Gravity as a force and as a curvature

[stevendaryl]

The Lagrangian you've written down is just the Lagrangian for a point particle of mass $m$ in a background metric $g_{\mu \nu}$, sure. But if $g_{\mu \nu}$ is curved, your Lagrangian is missing a term: the Einstein-Hilbert term $R g_{\mu \nu}$ (with some constant factor or other that depends on the units).

What do you mean "missing"? I'm specifically asking about the consistency without any terms coupling the particle motion to curvature, where the metric (and curvature) is non-dynamic.

If you don't include the missing term, your theory isn't complete; it doesn't constrain $g_{\mu \nu}$.

That's what I meant by "background". It's an arbitrary, fixed choice, that is unconstrained by the motion of particles. Yes, such a theory would not allow you to deduce the curvature, so it would certainly be incomplete in that sense. It's the same sort of thing as considering the 2D motions of point particles on the surface of a sphere. The shape of the sphere affects the motion of particles, but the motion of particles does not affect the shape of the sphere.

So what constrains it?

SR doesn't answer that question. That's what GR is for.

In other words, you're assuming the conclusion you're supposed to be proving, that "SR" allows curved spacetime.

I'm assuming that SR in curved spacetime, in which the metric is a "background" field, unaffected by particle motion, is a consistent theory. As you point out, it's an incomplete theory. To address that incompleteness requires going all the way to GR.

Maybe a better way of putting my objection to this would be to say that we appear to have a disagreement about what "SR" means:

* You think "SR" means "any theory that doesn't use the Einstein Field Equation".

No, I would say any theory that is locally Lorentzian (has the metric (+---) in which the metric is unaffected by mass/energy.

Or perhaps a better phrasing would be "any theory that doesn't include a dynamical equation for the metric". And then you simply ignore any issues about whether the theory is complete.

I'm not ignoring it--I'm explicitly saying that it's incomplete. But you said it was inconsistent, which doesn't seem to be true.

 

[PeterDonis]

What do you mean "missing"? I'm specifically asking about the consistency without any terms coupling the particle motion to curvature, where the metric (and curvature) is non-dynamic.

And then we can argue about what "consistency" means, and whether it requires completeness. But my real beef is with the use of the term "SR"; I'm not as much concerned about consistency or completeness, so I'll drop my claim about consistency and just focus on the use of the term "SR".

No, I would say any theory that is locally Lorentzian (has the metric (+---) in which the metric is unaffected by mass/energy.

In other words, you think "SR" is an appropriate name for a theory in which there are no global inertial frames, no invariance under global Lorentz transformations, where parallel transport is path dependent, etc.? That seems like a gross abuse of terminology to me; it would be like calling the intrinsic geometry of the 2-sphere "Euclidean" because, well, it's locally Euclidean and it's not affected by any other features of our theory.

 

[stevendaryl]

And then we can argue about what "consistency" means, and whether it requires completeness. But my real beef is with the use of the term "SR"; I'm not as much concerned about consistency or completeness, so I'll drop my claim about consistency and just focus on the use of the term "SR".

Everyone agrees that SR was formulated for flat spacetime. Einstein did not consider spacetime curvature in developing SR. Everyone agrees with that. So if you want to say that "SR in curved spacetime is an oxymoron", fine. That's not what you said, though. I was specifically trying to answer the question what would it mean to generalize SR to curved spacetime. Once you generalize something, you no longer have the same thing. So yes, it's no longer SR. However, when it comes to the actual predictions of SR, the predictions don't depend on spacetime being flat. If that were the case, then none of the equations of SR would have any relevance in our world. Even if there are no global inertial reference frames, then we can still have approximately inertial local reference frames, and those are good enough for the predictions of SR. So "SR generalized to curved spacetime" is simply making use of local reference frames in a systematic way.

In the same way, there probably is a natural generalization of Newtonian physics to curved space (I'm not sure about curved spacetime). You don't need to assume that space is flat, it is sufficient, to use Newtonian physics, to assume that in a small enough region, spatial curvature is unimportant to the predictions of that theory.

 

[stevendaryl]

In other words, you think "SR" is an appropriate name for a theory in which there are no global inertial frames, no invariance under global Lorentz transformations, where parallel transport is path dependent, etc.?

