I am also going to tell you to "read a textbook", but the textbook in question is actually the textbook for this problem: Moyer's Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation (http://descanso.jpl.nasa.gov/Monograph/series2/Descanso2_all.pdf). Specifically, for instance, Eq. 4-26 provides the formula for acceleration, including post-Newtonian relativistic corrections, in the solar system barycentric coordinate system. The equations in this volume are the equations that are used in orbit determination codes.

Hi V.T., thanks for the nice try, better than I've seen before, BUT on page 2-10 we learn that we are about to dive into the solution for light (massless particles) in the gravitational field of n massive bodies.  Most of the text is cluttered with reference frame transformations important only for stellar navigation, and absolutely obscuring any conceptual investigation.  The critical computation for for particles with mass is missing because it is not used by NASA.

In any case, even if the desired equation were buried in this 549 page tome on "Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation," I would doubtless find myself asking if some expert would please reduce n massive bodies to one (so that the "metric tensor" goes away and I have a plain metric back), specialize it for the Schwarzschild metric, and include both photons and regular particles.

If I try to simplify this to my intended scope, I'd almost certainly make a mistake, or even if I didn't people would think I had made a self-serving mistake.  Think of this like Independent Verification and Validation.  I need a formula from a GR expert, optimally tailored to the simple, conceptual case, so that I have not perverted it in any non-transparent way.

.. AND by the way, I really want, and certainly all the "masses" want (either ponderable or pondering) a tutorial explanation!  Exactly how does spacetime curvature influence trajectory?

You have a particle m traveling with 3-vector velocity (0,1000,0) (aka vx, vy, vz) let's say, at 3-vector position (5Ro,0,0) where Ro is the Schwarzschild radius of object M at (0,0,0).  These coordinates are valid for a remote observer.  Likewise use time for the remote observer.  Not the "proper" time of the object.  (Observers don't survive at 5xRo!)  I want to know what we observe by our clocks.  Schwarzschild R-units are fine since they are defined by circumference ratio to Ro and don't try to directly measure the squirrely radius.

What is d(vx, vy, vz)/dt for the particle m as a function of (x, y, z), Ro and M.  (presumably it is not a function of m if m

Robert: To be honest, I don't know what you mean by the sentence, "The 'metric tensor' goes away and I have a plain metric back". Even the metric of flat, empty spacetime, the Minkowski metric, is a tensor: it is a prescription to form the invariant inner (scalar) products of pairs of vectors. So what is it that you call a "plain metric" that would not be a tensor?

I am also wondering if your question itself may be hiding a small conceptual misunderstanding. You write, "I am looking for a way to take a particle position, mass and velocity [...] given a simple metric, particularly the Schwarzschild metric"... However, if the Schwarzschild metric is applicable, that means that the particle's own mass can be neglected; otherwise, the presence of this mass modified the gravitational field and it is no longer adequately described by the Schwarzschild metric of the other source.

If you look at Eq. 4-26 in Moyer's book again, you will notice that a) this equation is usable "as is", without concerning yourself with all the navigational details about coordinate systems, time standards, and the like; and that b) the equation describes the acceleration of a massive particle (the i-th particle) in the combined gravitational field of all the particles. So the moving particle is not assumed to be massless. And while the formula is general for an n-body system, nothing can prevent you from applying it specifically to the n=2 case, which simply means that you can drop the summation signs and just substitute i=1, j=2, k=1, l=2 and there, you have it: the trajectory of massive particle 1 in the combined gravitational field of particles 1 and 2.

Moyer's formula is applicable in cases where the post-Newtonian description is valid, i.e., weak gravity. Further, velocities are assumed to be significantly less than the speed of light. You also asked about photons. This case is actually discussed very neatly in Weinberg's 1972 book, Gravitation and Cosmology. Here, he takes the ultrarelativistic limit (approximating the photon as a massive particle whose kinetic energy is much larger than its mass) and shows how to write down the photon's orbit in the presence of a mass such as the Sun. The context is the gravitational deflection of light by the Sun, but the procedure is generic, as is the formula given by Weinberg's Eq. 8.5.4.

