, so complex space is eliminated. Problem solved. This is the way I believe it should be, and if I am allowed to use any coordinate system I choose, this is definitely the one that makes sense to me. If a freefalling observer were to fall all the way to the point singularity in this coordinate system, in a finite time according to his own watch, then since that is also where the point mass must exist, the freefaller will immediately strike the mass rather than falling further into some complex space. If the mass were not struck somehow, and the freefaller could keep going, he should simply pass through the point in real space and begin decelerating as he now travels away from the point while his watch continues to tick at a gravitationally time dilated rate, although to the distant observer, it would take him an infinite amount of time to reach the point singularity.

The transformations are now

dt' = dt / sqrt(1 + 2 m / r1)

dr' = dr sqrt(1 + 2 m / r1)

with tangent distances measured as (1 + 2 m / r1) greater locally. Locally, tangent and radial speeds of particles would be measured as

v_t' = v_t (1 + 2 m / r1)^(3/2)

v_r' = v_r (1 + 2 m / r1)

Of course, there are still plenty of other coordinate systems that create a point singularity, r1 = r sqrt(1 - 2 m / r) for instance. The only other rule that must apply is that r1 approximates r as r goes to infinity. If we want to describe -r1 as the same as r1 but in the opposite direction, so that r1 and -r1 must give the same time dilations and length contractions, then we could use r1 = sqrt(r^2 - (2 m)^2). Even if others might not agree, since there is theoretically no right or wrong in GR coordinate choices, I truly believe this condition of only allowing real space should be included in the coordinate choice. Other conditions might narrow it down further as well. Does anyone have any other conditions for the coordinate choice in mind that might help things make more sense to you?

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