I What is the Principle of Equivalence and how was it determined?

[loislane]

In principle this is true, yes.
The EP doesn't "equate" those two things in all respects. It just says that, for a region of spacetime that is small enough that the effects of curvature can be ignored, the two things can be treated the same, so you can understand a lot of the physics of "gravitational fields" by looking at the physics in a local inertial frame. For example, the gravitational redshift and gravitational time dilation can be understood purely in terms of the behavior of accelerated clocks in a local inertial frame, without any reference to spacetime curvature.

This kind of procedure is by no means limited to relativity; it's done all the time in physics. In order to understand some particular piece of physics, we ignore other pieces of physics that are, for practical purposes, negligible in the specific circumstances under consideration. If we always had to solve the entire equations of physics to understand one little problem, we'd never get anywhere.

So then you are saying the EP is just an approximation heuristic like it is indeed often used in physics?
I would say that since it is called a principle it is meant to be exact, besides if it is not an exact statement many things in GR wouldn't make much sense, like background independence, but I'm not a physicist so I can't assure what is meant by the EP.

 

[PeterDonis]

So then you are saying the EP is just an approximation heuristic like it is indeed often used in physics?

In the sense that in principle curvature is never exactly zero, you could call the EP an approximation, yes. Any local inertial frame, in which we assume that the EP holds and use the special relativistic laws of physics to make predictions, is in principle an approximation.

However, the EP is not a "heuristic", because the predictions you get when you use it to justify applying the SR laws of physics in a local inertial frame are not heuristic predictions. They are exact predictions using the laws of SR. The only approximation involved is the assumption that, to the accuracy of measurement, curvature effects are negligible, so we expect the predictions using the laws of SR in a local inertial frame to be correct to the accuracy of measurement. Empirically, we find that this is certainly true.

I would say that since it is called a principle it is meant to be exact

I believe the EP can be formulated mathematically as an exact statement, using the concept of "jet space" and "jet bundles". However, it's been quite a while since I looked into this, so I don't have a good reference handy.

if it is not an exact statement many things in GR wouldn't make much sense, like background independence

I'm not sure how any of this relates to background independence.

 

[Dale]

I was obviously referring to the impossibility of physically building an accelerometer that is "arbitrarily small", i.e. as small as one wishes, like as small as an atom, or a proton or the Plack length or smaller(arbitrarily, but finite nothing to do with infinitely). That's what made your assertion not empirically testable.

It still seems like you don't understand what is meant by "arbitrarily small" in this context. Or perhaps you are conflating present technological limitations with fundamental theoretical limitations. Yes, at any point there will always be some limit to what physical regimes we can experimentally probe. That doesn't mean that the theory itself is somehow not empirically testable, nor that statements made within the theory are non physical or non scientific.

If you wish to continue this line of assertions, then please produce a professional reference that explains and supports your characterization of the "arbitrarily small" concept as "out of the scope of physical science" or whatever.

 

[stevendaryl]

So then you are saying the EP is just an approximation heuristic like it is indeed often used in physics?
I would say that since it is called a principle it is meant to be exact, besides if it is not an exact statement many things in GR wouldn't make much sense, like background independence, but I'm not a physicist so I can't assure what is meant by the EP.

Well, the EP may be a heuristic, but it is the basis for a very precise scientific claim, which is that gravity only affects particles and fields through the metric tensor (and its derivatives). There is no separate "force" term for gravity in the equations of motion for particles and fields. What is commonly thought of as the "force" of gravity is actually seen to be the Christoffel coefficients used to describe motion in curvilinear coordinate systems. The equivalence principle tells us that gravity enters into the equations of motion in exactly the same way that fictitious forces such as centrifugal and coriolis forces do. The EP basically tells us that gravity is a manifestation of curved spacetime.

That doesn't uniquely determine General Relativity, because there are other theories of gravity that also satisfy the EP, incuding Newtonian gravity.

 

[loislane]

Well, the EP may be a heuristic, but it is the basis for a very precise scientific claim, which is that gravity only affects particles and fields through the metric tensor (and its derivatives).