I actually used Moyer's book and formulation to write precision orbit determination code. I also used Weinberg's approach, appropriately modified, in the context of modified gravity theories to calculate the deflection of light by a star. So I can assure you that these two references are precisely those that provide "simple" (to the extent that these can be simplified), actionable approaches to these problems at hand, readily usable in numerical simulations, for instance.

V.T., I appreciate your interest.  That was a long post.  I see that I stand corrected in my estimate based on section 2 as to scope, but  I am reproducing formula 4-26 attached below for those who are interested but maybe not enough to download the book and look for it.  It seems I did not mis-estimate the n-body aspect.  (By the way, the link above does not seem to work, but I found it at http://descanso.jpl.nasa.gov/Monograph/series2/Descanso2_all.pdf )

The page with the formula begins: "The point-mass Newtonian acceleration plus the point-mass relativistic perturbative acceleration of body i due to each other body j of the Solar System."  Again it is the n-body problem, and there is nothing tutorial or explanatory about it. 

I would like to avoid notation like "tensor" that most people interested in this answer would not be comfortable with.  I'm not convinced it is necessary.  Let's start with the familiar "flat space" Minkowski metric, which would be a diagonal on the 4x4 tensor:

ds2 = -c2dt2 +dx2 +dy2 +dz2

Now we can move on to the equivalent expression for Schwarzschild, which can be reasonably written as a plane section through the spherical space and still capture the relevant information for our purposes, from http://www.mathpages.com/rr/s6-02/6-02.htm formula (6):

(dτ)2 = { (r-2m)/r) } (dt)2 - { r/(r-2m) } (dr)2 - r2(dφ)2

Well of course we have got spherical coordinates, but not too many terms.  Unfortunately this is not too useful for calculation because it is written in relativistic units with c=1, but that can be fixed.  Anyway, while in spirit it might be a tensor, it still looks a lot like a single equation.

Obviously I'm aware of Kevin Brown's simplified explanations since I linked them above.  I've known about them for years.  They are still too oblique.  For one thing, Brown tends to compute whole orbits or in the case of light, total bending angle, which is not too useful for figuring out what is going on in a differential space, like Einstein's microscopic physics lab in which he discusses equivalence.

I have taken Brown's computation of orbits and shown it to be equivalent to a very simple acceleration formula derived from time dilation alone (without considering spatial curvature) at all radii beyond about 10 times the Schwarzschild radius.  Quite adequate for Mercury, for example.  But to get light bending, I have to double this.  What I calculated was a formula based on transverse velocity v, which is equation (19) in my 2011 paper on inertia https://www.researchgate.net/publication/233416418_Isotropy_equivalence_and_the_laws_of_inertia?ev=prf_pub .  If we let g be the relativistic acceleration of falling objects, then the radially directed acceleration of transversely moving objects gT is

gT = g (1 + vT2/c2)

In other words, where vT=c it is doubled.  Now this is very simple, and gives the correct result.  It does not use spatial curvature, only time dilation.  What gives?  I'm trying to figure that out.  If I have a side by side procedure of comparable complexity using the spatial curvature, then maybe I can figure it out.  There are also unresolved questions about equivalence I'd like to address with this.  And of course, thousands of people will be simply curious to see an explanation they can understand and follow about how to calculate using curvature.  You could be famous.  : )

Article Isotropy, equivalence and the laws of inertia

What's wrong with numerical integration of the geodesic equation? ODE solvers are quite evolved routines and work quite well.

Paul, this is for conceptual study and investigation.  Numerical solver is useless.

The first integral of the geodesic equation , that is,

g_ab v^a v^b = 1 , where v^a is the four-velocity and g_ab is the metric tensor, usually helps.  Constants of motion also help.

Maria, an intriguing comment, but can you provide a little more detail?  If I Google these terms, I get a rapidly expanding set of matricies without any conceptual explanation of what they are and where they come from ... for example at http://en.wikipedia.org/wiki/Solving_the_geodesic_equations .  Assume that I have only

(ds)2 = { (r-2m) / r) } (dt)2 - { r / (r-2m) } (dr)2 - r2(dφ)2

Assume that Vr=Vφ=0.  What is dVr/dt?

