General Relativity

[Amr Morsi]

Equivalence principle: is it exact? If yes, then why A. Einstein put more than one mathematical model for the GR in the beginning and waited for the proof? And, what about the negative sign that seems to have reversed the meaning of the Equivalence principle, as asked by many scientists?

[md65536]

Equivalence principle: is it exact? If yes, then why A. Einstein put more than one mathematical model for the GR in the beginning and waited for the proof? And, what about the negative sign that seems to have reversed the meaning of the Equivalence principle, as asked by many scientists?

I dunno but if no one else has an answer I'll stab at it...

 

Yes?

 

There are multiple mathematical models because there are multiple possible "shapes" of gravitational fields.

The typical example of an accelerating rocket is not exactly equivalent to typical gravitational fields, ie. those around spherical bodies.

I believe the rocket would be exactly equivalent to a homogeneous gravitational field maybe?

An observer in a rocket sitting on a (tiny) planet's surface might be able to detect subtle differences in the direction of gravitational force at different locations within the rocket, or the gravitational gradient between the top and bottom of the rocket.

 

I suppose the equivalence principle works exactly for different types of acceleration and corresponding gravitational fields. Gravity on Earth would be equivalent to some imaginable rocket whose parts are not all accelerating exactly the same.

 

 

[ajb]

The Weak Principle of Equivalence states all the laws of motion for freely falling particles are the same as in an unaccelerated reference frame. Mathematically this really means that space-time can be modelled as a smooth manifold. Roughly, in a small enough region around a feely falling particle space-time is flat.

 

The Strong Principle of Equivalence states all the laws of nature are the same in a uniform static gravitational field and the equivalent accelerated reference frame. This implies the weak, but has the stronger condition that in a small enough region in a uniform static gravitational field the laws of nature reduce to that of their special relativistic form. For example this puts constraints on how particles and fields can couple to the gravitational field.

 

I believe all the experimental evidence supports the weak principle.

[IM Egdall]

The Strong Principle of Equivalence states all the laws of nature are the same in a uniform static gravitational field and the equivalent accelerated reference frame.

 

The strong Equivalence Principle (EP) guided Einstein to his theory of general relativity, but is has its limitations. It only takes into account the warping of time, but does not include the warping of space in a gravitational field.

 

Consider the bending of starlight grazing the Sun. Time warp only per the strong EP predicts a bending of 0.875 arc seconds. (This is the same value predicted by Newton's theory.) But space warp predicts another 0.875 arc seconds for a total bending of 1.75 arc seconds.

 

Einstein's early gravitational equations based on the Strong EP included only the warping of time (circa 1912). In his field equations of 1915, he corrected this and included both the warping of time and space; giving the full 1.75 arcsecond prediction.

 

A number of tests confirm the larger value. In what I believe was the most accurate of these tests, a 1975 experiment measured a value of 1.75 ± 0.019 arc seconds. (K. R. Lang, Astrophysical Formulae, Vol. 1, p. 159.)

So the strong EP is a historically important but incomplete model of reality.

 

(Edited for spelling and clarity. )

Edited July 18, 2011 by I ME

[ajb]

The strong Equivalence Principle (EP) guided Einstein to his theory of general relativity, but is has its limitations. It only takes into account the warping of time, but does not include the warping of space in a gravitational field.

 

 

I don't follow why the strong equivalence principle only implies curvature of time? (I doubt you can honestly formulate it that way in general) Can you explain to me why this is so?

[DrRocket]

The Weak Principle of Equivalence states all the laws of motion for freely falling particles are the same as in an unaccelerated reference frame. Mathematically this really means that space-time can be modelled as a smooth manifold. Roughly, in a small enough region around a feely falling particle space-time is flat.

 

The Strong Principle of Equivalence states all the laws of nature are the same in a uniform static gravitational field and the equivalent accelerated reference frame. This implies the weak, but has the stronger condition that in a small enough region in a uniform static gravitational field the laws of nature reduce to that of their special relativistic form. For example this puts constraints on how particles and fields can couple to the gravitational field.

 

I believe all the experimental evidence supports the weak principle.

 

 

The guts of the matter is in your first statement, that spacetime is a pseudo-Riemannian manifold. From that assertion one can proceed.

