# Einstein’s equivalence principle for a collapsed star

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The Equivalence Principle says depicting gravity as a field associated with matter is equivalent to picturing it as twisting of space. Is there a space-twisting equivalent of a collapsed star as a shell of condensed matter with interior of intense gravitational field energy (found in the field description to satisfy the Einstein-GR equations)?

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The Equivalence Principle says depicting gravity as a field associated with matter is equivalent to picturing it as twisting of space.
That is not what the equivalence principle says. It says that that gravitational forces are locally indistinguishable from the inertial pseudo-forces that appear under uniform acceleration. The equivalence principle goes a long ways towards motivating general relativity's model of gravity being caused by the curvature of spacetime (not just space!), but does not itself refer to that curvature in any way.

Is there a space-twisting equivalent of a collapsed star as a shell of condensed matter with interior of intense gravitational field energy (found in the field description to satisfy the Einstein-GR equations)?
There is only one solution of the EFE for a static and spherically symmetrical mass distribution, and that is the Schwarzschild spacetime (and Kerr if the rotation is not negligible).

Wikipedia on the strong equivalence principle says this “suggests that gravity is entirely geometrical by nature (that is, the metric alone determines the effect of gravity)”. My formulation is loose, but the question remains.

The Schwarzschild and Kerr solutions are geometric warping of space with weak fields. Schwarzschild’s describes empty space outside a spherical distribution of matter; also it’s the first term in a weak field approximation. The Kerr solution includes rotation. Neither has an interior solution including matter. Kerr closed his remarkably short paper with "it would be desirable to calculate an interior solution to get more insight". The field solutions I’m concerned with have interior matter and ultra-strong field.

PeterDonis
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a collapsed star as a shell of condensed matter with interior of intense gravitational field energy
I don't understand what you're describing here. Are you trying to describe a black hole? A neutron star? The words "a shell of condensed matter with interior of intense gravitational field energy" don't make sense to me.

PeterDonis
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Neither has an interior solution including matter.
That's because, in the general case, there isn't a known closed-form expression for the interior of a body containing matter. The only known closed-form solution I'm aware of is for a sphere of constant density, which is highly unphysical. I can't find a quick online reference, but this solution is described in many GR texts, such as MTW.

Kerr closed his remarkably short paper with "it would be desirable to calculate an interior solution to get more insight".
Numerically (i.e., without achieving a closed-form expression), interior solutions can indeed be calculated, but the calculations require a lot more computer power than existed anywhere in the world when Kerr published his paper (1963, IIRC). The key thing you need is some relationship between the pressure (and other stresses in the material) and the density; in the case of a perfect fluid, with isotropic pressure and no other stresses, this relationship is called the equation of state. This reduces the number of variables to the point where the calculation becomes solvable.

Dale
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Wikipedia on the strong equivalence principle says this “suggests that gravity is entirely geometrical by nature (that is, the metric alone determines the effect of gravity)”. My formulation is loose, but the question remains.
Your formulation is so loose that it is incomprehensible to me.
Is there a space-twisting equivalent of a collapsed star as a shell of condensed matter with interior of intense gravitational field energy (found in the field description to satisfy the Einstein-GR equations)?
It isn't space, it is spacetime. The usual term is curve, not twist. By collapsed star do you mean a neutron star? The gravitational field energy is a frame-dependent thing that has limited applicability. It is only used to solve the EFE in special cases.

With all of that it is hard to decipher what your question is. The wiki quote seems correct to me, but I don't see the relation to your question. Can you try to clarify?

....it is hard to decipher what your question is. The wiki quote seems correct to me, but I don't see the relation to your question. Can you try to clarify?
Of course choose a frame of reference in which gravitational field energy is well-defined and in which to solve the field equations. Then the types of stationary solutions available are not only those of Oppenheimer-Volkoff with monotonic density peaking at the centre, but also solutions with material density in a shell and field energy rising through it to intense levels in the interior. By “intense levels” of field, I mean relativistic levels with energy in mass terms comparable to matter densities, as in Cameron’s neutron star models. Star structure computed like A G Cameron (1959) for neutron equation-of-state and mass under the TOV limit (2.0 solar masses, avoiding worries about collapse on a ‘black hole’) can be found in the form of a matter-shell + field interior.

The neutron EoS is not special, similar solutions exist for degenerate electron EoS (white dwarf stars) and probably exotic quark stars with neutron edges.

My question is – under the Equivalence Principle in strong form (because the field is ultra-strong), is there an equivalent geometric (curved space-time) description for such shell/field stars? It’s a well-posed question, unless you deny that the Einstein GR equations in the chosen reference frame can be satisfied by such structures.

PeterDonis
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solutions with material density in a shell and field energy rising through it to intense levels in the interior
This still doesn't make sense. Are you intending that there is a shell of material and vacuum inside it?