What I'm after here is for someone to mathematically justify that the EP serves as the basis for such claim(not entering into the claim itself). Considering not only the metric and it first derivatives, but also the second derivatives necessary to define the Riemann curvature. I just don't see how the EP says anything about the second derivatives of the metric.

What is commonly thought of as the "force" of gravity is actually seen to be the Christoffel coefficients used to describe motion in curvilinear coordinate systems. The equivalence principle tells us that gravity enters into the equations of motion in exactly the same way that fictitious forces such as centrifugal and coriolis forces do.

And this I can accept as the mathematical essence of the EP, I think it only reaches up to the Christoffel symbols, the first derivatives of the metric.

The EP basically tells us that gravity is a manifestation of curved spacetime.

And above I'm suggesting that it says less than that.

That doesn't uniquely determine General Relativity, because there are other theories of gravity that also satisfy the EP, incuding Newtonian gravity.

This would seem to confirm my point since in Newtonian gravity the EP cannot certainly say that gravity is a manifestation of curvature(Newtonian space is euclidean).

 

[stevendaryl]

What I'm after here is for someone to mathematically justify that the EP serves as the basis for such claim(not entering into the claim itself). Considering not only the metric and it first derivatives, but also the second derivatives necessary to define the Riemann curvature. I just don't see how the EP says anything about the second derivatives of the metric.

It doesn't. To apply the EP, you never need anything other than the first derivatives of the metric. In the book Gravitation by Misner, Thorne and Wheeler, the two ideas of GR are summarized by:

1. Gravity tells matter how to move.
2. Matter tells spacetime how to curve.

Point 1, how gravity affects matter, pretty much only involves the first derivatives of the metric tensor--the Christoffel coefficients. If you write down the equations of motion for flat spacetime, then incorporating gravity means replacing ordinary partial derivatives by covariant derivatives (which means sticking Christoffel coefficients in at various places). The second derivatives don't come into play.

On the other hand, point 2, how matter affects gravity, must involve second derivatives (or higher-order derivatives, although these don't appear in GR). The reason for this is that first derivatives of the metric can always be made to vanish locally by choosing an appropriate coordinate system. Obviously, you can't make matter disappear through a coordinate change. So if matter is to affect gravity, then it must be through higher-order derivatives.

This would seem to confirm my point since in Newtonian gravity the EP cannot certainly say that gravity is a manifestation of curvature(Newtonian space is euclidean).

That's not true. If you describe gravity in a coordinate-independent way, you are led to a geometric view of Newtonian gravity, as well. That's not the way it's usually presented, but Newtonian gravity is completely equivalent to a geometric theory in which gravity is a manifestation of spacetime curvature. Going back to my two points about GR, point number 1 is the SAME in both Newtonian gravity (expressed in geometric terms) and GR. It's point number 2 that differs between the two theories. In Newtonian gravity, it's mass (a scalar quantity) that affects gravity, while in GR it's the energy-momentum tensor, which is a tensor.

 

[loislane]

It doesn't. To apply the EP, you never need anything other than the first derivatives of the metric. In the book Gravitation by Misner, Thorne and Wheeler, the two ideas of GR are summarized by:

1. Gravity tells matter how to move.
2. Matter tells spacetime how to curve.

Point 1, how gravity affects matter, pretty much only involves the first derivatives of the metric tensor--the Christoffel coefficients. If you write down the equations of motion for flat spacetime, then incorporating gravity means replacing ordinary partial derivatives by covariant derivatives (which means sticking Christoffel coefficients in at various places). The second derivatives don't come into play.

On the other hand, point 2, how matter affects gravity, must involve second derivatives (or higher-order derivatives, although these don't appear in GR). The reason for this is that first derivatives of the metric can always be made to vanish locally by choosing an appropriate coordinate system. Obviously, you can't make matter disappear through a coordinate change. So if matter is to affect gravity, then it must be through higher-order derivatives.