Yes, if it's true that in a small enough region of spacetime, we can find approximately inertial frames, we have approximate invariance under Lorentz transformations, and parallel transport is approximately path-independent, and the speed of light is approximately the same in all approximate inertial frames.

We're talking about a generalization, not the original theory. If you generalize, you no longer have the original theory. It doesn't make sense to complain that things are different in the generalization--if that weren't the case, it wouldn't be a generalization.

That seems like a gross abuse of terminology to me; it would be like calling the intrinsic geometry of the 2-sphere "Euclidean" because, well, it's locally Euclidean and it's not affected by any other features of our theory.

For certain purposes, that's appropriate. It depends on which features of Euclidean geometry are important for whatever it is you are doing.

Look, relativistic physics has two parts: 1. An assumption about the geometry of spacetime, and 2. equations of motion for particles and fields moving and evolving on that geometry. If you are talking about generalizing the theory to curved spacetime, it seems to me that the only meaning that makes any sense would be to change 1. and leave 2. alone (to the extent possible).

 

[PeterDonis]

Yes, if it's true that in a small enough region of spacetime, we can find approximately inertial frames, we have approximate invariance under Lorentz transformations, and parallel transport is approximately path-independent, and the speed of light is approximately the same in all approximate inertial frames.

In other words, you think the only difference between "SR" and "GR" is whether or not we explicity include a dynamical equation for the metric. I see. I completely disagree, but I see.

Also, we're not talking about what to call a "small enough" region of spacetime, or what we should call the theory that describes it. We're talking about what to call the entire, curved spacetime, and what we should call the theory that describes *that*. I'm not disputing that we can use SR locally in a curved spacetime; I'm just disputing that the theory we use to describe the entire curved spacetime, globally, can be called "SR".

We're talking about a generalization, not the original theory. If you generalize, you no longer have the original theory.

And therefore, to avoid confusion, you don't call it by the same name as the original theory. That's why we invented a new name, General Relativity, for the generalization of special relativity to curved spacetime.

It doesn't make sense to complain that things are different in the generalization

That's not what I'm complaining about. I'm complaining about calling the generalization, which you admit is different from the original theory, by the same name as the original theory.

For certain purposes, that's appropriate.

In other words, you think that for certain purposes, it's appropriate to call the geometry of the Earth's surface--not a small piece of the Earth's surface, but the Earth's surface as a whole--"Euclidean". Again, this seems like a gross abuse of terminology to me.

Look, relativistic physics has two parts: 1. An assumption about the geometry of spacetime, and 2. equations of motion for particles and fields moving and evolving on that geometry. If you are talking about generalizing the theory to curved spacetime, it seems to me that the only meaning that makes any sense would be to change 1. and leave 2. alone (to the extent possible).

Which is exactly what GR does; you allow the geometry to be curved, but you keep all the non-gravitational equations the same except for the minimal changes needed to deal with curvature. (Basically this means changing partial derivatives to covariant derivatives and including $\sqrt{-g}$ in the integration measure.)

I'm not disputing anything about the physics of curved spacetime. I'm only disputing the use of the term "SR" to describe a theory set in a curved spacetime. As far as I can tell, pretty much everybody's usage is that once you're using curved spacetime, you're doing GR, not SR. For example, MTW spends a few chapters on what they call "Special Relativity", which are entirely set in flat spacetime. Then they spend a number of chapters on what they call "General Relativity" doing nothing but talking about how to generalize the geometric description of physics to curved spacetime, before they even mention the Einstein Field Equation.

Can you find any mainstream sources that adopt your usage instead of MTW's usage? That is, they talk about a theory set in curved spacetime, but without including any dynamics for the metric, and they use the term "Special Relativity" to describe such a theory? Remember I'm not talking about the local geometry of a small patch of a curved spacetime; I'm talking about the global geometry of the curved spacetime as a whole. (I'd also be curious to see if you can find any mainstream sources that call the global geometry of a 2-sphere "Euclidean".)

 

[PeterDonis]

So if you want to say that "SR in curved spacetime is an oxymoron", fine. That's not what you said, though.

You're right, it isn't. What I said was that "SR" is not an appropriate name for a theory set in curved spacetime. More precisely, it's not an appropriate name for a theory that actually deals with curved spacetime as a whole, globally. See further comments below.

