Einsteins-Rosen bridge different from schwarzschild black hole metric?

[hodei_cloud]

Hello, if you could help, I will be glad.

I am studying the Einsteins-Rosen bridge (a matematically solution of the black hole) and I thought that the Einsteins-Rosen bridge was what we found making the Schwarzschild metric a change in kruskal coordinates. But reading an scientific article it says that:

"Since the radial motion into a wormhole after passing the event horizon is physically different from the motion into a Schwarzschild black hole, Einstein-Rosen and Schwarzschild black holes are different."

If you could explain me why it´s different, and then what is Einsteins-Rosen bridge... (in a easy way?)

Thanks for the help

 

[bapowell]

What's the reference? "Einstein-Rosen" black hole is not a standard term.

 

[hodei_cloud]

What's the reference? "Einstein-Rosen" black hole is not a standard term.

I read it from:
http://arxiv.org/abs/0902.1994

 

[bapowell]

The Einstein-Rosen "black hole" is evidently the Kruskal extension with a particular topology imposed:

"For an Einstein-Rosen black hole with Rindler’s elliptic identification of the two antipodal future event horizons the interior (from the point of view of a distant observer; it really is the exterior region III) is static, while for a Schwarzschild black hole the interior (region II) is nonstatic. Rindler’s identification also guarantees that the Einstein-Rosen bridge is a stable solution to the gravitational field equations..." (bottom of pg. 5)

This topology makes the ER wormhole traversable, in contrast to the Schwarzschild wormhole which is causally closed off.

 

[hodei_cloud]

... ok. I think I understand it.

 

[PeterDonis]

This topology makes the ER wormhole traversable, in contrast to the Schwarzschild wormhole which is causally closed off.

I'm skeptical that this topology actually gives a valid solution of the Einstein Field Equation. The "discontinuity" at the horizon that the paper talks about is a coordinate artifact, which appears only because the author insists on using isotropic coordinates. In Kruskal coordinates, for example, nothing is discontinuous at the horizon. And in those coordinates, it's also obvious (at least to me) that you can't just arbitrarily eliminate region II (and region IV) and join together regions I and III at their respective horizons. That may be a valid topology ("valid" in the sense of "consistent if we only consider the topology"), but it doesn't seem to me that it's valid physically, because it's not a solution of the EFE. The solution of the EFE, when extended through the horizon, gives you region II, not region III.

Edit: I see the paper actually agrees with me. From the Discussion section, p. 6:

It has been shown that the Einstein-Rosen bridge metric (1) is not a solution of the vacuum Einstein equations but it requires the presence of a nonzero energy-momentum tensor source $T_{\mu \nu}$ that is divergent and violates the energy conditions at the throat of the wormhole

In other words, the solution only works if there is a sheet of lightlike stress-energy with infinite density at $r = r_g$ (i.e., where the "horizon" would be in the standard black hole solution). That doesn't seem physically reasonable to me.

 

[bapowell]

Agreed. I was trying to understand the distinction between the "ER" and Schwarzschild black hole according to the OP's reference. That is the stated difference, but I was scratching my head on it. Thanks for clarifying.

 

[Ben Niehoff]

I'm skeptical that this topology actually gives a valid solution of the Einstein Field Equation. The "discontinuity" at the horizon that the paper talks about is a coordinate artifact, which appears only because the author insists on using isotropic coordinates. In Kruskal coordinates, for example, nothing is discontinuous at the horizon. And in those coordinates, it's also obvious (at least to me) that you can't just arbitrarily eliminate region II (and region IV) and join together regions I and III at their respective horizons. That may be a valid topology ("valid" in the sense of "consistent if we only consider the topology"), but it doesn't seem to me that it's valid physically, because it's not a solution of the EFE. The solution of the EFE, when extended through the horizon, gives you region II, not region III.

In principle, I have to disagree with you. The EFE are only local equations; likewise the metric tensor is only a local object. So the EFE are agnostic about however you wish to cut a manifold up and glue the pieces together, provided that the gluing is smooth.*

However, ...

Edit: I see the paper actually agrees with me. From the Discussion section, p. 6:

In other words, the solution only works if there is a sheet of lightlike stress-energy with infinite density at $r = r_g$ (i.e., where the "horizon" would be in the standard black hole solution). That doesn't seem physically reasonable to me.

It appears in this case the gluing is not smooth, but has a curvature singularity along the "seam". So yes, this paper fails to really accomplish what it set out to do.

* Since the EFE are second-order PDEs, gluings must be at least $C^2$ in order for the EFE not to notice.

 

[PeterDonis]

the EFE are agnostic about however you wish to cut a manifold up and glue the pieces together, provided that the gluing is smooth.*

Yes, but, as you note, in this particular case the gluing is *not* smooth. I agree that if there were a way to realize the scenario in the paper with a smooth gluing, that would not violate the EFE.

 

[pervect]

The original paper appears to be: The Particle Problem in the General Theory of Relativity
Phys. Rev. 48, 73 – Published 1 July 1935

You can currently find it online at http://dieumsnh.qfb.umich.mx/archivoshistoricosmq/ModernaHist/Einstein1935b.pdf

The writers investigate the possibility of an atomistic theory of matter and electricity which, while excluding singularities of the field, makes use of no other variables than the gμν of the general relativity theory and the ϕμ of the Maxwell theory. By the consideration of a simple example they are led to modify slightly the gravitational equations which then admit regular solutions for the static spherically symmetric case. These solutions involve the mathematical representation of physical space by a space of two identical sheets, a particle being represented by a "bridge" connecting these sheets.

From a superficial reading, Einstein doesn't seem to think of the "bridge" as modelling a black hole. He seems to be imagining two event horizons (r=2m), which he regards as singular, joined together "by a hyperplane in which g vanishes", and he seems to regard this in the context of a different theory with modified field equations. I don't quite follow what he is trying to do, but it doesn't appear to be as if he's trying to describe a black hole.

I have, however, seen the phrase used to describe the geometry of black holes - MTW describes the geometry of a non-charged, non-rotating black hole as a closed, non-traversable, dynamically-evolving Einstein-Rosen bridge, for instance.

So I would say that the general idea of an "Einstein Rosen Bridge" is the geometry of two sheets glued together by a tube, but it was someone other than Einstein that suggested that this geometry had a close relation to the geometry of a Schwarzschild black hole.

 

[bapowell]

There was an early paper (behind paywall, sadly) by Fuller and Wheeler: http://journals.aps.org/pr/abstract/10.1103/PhysRev.128.919 that studies the Einstein-Rosen bridge (and refers to it as such). They studied the opening and closing of the Schwarzschild worm hole and determined that the throat is not open long enough for light to pass through.