Is there any topology behind factor 2 in Schwarzschild radius?

[Anton Rize]

I am exploring the topological reasons behind certain physical constants. In GR, is the factor of 2 in the Schwarzschild radius (Rs=2GM/c^2) ever treated as having a deeper topological significance?

 

[Demystifier]

No.

 

[martinbn]

Do you have any reasons to suspect that, or is this a shot in the dark?

 

[Anton Rize]

Do you have any reasons to suspect that, or is this a shot in the dark?

Thanks, that’s a fair question. The reason I asked is because I ended up with this factor of 2 showing up in a calculation of mine, and it looked a bit too perfect to just ignore. It made me wonder if there was any known geometric or topological explanation for it, or if the consensus is that it’s purely algebraic.

 

[Anton Rize]

No.

Laconic clean and neat. Love it! You must a blast at party's.

 

[Ibix]

I think you can just absorb the two into the definition of $G$. It's defined the way it is because we thought Newton's law of gravity was fundamental for centuries, but if we'd known it was a limit of relativity we'd probably have defined $G$ differently. Actually we'd probably define $\kappa=8\pi G/c^4$ from the Einstein Field Equations.

 

[PeterDonis]

I think you can just absorb the two into the definition of $G$.

No, you can't, because $G$ (or $8 \pi G / c^4$, or whatever units you choose for it) shows up in a lot of places other than the metric coefficients for Schwarzschild spacetime, and you can't just arbitrarily change it in all those places.

 

[PeterDonis]

In GR, is the factor of 2 in the Schwarzschild radius (Rs=2GM/c^2) ever treated as having a deeper topological significance?

Its significance is geometric: the area of the horizon of a Schwarzschild black hole is $4 \pi R_s^2 = 16 \pi G^2 M^2 / c^4$ if we use conventional units.

 

[PeterDonis]

we'd probably define $\kappa=8\pi G/c^4$ from the Einstein Field Equations.

The usual convention is $\kappa = G / c^4$ (or sometimes $G / c^2$, depending on whether you prefer mass or energy units for the stress-energy tensor), with the factor of $8 \pi$ being left explicit in the EFE. That's because the factor of $8 \pi$ is geometric and is not considered to be part of the choice of units.

 

[Ibix]

No, you can't, because $G$ (or $8 \pi G / c^4$, or whatever units you choose for it) shows up in a lot of places other than the metric coefficients for Schwarzschild spacetime, and you can't just arbitrarily change it in all those places.

I think fundamentally it only appears in the EFEs, doesn't it? Certainly the appearance in Newton and all GR metrics stems grom there.

 

[PeterDonis]

I think fundamentally it only appears in the EFEs, doesn't it?

The appearances of $G$ can be traced back to the EFE, yes. But that means that any attempt to redefine $G$ needs to take into account all of the solutions of the EFE, not just the Schwarzschild solution. And a redefinition such as you proposed would affect all of those solutions--and would make no sense in many of them (for example, FRW spacetime).

Even in the Schwarzschild solution, the physical meaning of the factor $G M / c^2$ is obtained by looking at the Newtonian limit, where we want to be able to derive the Newtonian equations that we already know are good approximations in that limit. And that all works only if we don't make the redefinition you proposed.

(Similar remarks apply to attempts to include the factor $8 \pi$ in the EFE in the definition of $G$--it messes up the correspondence in the Newtonian limit.)

 

[Anton Rize]

Thanks all — that answers my curiosity. I see now that in standard GR the “2” is fixed by the Newtonian limit and shows up in the horizon area as part of the usual geometry, not from any deeper topological principle. That was exactly what I wanted to check.

Even if that doesn’t fully satisfy the philosophical side of my curiosity, that’s fine — the universe isn’t obliged to match our expectations.
Thanks again.