I don't have that book but I would doubt that any book other than some basic popularization could use such hand-waving arguments. They simply aren't mathematically sustainable. Gravity, that is represented by Riemann curvature in GR cannot afffect matter in a mathematically different way from how matter affects gravity as long as you are talking about the same Riemann curvature in both statements. So the second derivatives are necessarily required if curvature(gravity) is involved.

According to your reasoning about how gravity(curvature according to GR) affects matter you make the presence of curvature depend on coordinate changes(wich you claim correctly that can't be done with matter). That is not mathematically sound when one understands the invariance of the curvature tensor.
When you say:"incorporating gravity means replacing ordinary partial derivatives by covariant derivatives (which means sticking Christoffel coefficients in at various places). The second derivatives don't come into play", it can hardly be more misleading because computing the Riemann curvature, that is gravity in the GR sense involves always second derivatives of the metric(derivatives of the christoffels) not simply sticking Christoffel coefficients.
Your point 1 could amount to what the EP says but that then doesn't include gravity as Riemann curvature.

 

[Dale]

What I'm after here is for someone to mathematically justify that the EP serves as the basis for such claim(not entering into the claim itself). Considering not only the metric and it first derivatives, but also the second derivatives necessary to define the Riemann curvature. I just don't see how the EP says anything about the second derivatives of the metric.

I am not sure what stevendaryl means above, but a good discussion of the EP can be found in chapter 4 here:
http://www.preposterousuniverse.com/grnotes/

 

[stevendaryl]

I don't have that book but I would doubt that any book other than some basic popularization could use such hand-waving arguments.

What argument? The two points I quoted is just a summary. It's not an argument.

They simply aren't mathematically sustainable. Gravity, that is represented by Riemann curvature in GR cannot afffect matter in a mathematically different way from how matter affects gravity as long as you are talking about the same Riemann curvature in both statements.

Let's take Newtonian gravity, for a simpler example than GR.

If you let \Phi be the gravitational potential, then gravity affects matter through the equation:

m \dfrac{d^2 x^j}{dt^2} = - m \dfrac{\partial \Phi}{\partial x^j}

Note: this involves the first derivative of \Phi

Matter affects gravity through the equation:

\sum_j \dfrac{\partial^2 \Phi}{(\partial x^j)^2} = 4 \pi G \rho

where \rho is the mass density. That involves the second derivative of \rho.

GR modifies both of these equations: Instead of the first equation, we have:

m \dfrac{d^2 x^\mu}{d \tau^2} = - m \Gamma^\mu_{\nu \sigma} \dfrac{dx^\nu}{d\tau} \dfrac{dx^\sigma}{d\tau}

where \Gamma^\mu_{\nu \sigma} is constructed from the first derivatives of the metric tensor.

Instead of the second equation, we have:

G_{\mu \nu} = 4 \pi T_{\mu \nu}

where G_{\mu \nu} is constructed from the second derivatives of the metric tensor, and T_{\mu \nu} is the energy-momentum tensor.

So what you're saying is mathematically unsustainable is a feature of both GR and Newtonian gravity. Which is a kind way of saying that you're completely wrong about this.

So the second derivatives are necessarily required if curvature(gravity) is involved.

Curvature does not (directly) influence the motion of particles.

According to your reasoning about how gravity(curvature according to GR) affects matter you make the presence of curvature depend on coordinate changes(which you claim correctly that can't be done with matter).

No, I said that the Christoffel coefficients are affected by coordinate changes. Curvature is covariant under coordinate changes.

That is not mathematically sound when one understands the invariance of the curvature tensor.

Look, there is nothing wrong with asking questions when you don't understand something. But don't pretend that you understand something when you don't.

When you say:"incorporating gravity means replacing ordinary partial derivatives by covariant derivatives (which means sticking Christoffel coefficients in at various places). The second derivatives don't come into play", it can hardly be more misleading because computing the Riemann curvature, that is gravity in the GR sense involves always second derivatives of the metric(derivatives of the christoffels) not simply sticking Christoffel coefficients.