I was specifically trying to answer the question what would it mean to generalize SR to curved spacetime. Once you generalize something, you no longer have the same thing. So yes, it's no longer SR.

Then we shouldn't call it by that name.

Even if there are no global inertial reference frames, then we can still have approximately inertial local reference frames, and those are good enough for the predictions of SR.

Agreed. Once again, I'm not disputing that SR is valid locally even if spacetime is globally curved.

So "SR generalized to curved spacetime" is simply making use of local reference frames in a systematic way.

Wait, what? SR can't describe a curved spacetime; it can only describe a flat spacetime. So the only thing you can use SR for in a curved spacetime is to describe a small local patch. That's not "SR generalized to curved spacetime"; it's "SR used in the only way it can be used in a curved spacetime". Using local reference frames does not actually include any effects of curvature; it just restricts everything to a small enough region that those effects can be ignored.

"SR generalized to a curved spacetime" is what you described in your other post, which I responded to just now; it means actually including the effects of curvature in the equations, not ignoring them. As I said in my previous post, it seems like the generally accepted name for such a theory is "GR".

Here's yet another way of looking at it. GR, as a branch of physics, has two parts: (1) finding useful solutions of the Einstein Field Equation, i.e., finding physically interesting curved spacetimes; (2) investigating the properties of a given solution--how physics looks in a particular curved spacetime. What you are calling "SR generalized to curved spacetime" is just part #2 of GR.

 

[stevendaryl]

In other words, you think the only difference between "SR" and "GR" is whether or not we explicity include a dynamical equation for the metric. I see. I completely disagree, but I see.

No, there are two differences between SR and GR: (1) GR takes place in curved spacetime, and (2) GR describes how matter/energy affects spacetime curvature. I'm saying that one could imagine taking the first step without the second, and that step could be called either SR + curvature, or GR - field equations. Whether you consider the intermediate step more like SR or more like GR is a matter of opinion. It's not GR in that there is no gravity, in the sense of an interaction between objects.

 

[stevendaryl]

Which is exactly what GR does; you allow the geometry to be curved, but you keep all the non-gravitational equations the same except for the minimal changes needed to deal with curvature. (Basically this means changing partial derivatives to covariant derivatives and including $\sqrt{-g}$ in the integration measure.)

I would say that GR is more ambitious than SR + curved spacetime. It also attempts to describe the origin and dynamics of curvature. The extra term does that.

Generalizing SR to curved spacetime can be thought of as simply DROPPING an assumption, which is that the metric is constant everywhere. Going to GR ADDS an assumption, which is an assumption about the interaction between mass/energy and spacetime curvature. So I think that GR is not simply the generalization of SR to curved spacetime.

 

[TrickyDicky]

I would say that GR is more ambitious than SR + curved spacetime. It also attempts to describe the origin and dynamics of curvature. The extra term does that.

Generalizing SR to curved spacetime can be thought of as simply DROPPING an assumption, which is that the metric is constant everywhere. Going to GR ADDS an assumption, which is an assumption about the interaction between mass/energy and spacetime curvature. So I think that GR is not simply the generalization of SR to curved spacetime.

I think this is a valid distinction. in fact originally Einstein was seeking simply the generalization of SR part. But what he found was that GR came with the dynamical equations.

Of course it is debatable what the use of a generalization of SR that leaves the gravitational interaction out would be.

 

[PeterDonis]

I'm saying that one could imagine taking the first step without the second

In an abstract, theoretical sense, yes, I agree that you could. However, as soon as you ask the question, *which* curved spacetimes are you going to use in this theory, how do you answer that question without already knowing the rest of GR, i.e., the Einstein Field Equation? Every curved spacetime we know of was obtained by looking for solutions of the EFE. Even if you think of the EFE as a sort of scaffolding that you throw away after it's been used to find your curved spacetime for you, it's still a part of the theory; you still need it to do physics.

that step could be called either SR + curvature, or GR - field equations. Whether you consider the intermediate step more like SR or more like GR is a matter of opinion.

I agree with this, since it's basically a question of terminology, and all questions of terminology come down to matters of opinion.

It's not GR in that there is no gravity, in the sense of an interaction between objects.