Star structure computed like A G Cameron (1959) for neutron equation-of-state and mass under the TOV limit (2.0 solar masses, avoiding worries about collapse on a ‘black hole’) can be found in the form of a matter-shell + field interior.
Can you give a specific reference?

under the Equivalence Principle in strong form (because the field is ultra-strong), is there an equivalent geometric (curved space-time) description for such shell/field stars?
I'm not sure why you bring in the EP here. The answer to the second clause of your question is that all of the solutions you are referencing are geometric descriptions; they're solutions to the Einstein Field Equation, and that's what a solution to the EFE is.

Dale
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Of course choose a frame of reference in which gravitational field energy is well-defined and in which to solve the field equations. Then the types of stationary solutions available are not only those of Oppenheimer-Volkoff with monotonic density peaking at the centre, but also solutions with material density in a shell and field energy rising through it to intense levels in the interior. By “intense levels” of field, I mean relativistic levels with energy in mass terms comparable to matter densities, as in Cameron’s neutron star models. Star structure computed like A G Cameron (1959) for neutron equation-of-state and mass under the TOV limit (2.0 solar masses, avoiding worries about collapse on a ‘black hole’) can be found in the form of a matter-shell + field interior.
Sounds like you just want the most general spherically symmetric metric. See equation 7.12 here: http://arxiv.org/abs/gr-qc/9712019

My question is – under the Equivalence Principle in strong form (because the field is ultra-strong), is there an equivalent geometric (curved space-time) description for such shell/field stars?
Yes, that is what is described by equation 7.12.

....

This still doesn't make sense. Are you intending that there is a shell of material and vacuum inside it?

I'm conceiving a thickish shell of material plus ordinary stellar matter (non-degenerate, normal density) inside, exerting relatively small pressure.

Can you give a specific reference?

References A G Cameron 1959 Astrophysical Journal, vol. 130, p.884 with recent solutions at tinyurl.com/PIRT-Moscow-15-MarshallWallis

I'm not sure why you bring in the EP here. The answer to the second clause of your question is that all of the solutions you are referencing are geometric descriptions; they're solutions to the Einstein Field Equation, and that's what a solution to the EFE is.
Cameron’s and the preceding Oppenheimer papers of 1939 are field calculations using the GR differential equations. Fitting interior solutions to the exterior Schwarzschild solution (for empty space) does require some differential geometry, but they remain solutions of a field description.

I’m setting a problem for the Equivalence concept – to come up with a geometric description of the shell neutron stars with intense field interiors. It clearly differs from the black-hole solution in that the matter remains in a compressed shell outside the Schwarzschild radius and the interior retains structure dominated by ultra-high gravitational field. A quite plausible configuration in GR (once one accepts that G-fields become repulsive at ultra-high intensity), our contribution is solving for different class of solutions from Cameron’s field solutions.

Dale
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once one accepts that G-fields become repulsive at ultra-high intensity
Please provide a professional scientific reference for this.

PeterDonis
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Cameron’s and the preceding Oppenheimer papers of 1939 are field calculations using the GR differential equations.
The word "field" has many different meanings in physics. One of them is "the spacetime metric", i.e., the geometry of spacetime. That's how the term is used in GR. We call the central equation of GR the Einstein Field Equation, but its solutions are spacetime geometries.

Fitting interior solutions to the exterior Schwarzschild solution (for empty space) does require some differential geometry, but they remain solutions of a field description.
Do you understand that the Einstein Field Equation is an equation in differential geometry?

the shell neutron stars with intense field interiors.
You still have not given a satisfactory description of what you are talking about here.

Please provide a professional scientific reference for this.
Apart from proposals to explain large-scale structure of the universe (‘dark energy’) let me cite

Borissova L.B. and Rabounski D.D..Field, Vacuum and the Mirror Universe. Moscow 2001; latest edn 2010 at https://www.researchgate.net/publication/264525205_Champs_Vide_et_Univers_Miroir [Broken], American research Press, 2010. ISBN, 978-1-59973-123-0
S. S. Gershtein,A. A. Logunov,M. A. Mestvirishvili Repulsive Forces in the Field Theory of Gravity Theoretical and Mathematical Physics Nov. 2005,Volume 145(2) 1604-1618; also A. A. Logunov, The Theory of Gravity, Nauka, Moscow (2001)

A repulsive regime for mega-gravity is of course conceivable on field theory, seeing the huge energy density as exerting a pressure; GR depicts gravitational energy as equivalent to a mass (negative, like binding energy; Cameron 1959) which repels material masses.