 

The "equivalence principle", strong or weak, was, as noted by IMe, useful to Einstein in a philosophical way as he struggled to formulate general relativity. But, like "general covariance" the meaning is more than a bit murky until the formulation in terms of pseudo-Riemannian geometry is presented. The principle served its purpose, but now is really extraneous except as it pertains to the history of development of general relativity and its pedagogical value to some people.

 

Things quite often become sticky and obscure when one tries to turn the phycist's approach to physics into something formal and axiomatic. The axiomitization of physics, proposed as a problem by Hilbert in 1900, remains open. Since Hilbert proposed the problem the situation has gotten farther from, not nearer to, resolution, with the advent of both relativity and quantum mechanics.

 

Einstein's approach to divining the laws of nature was not axiomatic, but rather relied on a deep, nearly mystical, insight into how natural laws ought to be formulated. It was not universally successful -- witness his opposition to modern quantum theory and failure to unify general relativity and electrodynamics. But when it worked, it worked very well indeed. This insight was the essence of Einstein's genius. Despite being relatively (pun intended) weak in mathematics, his physical intuition produced spectacular physics that has since been refined and polished by others. The refining and polishing provides clarity and obviates any logical requirement to follow Einstein's original twists and turns to reach a modern understanding of his theories.

[IM Egdall]

I don't follow why the strong equivalence principle only implies curvature of time? (I doubt you can honestly formulate it that way in general) Can you explain to me why this is so?

 

I think it goes something like this: In 1911 Einstein used the EP and a uniformly accelerating elevator with a beam of light traveling parallel to the direction of acceleration to predict gravitational redshift. This is equivalent to gravitational time dilation or the warping of time. Notice how there is no mention of the effect of space warp in this thought experiment.

 

Einstein also used the same thought experiment but with beam of light traveling perpendicular to the direction of acceleration to predict the bending of light in a gravitational field. But the value he got for the bending is only half the value predicted by his 1915 theory of general relativity. THis is becuase the EP only considers the warping of time, while general relativity considers both time and space warp.

 

I learned this from Hans Ohanian in Einstein's Mistakes:

 

"(Einstein's) 1911 calculation of the bending of rays of light, which was based on the Equivalence Principle, yielded a result half as large as the new calculation (in 1915) based on his new theory of gravititation. Einstein understood that this the reason for this discrepency was the new calculation included an extra deflection coming from the warping of space, whereas the 1911 calculation had effectively included only the warping of time . . . For the physicist enclosed in a box . . . the bending of a ray of light in an accelerated box is half as large as the bending in a box at rest in a gravitational field."

 

Einstein's Equivalence Principle is fully compatible with Newton's gravitational theory; and the elevator in free-fall thought experiment takes into account time warp only; ignoring the effects of the warping of space. But if we evaluate Einstein's gravitational field equations for time warp only and ignore the effects of space warp, we get solutions for gravitational fields which are identical to Newton's.

 

Assume a uniform density, non-rotating, uncharged spherical stellar object (ala Schwarzschild geometry). If we then compute the global change in the spacetime interval in the neighborhood of this stellar object including the curvature of time only, we get the following equation from both Newton's and Einstein's gravitational theories:

 

Spacetime Interval in Gravitational Field – Time warp only (Newton and Einstein)

 

ds² = (dx² + dy² + dz²) - (1 – 2GM/r) (dt)²

 

where G is the gravitational constant,

 

M is the mass of a Newtonian star, and

 

r is the distance from the star to a particular point in space.

 

If we include the curvature of time and space, we get the full solution of Einstein's field equations, which no longer agrees with Newton's theory:

 

Spacetime Interval in Gravitational Field – Time and space warp (Einstein only)

 

ds² = (1+2GM/r) (dx² + dy² + dz² ) - (1 – 2GM/r) (dt²)

 

(Derivations for these equations can be found in Schutz, Gravity from the Ground Up, p. 226.

Edited July 20, 2011 by I ME

[ajb]

Ok, taking the metric to be of the form [math]g_{00} = - (1 - \frac{2GM}{r})[/math] is known as the Newtonian limit. I now understand what you mean.

[Amr Morsi]

By the way, if you examined GR Equations, you will find that free falling local frame of reference is not as an inertial frame, even if you applied it on an external basis.

 

This is not the point. The point is Exactness (I am not saying the correctness) of GR and the interpretation of the EP. The negative sign put reverses (or may) the meaning, and this was mentioned (If I am not wrong) in Einstein's Papers.