This seems like another weird use of words to me. Suppose we're using Schwarzschild spacetime as our curved spacetime. Even if we eliminate all discussion of the fact that this spacetime is a solution of the EFE, there's still gravity present: roughly speaking, geodesics in this spacetime "bend inward" towards the center. It's true that there's nothing requiring us to think of this "gravity" as "an interaction with the central object", but that's true in what you are calling "GR" (i.e., including the EFE) as well. "Gravity" is just a manifestation of spacetime curvature in GR (i.e., GR with the EFE); it's not an "interaction between objects".

 

[PeterDonis]

originally Einstein was seeking simply the generalization of SR part.

If you mean, he was originally seeking the generalization of SR to curved spacetime, no, that's not correct. He was originally seeking a way to incorporate gravity into SR. His first theoretical efforts along those lines were based on the equivalence principle, starting with what he called the "happiest thought" of his life in 1907, that a person in free fall will not feel his own weight. At that time, I believe he thought he could mock up a theory of gravity in general by emulating it with acceleration; he hadn't yet realized that more was needed.

At that point he wasn't even thinking in terms of spacetime at all; the spacetime concept came from his old professor, Minkowski, and Einstein took several years to even see the value of it (possibly he was influenced by the fact that Minkowski called him a "lazy dog" while he was in school and said he would never amount to anything in physics). It was only after those several years that he realized that the geometric interpretation was a fruitful line of inquiry, and got Marcel Grossman to teach him Riemannian geometry. His efforts from then on were focused on getting the right field equation; there was never, as far as I can tell, any point at which he was working on a theory that just added curvature to SR, without also including the dynamics that governed the curvature.

 

[TrickyDicky]

If you mean, he was originally seeking the generalization of SR to curved spacetime, no, that's not correct. He was originally seeking a way to incorporate gravity into SR...
... there was never, as far as I can tell, any point at which he was working on a theory that just added curvature to SR, without also including the dynamics that governed the curvature.

I wasn't thinking as far back when I said originally, I thought obvious that I meant from the time he associated gravity to curvature and borrowed from Minkowski the Lorentzian manifold formalism, which happened around 1912.
As late as October 1915 as commented in a previous post he still didn't have the right EFE so I don't think he could foresee much about the dynamics derived from it. He was seeking a purely geometrical theory, not a dynamic field theory in the sense it is now considered.

 

[PeterDonis]

As late as October 1915 as commented in a previous post he still didn't have the right EFE

Right; he didn't get it until November 1915 when he realized he needed to add the trace term.

so I don't think he could foresee much about the dynamics derived from it.

Actually, he could already foresee a lot. Even the incorrect EFE he had in October (which was actually the one he had had as early as 1913, IIRC, but had dropped and then come back to), which was basically $R_{ab} = T_{ab}$ (i.e., everything but the trace term), was enough to tell him that Ricci curvature only, not Weyl curvature, was what coupled to the SET, which already gives you a lot of the right features of the dynamics.

But however that may be, this...

He was seeking a purely geometrical theory, not a dynamic field theory in the sense it is now considered.

...is just wrong; the fact that he didn't yet have the exactly correct dynamics does not mean he wasn't looking for a theory that included dynamics. If he had just been looking for a purely geometrical theory, he had one in 1913. Why didn't he publish it then and say it was done? Because he wasn't satisfied that he had the right dynamics, i.e., the right field equation. And when did he actually publish GR? When he knew he had the right dynamics.

 

[TrickyDicky]

Actually, he could already foresee a lot. Even the incorrect EFE he had in October (which was actually the one he had had as early as 1913, IIRC, but had dropped and then come back to), which was basically $R_{ab} = T_{ab}$ (i.e., everything but the trace term), was enough to tell him that Ricci curvature only, not Weyl curvature, was what coupled to the SET, which already gives you a lot of the right features of the dynamics.

But however that may be, this...



...is just wrong; the fact that he didn't yet have the exactly correct dynamics does not mean he wasn't looking for a theory that included dynamics. If he had just been looking for a purely geometrical theory, he had one in 1913. Why didn't he publish it then and say it was done? Because he wasn't satisfied that he had the right dynamics, i.e., the right field equation. And when did he actually publish GR? When he knew he had the right dynamics.

