Gravity as a force and as a curvature

[shounakbhatta]

Hello All,

As far as the Newtonian mechanics and Einstein's GR is concerned, I am a little bit confused in the following things:

(a) Concerning the bending of light due to gravity: Some lectures and opinions show that light bends due to the force of gravity as shown in the event of a solar eclipse, light bends and takes a different route before reaching the observer. Another argument shows that as photons are mass less, hence how can they be affected by gravity? The curvature of spacetime allows the photons to take the curved path. WHICH ONE IS CORRECT?

In case we use F= G m1m2/r^2 and use photons, substituting m=0 for mass less photons, we get F=0. From that can we tell that for photons there is no force?

(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?

Thanks.

 

[tiny-tim]

hello shounakbhatta!

Some lectures and opinions show that light bends due to the force of gravity as shown in the event of a solar eclipse, light bends and takes a different route before reaching the observer.

no, light does not bend

light does not take a bent path

light always takes a straight path through space-time

but the straight path isn't always what you might think it is

(for example, if you look "down" on the solar system from a long way "above", then the path that the light takes looks bent to you)

mass has nothing to do with it …

the light follows a straight path through space-time because it has no reason to do otherwise ​

 

[dauto]

You're making two mistakes.
1st, Even in Newtonian mechanics it is NOT correct to assume that a massless particle does not accelerate. True, you get F=0. Try plugging that into Newton's 2nd law to find the acceleration. What did you get?

2nd, In GR it is NOT correct to assume that gravity only acts on massive objects.

 

[.Scott]

Photons have mass, but they have no rest mass.
But even if there is a particle with no mass (or negative mass) it would still be affected by gravity. Although the gravitational force applied to an object is proportional to its mass, so is the amount of force required to accelerate it.

 

[Dale]

In case we use F= G m1m2/r^2 and use photons, substituting m=0 for mass less photons, we get F=0. From that can we tell that for photons there is no force?

First, this argument is based on Newtonian gravity which has been shown experimentally to be incorrect regarding light and gravity. Second, in the context of Newtonian gravity the argument is incorrect. In Newtonian gravity the acceleration of an object is independent of its mass, so a massless object accelerates at the same rate as a massive object. True, the force is 0, but by F=ma it is clear that it takes 0 force to accelerate an object with 0 mass, so you have to determine the acceleration by some other means than a calculation of the force.

 

[shounakbhatta]

Hello,

Tiny Tim, thanks for your answer. I have one question:photons have relativistic mass?

Now what we get an explanation during solar eclipse, is that the star light bends as the massive object like sun causes a bend, a dip in the spacetime curvature. Is that so? Please correct me as it might seem rudimentary questions at the initial level.

 

[Ibix]

In general relativity, the source of gravity is the stress-energy tensor, which you can think of as a 4x4 matrix. One element of that is the rest mass, the "m" in Newton's equations. That is zero for a photon, but not all of the components are zero. Some people describe this as .Scott did, as the photon having "relativistic mass"; others don't like the terminology since it's also used for $\gamma m$ and $\gamma^3m$ at least.

To the extent that I'm qualified to have an opinion, I prefer not to use the term "relativistic mass" at all, and I seem to be in the majority (at least round here). But, whatever language you use, photons have non-zero components in their stress-energy tensors. Just not the classical mass element of it.

On the curvature of spacetime, "a dip" is probably not the best way to describe it. But it is often depicted that way. Don't read too much into that, though.

 

[A.T.]

Now what we get an explanation during solar eclipse, is that the star light bends as the massive object like sun causes a bend, a dip in the spacetime curvature. Is that so?

The often shown "dip" is just the spatial curvature, which accounts for half of the light bending. See the lower picture here:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

The other half of light bending is due to gravitational time dilation (or "time curvature"): Light at different distances from the mass propagates at different speeds when measured with the same clock, so a wave front propagating tangentially changes direction. This is similar to light bending in a medium with varying refractive index:
http://www.nature.com/nchina/2013/131106/full/nchina.2013.113.html

 

[tiny-tim]

hello shounakbhatta!