I've explained this as best I can: If you want to compute the path of a particle under the influence of gravity, then you use the geodesic equation. That doesn't involve curvature, it only involves the Christoffel coefficients, which are constructed from the first derivatives of the metric tensor. If you want to compute the effect of matter, momentum and energy on gravity, then the appropriate equation is the field equations, which does involve the curvature tensor (and therefore, second derivatives of the metric tensor).

Your point 1 could amount to what the EP says but that then doesn't include gravity as Riemann curvature.

That's right. The EP basically says that test particles move on geodesics. The EP by itself doesn't say anything about curvature.

The EP is basically about the effect of gravity on the equations of motion of particles and fields. That does not involve curvature. Curvature is involved when you look at the other half of the story: how do particles and fields affect gravity.

 

[stevendaryl]

I've explained this as best I can: If you want to compute the path of a particle under the influence of gravity, then you use the geodesic equation. That doesn't involve curvature, it only involves the Christoffel coefficients, which are constructed from the first derivatives of the metric tensor. If you want to compute the effect of matter, momentum and energy on gravity, then the appropriate equation is the field equations, which does involve the curvature tensor (and therefore, second derivatives of the metric tensor)

There is a sense in which the two questions: how does gravity affect matter and fields, and how does matter and fields affect gravity, aren't actually separate in GR. I believe that it is possible to derive the prediction that test particles follow geodesics from the field equations.

 

[stevendaryl]

I am not sure what stevendaryl means above, but a good discussion of the EP can be found in chapter 4 here:
http://www.preposterousuniverse.com/grnotes/

If you have equations of motion that are valid in flat spacetime (say, for charged particles moving in an electromagnetic field), then the EP basically says that if we replace partial derivatives by the appropriate covariant derivatives (I know that prescription is a little ambiguous, because of operator ordering problems, but it is a good heuristic) then we get equations of motion that are valid in the presence of gravity (at least to the extent that we can treat the gravity as approximately unaffected by the particles and fields under consideration; so we can use the prescription to describe the motion of rocks and light near the Earth, but not to describe the motion of the Moon, which is large enough to have a significant gravitational effect).

 

[atyy]

What I'm after here is for someone to mathematically justify that the EP serves as the basis for such claim(not entering into the claim itself). Considering not only the metric and it first derivatives, but also the second derivatives necessary to define the Riemann curvature. I just don't see how the EP says anything about the second derivatives of the metric.

There are several versions of the EP.

The heuristic one is says something about the local laws of physics, and fails for the "nonlocal" laws of physics, eg. http://arxiv.org/abs/0806.0464.

The non-heuristic one is called (universal) minimal coupling, and it is essentially exact for the known laws of physics, eg. http://arxiv.org/abs/0707.2748.

Although I don't understand it, there is an argument that universal minimal coupling can be derived eg. http://arxiv.org/abs/1007.0435v3 (section 2.2.2).

 

[PeterDonis]

Point 1, how gravity affects matter, pretty much only involves the first derivatives of the metric tensor

This is only true within a local inertial frame. Once you go beyond a local inertial frame, tidal gravity affects the relative motion of neighboring geodesics, and tidal gravity involves second derivatives of the metric tensor.

Curvature does not (directly) influence the motion of particles.

Yes, it does. See above.

What curvature does not affect directly is covariant derivatives at a particular event, as you point out elsewhere. Those are only affected by the connection coefficients, i.e., by first derivatives of the metric. But when you start trying to parallel transport vectors and tensors from one event to another, knowledge of the covariant derivative at one event is not sufficient. You need to know the curvature tensor, because parallel transport is path dependent, and the curvature tensor tells you the path dependence.

The reason the EP doesn't involve curvature is that it only talks about a single event, or more precisely a sufficiently small region of spacetime around a single event. But that doesn't mean curvature doesn't affect the motion of particles at all. It only means it doesn't affect them (to the accuracy of measurement) within a sufficiently small region of spacetime around a single event.

 

[PeterDonis]

If you want to compute the path of a particle under the influence of gravity, then you use the geodesic equation.

But this requires a choice of coordinate chart (and not just within a single local inertial frame--see below). I don't think that is sufficient to support the broader claim that "curvature doesn't affect the motion of particles".

