# Meaning of the equivalence principle in General relativity

It is known that equivalence principle is good for understaning of general relativity (GR). This means comparison of the elevator and falling in homogenic gravitational field.

But, I here somewhere something like "in the higher lever of GR, let us forget on principle of equivalence". How it is with this? Does GR does not need equivalence prinicple? Does equivalence principle is contradictory in some higher levels of GR?

Dale
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But, I here somewhere something like "in the higher lever of GR, let us forget on principle of equivalence".

• exponent137
Nugatory
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I hear somewhere something like...

What exactly did you read and exactly where did you read it? Without that information there is no sensible way of answering your question - we don't know whether you read something incorrect or you read something correct and misunderstood.

• exponent137
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stevendaryl
Staff Emeritus
It is known that equivalence principle is good for understaning of general relativity (GR). This means comparison of the elevator and falling in homogenic gravitational field.

But, I here somewhere something like "in the higher lever of GR, let us forget on principle of equivalence". How it is with this? Does GR does not need equivalence prinicple? Does equivalence principle is contradictory in some higher levels of GR?

Here's the way I think of it: When you are on board an accelerating rocket (or if you are using a rotating coordinate system), there are apparent forces that disappear when using Cartesian coordinates. These include "g-forces", "centrifugal forces" and "coriolis forces". If you analyze these "fictitious forces" or "inertial forces" or whatever you want to call them, you will find that they are due to the use of curvilinear coordinates. (Technically, these forces involve the connection coefficients, which describe how the basis vectors in curvilinear coordinates vary from point to point). The equivalence principle is simply the hypothesis that the force of gravity is an inertial force of exactly the same type as centrifugal forces--that is, it's a manifestation of using curvilinear coordinates.

That insight--that there is no actual "force" of gravity, but that there are only terms involving connection coefficients, is baked into General Relativity. I don't think that the Equivalence Principle plays any role beyond being the inspiration for that insight.

• Ravi Mohan, pervect, vanhees71 and 1 other person
Here's the way I think of it: When you are on board an accelerating rocket (or if you are using a rotating coordinate system), there are apparent forces that disappear when using Cartesian coordinates. These include "g-forces", "centrifugal forces" and "coriolis forces". If you analyze these "fictitious forces" or "inertial forces" or whatever you want to call them, you will find that they are due to the use of curvilinear coordinates. (Technically, these forces involve the connection coefficients, which describe how the basis vectors in curvilinear coordinates vary from point to point). The equivalence principle is simply the hypothesis that the force of gravity is an inertial force of exactly the same type as centrifugal forces--that is, it's a manifestation of using curvilinear coordinates.

That insight--that there is no actual "force" of gravity, but that there are only terms involving connection coefficients, is baked into General Relativity. I don't think that the Equivalence Principle plays any role beyond being the inspiration for that insight.

Are there any examples where Principle of equivalence (PE) is not enough for calculation of GR phenomena? That means that, are there examples where curved space-time cannot be built up from patches of small almost uncurved space-times?

PE is important also because it is possible to simulate curved space-time with elevator in uncurved space-time. Are there any examples where curved space-time cannot be simulated with uncurved space-time?

stevendaryl
Staff Emeritus
Are there any examples where Principle of equivalence (PE) is not enough for calculation of GR phenomena? That means that, are there examples where curved space-time cannot be built up from patches of small almost uncurved space-times?

PE is important also because it is possible to simulate curved space-time with elevator in uncurved space-time. Are there any examples where curved space-time cannot be simulated with uncurved space-time?

Well, the details of how gravity varies with location cannot be imitated by using accelerated reference frames. For example, gravity near a planet decreases with distance from the center according to the inverse-square law, but that wouldn't be true for the fictitious gravitational force inside an accelerated rocket.

But in the limit of very small regions of spacetime, so small that there is negligible change in gravity within the region, it can be approximated by an accelerated frame.

• exponent137
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PE is important also because it is possible to simulate curved space-time with elevator in uncurved space-time. Are there any examples where curved space-time cannot be simulated with uncurved space-time?
it's the other way around. There are no situations in which curved space-time can be exactly simulated by uncurved spacetime (although you can often get a pretty good approximation by neglecting the tidal effects that are always present in curved spacetime but not flat spacetime).

Even the equivalence between standing on the surface of the earth and in a spaceship accelerating at 1G is an approximation - sufficiently accurate measurements would find tidal effects on earth but not in the ship. For example, if I hold two objects in my outstretched arms and drop them, they will follow parallel paths to the floor on the ship but slightly converging paths on earth.