(just got up :zzz:)

photons have relativistic mass?

some people call the energy of a particle its relativistic mass, so in that sense, yes

Now what we get an explanation during solar eclipse, is that the star light bends as the massive object like sun causes a bend, a dip in the spacetime curvature. Is that so? Please correct me as it might seem rudimentary questions at the initial level.

i prefer to call it an increase in curvature …

space everywhere has some curvature (there's no place with zero gravity! )

the stronger the gravity, the stronger the curvature, and the more "parallel" lines deviate from each other (much like "parallel" lines of longitude on the earth) … this is also known as geodesic deviation

(technically, i shouldn't have been saying "straight", i should have been saying "geodesic" … but "straight" is easier! )

 

[pervect]

Hello All,

As far as the Newtonian mechanics and Einstein's GR is concerned, I am a little bit confused in the following things:

(a) Concerning the bending of light due to gravity: Some lectures and opinions show that light bends due to the force of gravity as shown in the event of a solar eclipse, light bends and takes a different route before reaching the observer. Another argument shows that as photons are mass less, hence how can they be affected by gravity? The curvature of spacetime allows the photons to take the curved path. WHICH ONE IS CORRECT?

The GR analysis better fits observations. For instance, famously, one of the early tests of GR was a test of "extra' light deflection.

I suspect that the issue is that you don't fully understand the GR paradigm of "curved space-time". Which is rather understandable, unless you've studied it - and it's not generally taught until college. AT has some nice diagrams that can help, if you are familiar enough with special relativity to understand space-time diagrams. If you are not familiar enough with SR to understand space-time diagrams, it seems unlikely that you can understand the significance of the space-time diagrams being drawn on a curved surface (which is what curved space-time means, operationally). So you'd need to study SR to the level at which you understood space-time diagrams first, before the GR analysis will make any sense to you.

In case we use F= G m1m2/r^2 and use photons, substituting m=0 for mass less photons, we get F=0. From that can we tell that for photons there is no force?

This is an attempt at a purely Newtonian analysis, it has nothing to do with GR. In GR, it's not "mass" that causes gravity - as I'm sure other posters have mentioned, it's the stress-energy tensor. In terms of the Newtonian analysis, it's a bit unclear to me how the "expected" Newtonian light prediction (the one that was half the GR value) was calculated. My personal answer would be that the Newtonian formulation of light deflection is known to be wrong anyway, so I'm not too interested in the details of how it was formulated.

(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?

Thanks.

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).

 

[Maxila]

Hello All,

As far as the Newtonian mechanics and Einstein's GR is concerned, I am a little bit confused in the following things:

(a) Concerning the bending of light due to gravity: Some lectures and opinions show that light bends due to the force of gravity as shown in the event of a solar eclipse, light bends and takes a different route before reaching the observer. Another argument shows that as photons are mass less, hence how can they be affected by gravity? The curvature of spacetime allows the photons to take the curved path. WHICH ONE IS CORRECT?

In case we use F= G m1m2/r^2 and use photons, substituting m=0 for mass less photons, we get

I believe for many people some confusion is because space-time is often referenced in a way that may indicate it to be some kind of independent entity or a structure of substance, rather than it simply being coordinates, location(s). Einstein specifically said in the last appendix he added to his book “Relativity The Special and General Theory” that, "Space-time does not claim existence on its own, but only as a structural quality of the field." [Page 155 under Appendix 5. Year 1952].

In the presence of a gravitational field space-time coordinates are curved, that is literately what is meant when you read, or see someone say, “light travels a straight path in curved space-time. It means light follows the curved coordinate path defined by space-time and there is no other path it can follow, that is the straightest (shortest) possible route, unlike the curve of a road that may ‘cut-off’ and a distance traveled can be shortened. That also shows why a photons lack of mass is irrelevant to why gravity effects its path (in addition to gravity effecting the path of any mass, it effects the very fabric, or space-time coordinates, of space itself.

 

[WannabeNewton]

I believe for many people some confusion is because space-time is often referenced in a way that may indicate it to be some kind of independent entity or a structure of substance, rather than it simply being coordinates, location(s).

This is terribly incorrect I'm afraid. Space-time has a geometric structure and kinematics/dynamics on space-time have invariant geometric forms all of which are several strata above coordinate systems. In fact coordinates are only computational tools, nothing more, and in principle never have to be introduced; we can talk about general relativity and space-time physics without ever mentioning coordinates-it will be quixotic and highly inefficient at times but that doesn't mean it can't be done in principle. I can refer you to general relativity textbooks that rarely if ever introduce coordinates-most of the calculations in the texts will be done using the coordinate-free abstract index and/or index-free calculus and most definitions will be set in this geometric language. It's far more elegant albeit far less practical, sometimes intractable, for down to Earth problems such as Perihelion precession or ultra-relativistic gravitational scattering.