It seems to me you are mixing accurate with inaccurate historical facts. From 1913 to 1915 Einstein followed a wrong path due precisely to his not acceptance of a dynamical theory of relativity in the way we know it currently of background Independence, on the ground of what is called "the hole argument". He realized his error late in 1915.
If you google "hole argument" you'll find plenty of documents about it.

 

[PeterDonis]

From 1913 to 1915 Einstein followed a wrong path due precisely to his not acceptance of a dynamical theory of relativity in the way we know it currently of background Independence, on the ground of what is called "the hole argument".

I've bolded the crucial phrase; not accepting a dynamical theory "in the way we know it currently" is *not* the same as not looking for a dynamical theory at all. He never stopped looking for a dynamical theory; he was always looking for field equations. He just switched from looking for generally covariant field equations to looking for field equations based on different criteria (because, as you say, he was for a while convinced by the "hole argument" that general covariance was the wrong criterion); then he switched back.

 

[russelljbarry15]

The fact mass bends space is the right answer and photons are massless, the light is just following what it thinks is a straight line. You have to solve Einsteins field equations using the Schwarzschild Solution. You can look up the metrics for this solution. The deflection is 4m/r where r is the closest pass to the center of the mass . The geometric mass m = GM/c^2 where G is Newtons gravitational constant, and M is the mass of the body it passes and of course c is the speed of light. I have not seen this done using Newtons equations although that does not mean someone has not done it, since Einsteins Field equations are based on Newtons gravitational equation for the gravitational potential. The Newton equation you would have to use is (hate the math editor, so I will write it out) (del)^2(phi)=4(pi)G(rho). (phi) is the gravitational potential and (rho) is the mass density. For slow moving particles Einsteins equations give this equation. But we are working with light so we are in the relativistic frame, so this equation will not work, without an error. Newtons equation does not take into account action at a distance. In other words in Newtons equation gravity implies an infinite speed of signal transmission or in other words gravity acts instantaneously which is not compatible with the principle of relativity. This is the straight answer to your question. In the case of the sun, a deflection of 1.75 in. of arc is predicted. This was off the top of my head so you might want to check it. And I do not think you can use Newtons equations, as I have shown they do not give the right prediction for the time it takes for gravity to act on another body.

 

[Dale]

Terminology does matter. It is how we communicate. If I use a word to mean one thing and you use the same word to mean another then we will talk but never understand.

You cannot communicate at all without agreeing on terminology.

However, I do take your point about equations. The key equation of GR is the Einstein field equation:
$G_{\mu\nu}+g_{\mu\nu}\Lambda=8\pi T_{\mu\nu}$

That equation describes gravity and curvature. There are many accelerations that have nothing to do with the EFE, and those are handled by SR.

 

[tiny-tim]

SR can handle … accelerating observers … See this FAQ for details:
http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

i don't see anything in that FAQ about accelerating observers …

the final section about the uniformly accelerating rocket doesn't consider an observer in the rocket (at least, not an observer of anything that isn't in his own tiny rocket)

(i don't dispute that SR can handle observers who are rotating but not accelerating, ie rotating relative to an inertial frame)

 

[Dale]

In SR "according to the observer" is just shorthand for "according to a reference frame where the observer is always at rest". If you can handle a non-inertial frame then you can handle a non-inertial observer simply by choosing any observer at rest in that frame. The SR accelerating frame is the Rindler frame and Rindler observers undergo continuous proper acceleration while remaining at rest in the Rindler frame.

Basically, you can use all of the tensors from GR in SR if you like. GR does not have a monopoly on tensors. The key distinction between GR and SR is not tensors, it is the EFE.

 

[WannabeNewton]

(i don't dispute that SR can handle observers who are rotating but not accelerating, ie rotating relative to an inertial frame)

Rotation is absolute. I don't need an inertial frame to talk about whether or not I'm rotating. I can both physically and mathematically characterize rotation entirely independent of any inertial frame or coordinate system. Furthermore we can easily describe observers at rest in a rotating frame in SR who have acting on them centrifugal forces.

Like I said, SR is a generalization of non-gravitational Newtonian mechanics. Would you argue that Newtonian mechanics can't handle accelerating or rotating frames? I would think not.