• vanhees71 and exponent137
Well, the details of how gravity varies with location cannot be imitated by using accelerated reference frames. For example, gravity near a planet decreases with distance from the center according to the inverse-square law, but that wouldn't be true for the fictitious gravitational force inside an accelerated rocket.

But in the limit of very small regions of spacetime, so small that there is negligible change in gravity within the region, it can be approximated by an accelerated frame.
I thought very small regions of spacetime, this means differentialy small ones.
Does this anyway means that tidal forces need new math which cannot be covered by very small accelerated rockets?

Nugatory
Mentor
I thought very small regions of spacetime, this means differentialy small ones.
There's no such thing as "differentially small" when dealing with tidal effects. Tidal effects appear when we consider the trajectories of two objects in slightly different positions, meaning that there is always a finite distance between them and no patch covering both can be made arbitrarily small.
Does this anyway means that tidal forces need new math which cannot be covered by very small accelerated rockets?
Tidal forces don't need any new math. We've used Newtonian gravity for centuries to calculate tidal effects in the weak-field approximation, and we have general relativity any time that we need an exact calculation.
It's never been possible, using either GR or Newtonian mechanics, to analyze any problem in terms of small accelerated rockets. For example, consider two people on opposite sides of the earth - they're accelerating in opposite directions at 1G, yet the distance between them remains constant. There's no way of doing that with rockets.

It sounds as if you are giving the equivalence principle much more weight than it deserves - it's a powerful analogy for how GR models gravity, but it's not a fundamental part of the theory. Stevendaryl says it well:
That insight--that there is no actual "force" of gravity, but that there are only terms involving connection coefficients, is baked into General Relativity. I don't think that the Equivalence Principle plays any role beyond being the inspiration for that insight.

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• exponent137
stevendaryl
Staff Emeritus
I thought very small regions of spacetime, this means differentialy small ones.
Does this anyway means that tidal forces need new math which cannot be covered by very small accelerated rockets?

I'm not sure exactly what kind of answer you're looking for, but the situation with the equivalence principle and General Relativity is very similar to the situation with trying to approximate a curved surface by flat pieces.

If you are trying to construct a sphere using flat pieces of cardboard, you can't do it, exactly. But you can approximate it. A soccer ball, for example, approximates a sphere using flat hexagons and pentagons. Using more pieces, you can get something whose shape is closer and closer to a sphere. Using flat pieces, you'll never get something that is exactly like a sphere, but for certain purposes, such as computing distances, the approximation might be good enough.

It's similar with General Relativity. You can approximate spacetime by using lots of little regions that are treated as "flat" (no tidal forces).

• exponent137
That insight--that there is no actual "force" of gravity, but that there are only terms involving connection coefficients, is baked into General Relativity. I don't think that the Equivalence Principle plays any role beyond being the inspiration for that insight.

Isn't the equivalence principle used to formulate laws of physics in curved spacetime by the Principle of Minimal Coupling, so in that sense is more than just an inspiration ?

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• exponent137
vanhees71
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The equivalence principle (usually the weak one) is used to heuristically motivate General Relativity, and it was the heuristics used by Einstein, and it took even him 10 years to get the theory straight!

We are in the lucky position that Einstein has done this work and can now very easily formulate the equivalence principle more precisely by using GR itself to do so. I'd formulate it in the following way: spacetime is described by a pseudo-Riemannian manifold with a fundamental form ("pseudo-metric") of signature (1,3) (or (3,1) if you prefer the east-coast convention as most GR guys do). This implies that at any space-time point there is a coordinate system, where in this point, i.e., locally the laws of special relativity hold. In other words, the tangent space on each spacetime point is a Minkowski space which admits the construction of local inertial frames (these are the free falling non-rotating local reference frames).

It's very important to stress the locality in the above prescription of the equivalence principle. Forgetting it gives rise to a lot of misunderstandings and quibbles. E.g., there's often the question, whether a free-falling charge emits electromagnetic radiation or not. Superficially you argue with the equivalence principle that you can always find a momentary rest frame of the particle which is locally inertial, and thus you have just a charge at rest which doesn't radiate. In fact that argument is flawed, because the question is about the charged particle including its entire electromagnetic field around it, and this is a highly non-local object, and you can't argue with the equivalence principle at all, but you have to solve Maxwell's equations in the given spacetime where the particle is in free fall in (i.e., moving along a geodesic world line).

• exponent137