 

[Maxila]

This is terribly incorrect I'm afraid. Space-time has a geometric structure and kinematics/dynamics on space-time have invariant geometric forms all of which are several strata above coordinate systems. In fact coordinates are only computational tools, nothing more, and in principle never have to be introduced; we can talk about general relativity and space-time physics without ever mentioning coordinates-it will be quixotic and highly inefficient at times but that doesn't mean it can't be done in principle. I can refer you to general relativity textbooks that rarely if ever introduce coordinates-most of the calculations in the texts will be done using the coordinate-free abstract index and/or index-free calculus and most definitions will be set in this geometric language. It's far more elegant albeit far less practical, sometimes intractable, for down to Earth problems such as Perihelion precession or ultra-relativistic gravitational scattering.

I still see defining a structure of space, a manifold, topological space, being paramount to the locations that define them, and in agreement with the quote of Einstein "Space-time does not claim existence on its own, but only as a structural quality of the field." However I stand corrected in implying space-time is a coordinate system and redact that implication.

 

[shounakbhatta]

Thank you very much for the wonderful answers. Pervect, it was great to read your explanations. Well, tiny tim, can I conclude, that in a curved spacetime, light follows the geodesic. For us it 'appears to bend'. Right?

Also, taking the classical concept of a field affecting the photos, gravitational fields, never bends or draw photons inward. Right?

 

[A.T.]

Also, taking the classical concept of a field affecting the photos, gravitational fields, never bends or draw photons inward. Right?

No. Read what DaleSpam wrote:

In Newtonian gravity the acceleration of an object is independent of its mass, so a massless object accelerates at the same rate as a massive object.

However, this classical prediction for "light particles" is only half of what is actually observed. The other half is due to the spatial-distortion, which is only present in General Relativity. So the observational test of Newtonian "light particles" vs. General Relativity is the amount of light bending, not it's existence.

 

[tiny-tim]

hi shounakbhatta!

Well, tiny tim, can I conclude, that in a curved spacetime, light follows the geodesic. For us it 'appears to bend'. Right?

right!

the geodesics 'appear' to bend, in the sense that geodesics that are initially parallel will get closer, then get further again

but they aren't actually bent, any more than lines of longitude on the Earth are bent!

Also, taking the classical concept of a field affecting the photos, gravitational fields, never bends or draw photons inward. Right?

i think this is a semantic issue

you can separate the field equations into two parts, which happen to have equal effects: a gravitational part and a curvature part …

on that view, there's a gravitational force on the photon, and it has an effect additional to that of curvature​

but if you regard gravity as not being a force, but as part of the general structure of space-time, then of course the question is meaningless (ie, no gravitational field anyway): everything is governed by curvature

 

[Dale]

you can separate the field equations into two parts, which happen to have equal effects: a gravitational part and a curvature part …

What are you describing here? Are you talking about separating the Christoffel symbols into two groups somehow or are you talking about something else?

 

[tiny-tim]

What are you describing here? Are you talking about separating the Christoffel symbols into two groups somehow or are you talking about something else?

i was thinking about the way the general relativity equations can be regarded as a correction to the special relativity equations (which give only half the correct result, as i believe einstein originally obtained before GR) …

perhaps I'm misremembering my history of physics, but i thought there was a pre-GR way of applying SR to curved space-time: if so, you could say that that is the curvature contribution, and the GR "correction" is the gravitational contribution

 

[Dale]

Hmm, OK, I am not familiar with that approach, but then I never spent much effort investigating historical models.

 

[PeterDonis]

i thought there was a pre-GR way of applying SR to curved space-time

I haven't seen anything like this in what I've read of pre-GR work in relativity (not that I have read everything there is to read by any means). Also, trying to apply SR to curved spacetime seems inconsistent to me, because curved spacetime breaks a basic assumption underlying SR, the assumption that free-falling observers who start out at rest relative to each other will stay at rest relative to each other.