The EP basically says that test particles move on geodesics.

I think this is too broad. The EP only talks about what happens within a local inertial frame.

 

[stevendaryl]

This is only true within a local inertial frame. Once you go beyond a local inertial frame, tidal gravity affects the relative motion of neighboring geodesics, and tidal gravity involves second derivatives of the metric tensor.

If you assume that test particles follow geodesics, then tidal effects follow from that assumption. So you don't need to assume any kind of coupling to second (or higher-order) derivatives of the metric. That's why I inserted the word "directly" in the statement about the effect of gravity on the equations of motion.

What curvature does not affect directly is covariant derivatives at a particular event, as you point out elsewhere. Those are only affected by the connection coefficients, i.e., by first derivatives of the metric. But when you start trying to parallel transport vectors and tensors from one event to another, knowledge of the covariant derivative at one event is not sufficient. You need to know the curvature tensor, because parallel transport is path dependent, and the curvature tensor tells you the path dependence.

What I'm saying is that if you know, for a particular coordinate system in a particular small region of spacetime, what the connection coefficients are at each point in that region, that's all you need to know to predict the motion of test particles or the evolution of small-amplitude fields. Curvature is computable from that knowledge, so it's not an additional piece of information.

 

[stevendaryl]

I think this is too broad. The EP only talks about what happens within a local inertial frame.

Well, if you know what happens in every local inertial frame, then doesn't that imply what happens globally? Under the assumptions that:

1. We're talking about test particles and weak fields whose effect on gravity is negligible, and
2. There are no direct couplings of the equations of motion to curvature or higher-order derivatives of the metric.

Nonminimal coupling can never be ruled out except experimentally, but I would say that if there are nonminimal couplings, that to me means that the EP does not hold for situations in which nonminimal coupling is relevant. Or to put it another way, to me, the impact of the EP is the claim that there is minimal coupling of matter and fields to gravity.

 

[loislane]

Let's take Newtonian gravity, for a simpler example than GR.

If you let \Phi be the gravitational potential, then gravity affects matter through the equation:

m \dfrac{d^2 x^j}{dt^2} = - m \dfrac{\partial \Phi}{\partial x^j}

Note: this involves the first derivative of \Phi

Matter affects gravity through the equation:

\sum_j \dfrac{\partial^2 \Phi}{(\partial x^j)^2} = 4 \pi G \rho

where \rho is the mass density. That involves the second derivative of \rho.

GR modifies both of these equations: Instead of the first equation, we have:

m \dfrac{d^2 x^\mu}{d \tau^2} = - m \Gamma^\mu_{\nu \sigma} \dfrac{dx^\nu}{d\tau} \dfrac{dx^\sigma}{d\tau}

where \Gamma^\mu_{\nu \sigma} is constructed from the first derivatives of the metric tensor.

Instead of the second equation, we have:

G_{\mu \nu} = 4 \pi T_{\mu \nu}

where G_{\mu \nu} is constructed from the second derivatives of the metric tensor, and T_{\mu \nu} is the energy-momentum tensor.

So what you're saying is mathematically unsustainable is a feature of both GR and Newtonian gravity. Which is a kind way of saying that you're completely wrong about this.

You build your point on an identification of the Newtonian gravitational potential with the metric tensor in GR. Without proof this is what I call hand-waving in the mathematical sense.
And certainly I don't pretend to know physics, that's why I am asking in this forum. There's simply some mathematical facts I happen to be acquainted with. But you certainly seem to pretend you do understand everything in this particular issue. If only that were true how fortunate we'd all be. Experience tells me the chances of that are scarce.

 

[stevendaryl]

This is only true within a local inertial frame. Once you go beyond a local inertial frame, tidal gravity affects the relative motion of neighboring geodesics, and tidal gravity involves second derivatives of the metric tensor.

I think that there is a very close analogy with Newtonian gravity. In Newtonian gravity, you have a gravitational potential \Phi, and the path of a test particle is given by:

m \dfrac{d^2 x^j}{dt^2} = -m \partial_j \Phi

The motion of the particle only depends on the first derivative of \Phi

Newtonian gravity certainly has tidal effects, which involve the second derivatives of \Phi. But the existence of tidal effects follows from the above equation of motion, except in the special case in which \nabla \Phi is a constant vector.