The word "field" has many different meanings in physics. One of them is "the spacetime metric", i.e., the geometry of spacetime. That's how the term is used in GR. We call the central equation of GR the Einstein Field Equation, but its solutions are spacetime geometries.
Semantic points should not divert us – theoretical physics uses the GR field equations to model the LIGO gravitational wave pulse from an inferred inspiralling binary system of real masses losing gravitational field energy via travelling waves. Far from asking that of differential geometry, I posed the question: how does the Equivalence Principle apply to neutron star solutions with relativistic gravity-fields, which concretely have field-filled interiors with relatively little (neutron) matter? What is the space-time geometry description, and can it tell us whether the shell/field solutions are preferred to Cameron’s monotonic ones?

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Dale
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S. S. Gershtein,A. A. Logunov,M. A. Mestvirishvili Repulsive Forces in the Field Theory of Gravity Theoretical and Mathematical Physics Nov. 2005,Volume 145(2) 1604-1618; also A. A. Logunov, The Theory of Gravity, Nauka, Moscow (2001)
The first one is not in English, but the second one is also available on arXiv: http://arxiv.org/abs/gr-qc/0505070

It discusses an alternative theory of gravity with which I am not familiar and that does not seem to be widely accepted. However, even in that theory they give an effective metric in EQ 6 which answers your question.

PeterDonis
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Semantic points should not divert us
I agree. But that being the case, why do you keep insisting that "field" means something different from "geometry of spacetime", when in fact both words refer to exactly the same theoretical model?

how does the Equivalence Principle apply to neutron star solutions with relativistic gravity-fields, which concretely have field-filled interiors with relatively little (neutron) matter?
I have no idea what objects you are talking about here. Please give a specific reference that describes such objects, as has already been requested twice now (the links you have provided don't say anything about such objects). We can't possibly answer your question if we don't know what you are talking about.

pervect
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It's possible to present a fairly simple explanation of Einstein's field equations without even mentioning the principle of equivalence. While the principle of equivalence was helpful to Einstein in developing the field equations, it is not necessarily central in describing them. A good but very basic pedagogical paper on the geometrical approach to Einstein's equation is Baez & Bunn's "The Meaning of Einstein's Equation", available online at http://math.ucr.edu/home/baez/einstein/einstein.html.

Some highlights - only the most important points are mentioned, it would be best to read the entire paper.

Einstein's equation can be expressed as a statement about the relative acceleration of very close test particles in free fall. Let us clarify these terms a bit. A test particle' is an idealized point particle with energy and momentum so small that its effects on spacetime curvature are negligible. A particle is said to be in free fall' when its motion is affected by no forces except gravity. In general relativity, a test particle in free fall will trace out a geodesic'. This means that its velocity vector is parallel transported along the curve it traces out in spacetime. A geodesic is the closest thing there is to a straight line in curved spacetime.

Again, all this is easier to visualize in 2d space rather than 4d spacetime. A person walking on a sphere following their nose' will trace out a geodesic -- that is, a great circle. Suppose two people stand side-by-side on the equator and start walking north, both following geodesics. Though they start out walking parallel to each other, the distance between them will gradually start to shrink, until finally they bump into each other at the north pole. If they didn't understand the curved geometry of the sphere, they might think a force' was pulling them together.

Similarly, in general relativity gravity is not really a force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial.
and, given these preliminaries, Einstein's field equations can be interpreted as implying the following:

We promised to state Einstein's equation in plain English, but have not done so yet. Here it is:

Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the
direction at that point, plus the pressure in the
direction, plus the pressure in the [PLAIN]http://math.ucr.edu/home/baez/einstein/img17.gifdirection. [Broken]
As noted in the appendix, though, this simplified interpretation is perhaps a bit over-simple. The actual math requires that one knows what the Riemann and Ricci tensors are, and more importantly, how they transform. The simplified presentation picks out one of the simplest possible cases, and assumes that one can transform a given physical situation into this particularly simple case. But it doesn't describe the needed transformation laws. That requires tensor calculus, and probably a bit of differential geometry, much more advanced topics than the rest of the paper.

Equation (9) will be true if any one component holds in all local inertial coordinate systems. This is a bit like the observation that all of Maxwell's equations are contained in Gauss's law and [PLAIN]http://math.ucr.edu/home/baez/einstein/img104.gif. [Broken] Of course, this is only true if we know how the fields transform under change of coordinates. Here we assume that the transformation laws are known.
It's not entirely clear to me if the OP's description of the EFE's is the same as Baez & Bunn's description. There are two basic possibilities. The two theories predict the same experimental results, in which case they can be regarded as different interpretations of the same theory, or they don't, in which case they are different theories.

I don't think there's enough information at this point to say for sure whether or not the OP is interpreting Einstein's equation differently, or is in fact using some theory of gravity that's not General Relativity.

It's also not clear to me how to overcome the language difficulties, I am hoping that a simple presentation of the geometrical view in geometrical language will be of some help.

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PeterDonis
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In the absence of any further specifics from the OP, this thread is closed.