 

[tiny-tim]

ah, i did a google search and decided i must have been thinking of this, from http://en.wikipedia.org/wiki/Histor...e_development_of_the_Einstein_field_equations …

When Einstein realized that general covariance was actually tenable, he quickly completed the development of the field equations that are named after him. However, he made a now-famous mistake. The field equations he published in October 1915 were

$R_{{\mu \nu }}=T_{{\mu \nu }}$,,

where $R_{{\mu \nu }}$ is the Ricci tensor, and $T_{{\mu \nu }}$ the energy-momentum tensor. This predicted the non-Newtonian perihelion precession of Mercury, and so had Einstein very excited. However, it was soon realized that they were inconsistent with the local conservation of energy-momentum unless the universe had a constant density of mass-energy-momentum. In other words, air, rock and even a vacuum should all have the same density. This inconsistency with observation sent Einstein back to the drawing board. However, the solution was all but obvious, and in November 1915 Einstein published the actual Einstein field equations:

$R_{{\mu \nu }}-{1 \over 2}Rg_{{\mu \nu }}=T_{{\mu \nu }}$,

where R is the Ricci scalar and $g_{{\mu \nu }}$ the metric tensor. With the publication of the field equations, the issue became one of solving them for various cases and interpreting the solutions. This and experimental verification have dominated general relativity research ever since.​

… but both parts are due to curvature, so it seems you can't separate out the gravity

 

[pervect]

I view the extra light deflection in GR as arising from the spatial part of the curvature, but I'm not sure there is general agreement about this

One of the issues is what does "the spatial part of the curvature" even mean? To make sense of this, you need to slice space-time into space+time, for generality one needs to specify that the slicing is done in the usual means by the time-like Killing vector.

Given that slicing, there are two approaches I'm aware of for arguing that the extra light deflection is due to the spatial part of the curvature

One is to use the Riemann tensor, and it's Bel decomposition into the electrogravitic, magnetogravitic, and topogravitic parts. Then one can show that the topogravitic local deflection is equal to the electrogravitic local deflection. I find this rather elegant, but I'm not sure I've seen it published anywhere. In this approach, The topogravitic part of the Bel decomposition of the Riemann tensor is what one means by "spatial curvature".

Another way is to use the non-tensor PPN parameter gamma, and note that it can be interpreted as "the amount of spatial curvature per unit mass". This description doesn't really define what "spatial curvature" is, but it is published (in MTW, Wiki also uses this).

 

[shounakbhatta]

Excellent answer tiny-tim !

Thanks everybody for the explanations.

 

[stevendaryl]

I haven't seen anything like this in what I've read of pre-GR work in relativity (not that I have read everything there is to read by any means). Also, trying to apply SR to curved spacetime seems inconsistent to me, because curved spacetime breaks a basic assumption underlying SR, the assumption that free-falling observers who start out at rest relative to each other will stay at rest relative to each other.

I would think that "SR in curved spacetime" would just be General Relativity without the field equations. That is, GR in a fixed, background geometry. So it would take into account the effects of curvature on equations of motion (mostly, replacing partial derivatives by covariant derivatives), but would not take into account the effect of mass/energy on curvature.

 

[PeterDonis]

I would think that "SR in curved spacetime" would just be General Relativity without the field equations. That is, GR in a fixed, background geometry. So it would take into account the effects of curvature on equations of motion (mostly, replacing partial derivatives by covariant derivatives), but would not take into account the effect of mass/energy on curvature.

First of all, such a theory would be inconsistent, because the equations of motion for matter are connected to the Einstein Field Equation; you can't separate them. (This is because the EFE obeys the Bianchi identities, and the Bianchi identities applied to the stress-energy tensor give equations of motion for the matter that the stress-energy tensor describes.)

Second, such a theory wouldn't be SR, because as soon as you allow curved spacetime to affect the motion of objects, you break a basic assumption of SR, as I said in my previous post: if you have curved spacetime, you have free-falling objects that start out at rest relative to each other but don't stay at rest relative to each other, and SR can't handle that.

You can certainly do SR in curvilinear *coordinates*, which requires you to deal with covariant derivatives instead of partial derivatives; but spacetime itself must still be flat if you're doing SR (i.e., you can have nonzero connection coefficients, but the Riemann curvature tensor must still be zero).

 

[tiny-tim]

… as soon as you allow curved spacetime to affect the motion of objects, you break a basic assumption of SR, as I said in my previous post: if you have curved spacetime, you have free-falling objects that start out at rest relative to each other but don't stay at rest relative to each other, and SR can't handle that.

can't it?

we regularly do Newtonian mechanics on a 2D curved surface, and i see no reason why we can't equally do it on a 3D "curved volume" embedded in 4D flat space

(in which there is geodesic variation, ie free-falling objects that start out at rest relative to each other don't stay at rest relative to each other)

can't we similarly do einsteinian SR mechanics on a 4D "curved volume" embedded in 5D flat Minkowski space (ok, i agree I'm not sure what that would be, i suppose it would have signature -++++ and some super-complicated Lorentz group)?

it would be ordinary flat -++++ SR, with everything constrained to move in a particular 4D volume: it would have geodesic variation but no GR

 

[PeterDonis]

can't we similarly do einsteinian SR mechanics on a 4D "curved volume" embedded in 5D flat Minkowski space

I'm not sure that an arbitrary 4-D curved Lorentzian manifold can always be embedded in flat 5-D Minkowski space. But I think there are some that can, so I think we can let that pass for this discussion.