 

[stevendaryl]

You build your point on an identification of the Newtonian gravitational potential with the metric tensor in GR. Without proof this is what I call hand-waving in the mathematical sense.

Look, it's either one or the other. Either you actually pick up a GR textbook, and learn that subject, or else you have to settle for hand-wavy arguments. If you're not willing to learn the math, then handwavy is the best you can get. There is no notion of "proof" outside of rigorous reasoning, which you can only learn from actually studying the subject.

And certainly I don't pretend to know physics, that's why I am asking in this forum.

This forum is not appropriate for learning a technical subject. Use a textbook for that.

 

[stevendaryl]

But you certainly seem to pretend you do understand everything in this particular issue.

I don't claim to understand everything, or even most things about GR, but the issues that you are stumbling over are not advanced topics in GR, they are the basics that you learn in the first course on GR.

 

[loislane]

Well, if you know what happens in every local inertial frame, then doesn't that imply what happens globally? Under the assumptions that:

1. We're talking about test particles and weak fields whose effect on gravity is negligible, and
2. There are no direct couplings of the equations of motion to curvature or higher-order derivatives of the metric.

Nonminimal coupling can never be ruled out except experimentally, but I would say that if there are nonminimal couplings, that to me means that the EP does not hold for situations in which nonminimal coupling is relevant. Or to put it another way, to me, the impact of the EP is the claim that there is minimal coupling of matter and fields to gravity.

If you assume minimal couplig from the start you are assuming the EP in exact(not just approximate) form. The problem with that assumption is that as commented by PeterDonis it leads you to a completely coordinate dependent formulation of the geodesic equation.

 

[stevendaryl]

If you assume minimal couplig from the start you are assuming the EP in exact(not just approximate) form.

Exactly. To me, the EP is the claim that gravity effects particles via minimal coupling.

 

[loislane]

I think that there is a very close analogy with Newtonian gravity. In Newtonian gravity, you have a gravitational potential \Phi, and the path of a test particle is given by:

m \dfrac{d^2 x^j}{dt^2} = -m \partial_j \Phi

The motion of the particle only depends on the first derivative of \Phi

Newtonian gravity certainly has tidal effects, which involve the second derivatives of \Phi. But the existence of tidal effects follows from the above equation of motion, except in the special case in which \nabla \Phi is a constant vector.

I think you are relying too much on Newtonian gravity, you do know is not exactly correct, don't you?

 

[PeterDonis]

To me, the EP is the claim that gravity effects particles via minimal coupling.

I agree with this formulation (the only potential quibble I would have would be to say "spacetime geometry" instead of "gravity"--"how spacetime tells matter how to move", so to speak). My previous comments weren't really about the physics but about ordinary language terminology. I agree with everything you have said about the physics.

The problem with that assumption is that as commented by PeterDonis it leads you to a completely coordinate dependent formulation of the geodesic equation.

Christoffel symbols are coordinate dependent, but covariant derivatives are not; they are proper tensorial objects. So the geodesic equation expressed in terms of covariant derivatives is properly covariant.

As I said above, I was not really commenting about the physics; I was commenting about terminology. I thought the phrase "curvature does not affect the motion of particles" might be misleading. But the "minimal coupling" formulation is saying the same thing, just in different words. The physics is the same either way.

 

[stevendaryl]

I think you are relying too much on Newtonian gravity, you do know is not exactly correct, don't you?

The particular points being discussed are true of both Newtonian gravity and General Relativity.

 

[PeterDonis]

I think you are relying too much on Newtonian gravity

No, he's not. He is saying that, if you want more than what has already been said in this thread, you are basically asking us to provide you a textbook on GR in the limited space of a PF thread. That's not going to happen. If you want more details, please consult a textbook.

 

[PeterDonis]

At this point the OP's question has been more than thoroughly answered. Thread closed.