 

[Nugatory]

i don't see anything in that FAQ about accelerating observers …

Hmmm... yes, and that's an argument for fixing the FAQ. The essential difference between SR and GR is that GR works in non-flat spacetimes (curvature tensor doesn't vanish, regardless of course of choice of coordinates) as well as flat spacetimes (curvature tensor does vanish, again regardless of choice of coordinates). Historically, however, this fact has been obscured for several reasons:
- The equivalence principle established an intuitive connection between acceleration and gravitation before the formulation of GR.
- Before the formulation of GR, there was very little motivation to distinguish between coordinate systems that were funny because of how they were defined (for example, the non-intertial Rindler coordinates that lead to a horizon in a perfectly ordinary flat spacetime that is just as easily spanned by by bone-stock Minkowski x/y/z/t coordnates) as opposed to funny because the underlying spacetime curvature effects.
- SR is nearly always explained using inertial frames because they're generally simpler.

 

[Naty1]

PeterDonis: post #43 You posted...

...was enough to tell {Einstein} that Ricci curvature only, not Weyl curvature, was what coupled to the SET, which already gives you a lot of the right features of the dynamics.

ok, the Ricci tensor measures the kind of curvature that is produced by local masses... the relative volume change of a geodesic ball etc,etc.

How does one obtain/derive that physical insight? I know its true, but so far the insight eludes me.

 

[PeterDonis]

the Ricci tensor measures the kind of curvature that is produced by local masses... the relative volume change of a geodesic ball etc,etc.

Yes. This is just a matter of geometry; see below.

How does one obtain/derive that physical insight? I know its true, but so far the insight eludes me.

I'm not sure exactly what thought process led Einstein to the (incorrect but close) 1913 field equation, $R_{ab} = T_{ab}$. But the definition of the Ricci tensor is purely geometric; if you've accepted that spacetime is a geometric object, it's just a simple geometric fact that the Ricci tensor is what describes things like the volume change of a geodesic ball. And the fact that that volume change should be driven by the presence of matter is just Newtonian gravity: gravity "pulls" geodesics inwards towards the center of the gravitating mass. So I think there's a fairly straightforward line of reasoning that gets you to some close relationship between the Ricci tensor and the stress-energy tensor.

 

[WannabeNewton]

How does one obtain/derive that physical insight? I know its true, but so far the insight eludes me.

The Ricci tensor completely determines the evolution of the expansion scalar of a shear-free, twist-free time-like congruence. The expansion scalar of course is nothing more than the volume change of a sphere Lie transported along an integral curve of the congruence. See Raychaudhuri's equation.

 

[Naty1]

See Raychaudhuri's equation.

ok, that helps...no wonder it did not seem 'obvious'...a bit beyond my ability to 'intuit' without such mathematical hipper dipper.

For those interested:

http://en.wikipedia.org/wiki/Raychaudhuri's_equation#Intuitive_significance

 

[Naty1]

One of the OP's questions:

(b) Concerning the red shift and blue shift: It shows that photons coming out of gravitation zone, looses energy and hence the wavelength becomes longer and causes red shift and vice-verse causing the blue shift. Here the energy of the photon is affected by gravity while IN THE ABOVE CASE IT IS NOT. AM I GETTING SOMETHING WRONG?

pervect replies in post #10:

In the Newtonian analysis, photons do gain or loose energy as it falls. In the GR analysis, the "energy-at-infinity" of the photon is a constant as it falls.

With the GR defintion of "energy-at-infinity" in the GR analysis it's the clocks that measure the photon's frequency that are affected and which cause the red and blue shift.

There is another meaning of "energy" in GR, the locally measured energy. This does red and blue shift as the photon falls, and is probably more similar to the Newtonian notion of energy with which you are familiar. But the local energy is not a conserved quantity, while the "energy-at-infinity" is conserved (well, there's some fine print - it's conserved for those space-times in which it can be defined, like static space-times).

Am I correct in understanding this description applies to both the OP's scenarios?

Seems like a given static observer at a given altitude who observes a different energy in the radially infalling scenario of the OP also observes photons to have slightly different energies, different frequencies, during the transit near a massive object.

Am I also correct in understanding that the 'energy at infinity' concept doesn't apply to cosmological scenarios... where there is expanding space?