However, whatever theory you come up with along these lines will not be SR; it will have similiarities with SR, but it won't be SR. The equivalent of the 4-D Riemann curvature tensor won't be zero; only the 5-D Riemann curvature tensor will be zero. But physically, the 4-D Riemann curvature tensor is what we observe, and what has to be zero for the theory to be "SR".

In other words, the fact that the theory has an underlying 5-D flat manifold is physically unobservable; but "SR" requires a *physically observable* flat manifold, not just an unobservable one in the theory.

 

[stevendaryl]

First of all, such a theory would be inconsistent, because the equations of motion for matter are connected to the Einstein Field Equation; you can't separate them. (This is because the EFE obeys the Bianchi identities, and the Bianchi identities applied to the stress-energy tensor give equations of motion for the matter that the stress-energy tensor describes.)

Are you sure about it being inconsistent?

It is certainly the case that in GR, you can derive the equations of motion from the field equations. But in SR, you can also derive them from a Lagrangian. For example,

$\mathcal{L} = \frac{1}{2} m g_{\mu \nu} \dfrac{d x^\mu}{d \tau} \dfrac{d x^\nu}{d \tau}$

Using variational techniques, you get the equations of motion:

$m \dfrac{d^2 x^\mu}{d \tau^2} g_{\mu \nu} + m \dfrac{d x^\mu}{d \tau} \dfrac{d x^\lambda}{d \tau} \dfrac{\partial g_{\mu \nu}}{\partial x^\lambda} = \frac{1}{2} m \dfrac{d x^\nu}{d \tau} \dfrac{d x^\lambda}{d \tau} \dfrac{\partial g_{\lambda \nu}}{\partial x^\mu}$

What is inconsistent about that?

Of course, that's kind of a trivial theory, with point-masses floating in a background curved spacetime, but it seems to be enough to show that your claim is wrong, or that I'm misunderstanding it.

Now, maybe it's true that once you add fields (such as an electromagnetic vector potential $A^\mu$) it becomes inconsistent, but I don't see that, either.

Second, such a theory wouldn't be SR...

Yes, it would be the generalization of SR to curved spacetime.

 

[stevendaryl]

[deleted]

 

[PeterDonis]

in SR, you can also derive them from a Lagrangian.

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).

If you include the missing term, varying the Lagrangian just gets you the Einstein Field Equation with the RHS being the stress-energy tensor for a point particle (and then you have to deal with mathematical singularities at the point particle's location, but that can be finessed by making it a "world tube" of finite density and radius). You appear to agree that this theory is "GR", not "SR".

If you don't include the missing term, your theory isn't complete; it doesn't constrain $g_{\mu \nu}$. So what constrains it? What tells you what $g_{\mu \nu}$ is? Your answer is basically, you just declare by fiat what it is. But then you're basically declaring by fiat that "SR" allows you to use a curved spacetime, because you can declare by fiat that $g_{\mu \nu}$ is curved.

In other words, you're assuming the conclusion you're supposed to be proving, that "SR" allows curved spacetime. To prove that, you would have to present some physical law, compatible with "SR", that admits a curved metric $g_{\mu \nu}$ as a solution. The only such law we know of is the Einstein Field Equation--but you admit that's "GR", not "SR".

it would be the generalization of SR to curved spacetime.

But that's GR, not SR!

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". 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. (Which in itself is not necessarily fatal, btw--"SR" with flat spacetime also ignores the issue of what physical law makes the spacetime flat.)

* I think "SR" means that spacetime must be flat, because to me a fundamental part of "SR" is the absence of geodesic deviation; if you have geodesic deviation, you don't have "SR", because geodesic deviation changes everything we normally think of as "SR"--no more global inertial frames, no more path independence of parallel transport, no more invariance under global Lorentz transformations, etc. Calling such a theory "SR", to me, is an abuse of terminology, and saying "well, it's still SR because we aren't including a dynamical equation for the metric" seems like sophistry, because the theory still has all the added complexity of curved spacetime, which is where most of the hard parts of GR reside.

 

[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".)