A bizarre thing happens if one just "solves" the equation .. dr/dt = 1, a constant, obviously not correct.  How conceptually does one use a metric equation such as that to derive a change in motion?  I am not asking for a computer program to give a number, or some equation to blindly plug into as if it were a computer program.  I'm asking for a physical understanding of motion in a space with that characteristic.

To R. Shuler

Explicit calculations can be found at:

Annals of Mathematics, Vol.40, p.922 (1939) by A. Einstein (spherical symmetry)

Proc. Cambridge Phylos. Soc. Vol.58, p.338 (1961) by Raychaudhuri and Som (cylindrical symmetry)

In principle it's just a matter of integrating the geodesic equation. In practice though, one wouldn't in general expect to get an analytic solution (the Christoffel symbol terms can be quite formidable, even for a 'simple' metric such as the Schwartzschild metric). You would, unfortunately, have to resort to numerical solution.  )-:

Eric,

It seems unreasonable to adopt a theory which cannot be analytically compared to any other similar theory.  For example, I have a very clear and simple derivation of 2x light bending in a uniform gravitational field.  Half of the effect is from refraction due to the motion of the light and is proportional to v2/c2 where in the case of the light v=c.  The other half is due to a Hamiltonian analysis which uses time dilation as a proxy for potential, justified by the Planck relation. 

How do I compare this to GR to see if it is producing the result in the same way or differently?

In particular, I seem to be able to produce correct light bending and precession with a spatial curvature component.  This is what I want to investigate.  Numerical solutions are useless.

Maria,

Is this what you are referring to? http://www.cscamm.umd.edu/tiglio/GR2012/Syllabus_files/EinsteinSchwarzschild.pdf

It is a complex treatment of multiple masses and extreme conditions (such as near the Schwarzschild radius).  It appears to be a paper about whether black holes can form.  Not about how to compute trajectories in a simple gravitational field.

I found the following on "geodesic equation": http://en.wikipedia.org/wiki/Null_geodesic

It appears the section "Equivalent mathematical expression using coordinate time as parameter" regards using the coordinates of an observer (not the accelerated particle).  Is that correct?

I find the superscripts unclear.  Does xμ refer to (x0,x1,x2,x3)? (see statement near top that greek indices take values [0,1,2,3])  Then which is time, x0?  Is that proper time?  What is the meaning of d2x0/dt2 in that case?

How do I get the Γ?

Does anyone have a link to some place with the indices all iterated out?  (I've looked in many texts but never seen it)

I think what I am asking for is to see the following written out for the case of a 2D (plane) section through a Schwarzschild metric:

That's all.  The geodesic equation appears to give differential acceleration.  I need both the case of a slow moving particle, and also a photon.

"How do I compare this to GR to see if it is producing the result in the same way or differently?" Your question leaves me wondering. If I compute the time of flight of an electron using classical mechanics and compare with the "same" calculation using quantum mechanics I get the same result (number). Do you consider these the same or different ways of computing time of flight of an electron? I ask because the geodesic equation does not include refraction of light which I take as a wave phenomena. 

Hi Paul,

Excellent and insightful question.  For the method I published in 2011 (the inertia paper posted on my profile) I did in fact use the save property of particles to compute trajectory.  Thus turns out to be unimportant for planets because it has an effect proportional to v2/c2 where v is the velocity transverse to the gravitational acceleration vector.

On the one hand, when I read GR texts I see nothing specifically about refraction as you say, and am inclined to deduce that it is not in the theory.  This would make the theory immediately incorrect because from a physical reasoning perspective, the light should bend this additional amount (i.e. 3x).  This is what Hamilton's 1936 paper was about, more or less.

However, there is a devil in the historical details.  Einstein had made his 1911 computation based *only* on light bending, with the idea that gravity would not directly accelerate a zero-rest-mass particle.  So we find ourselves asking what was Albert thinking when he dropped refraction bending?  Maybe he noticed the 2x that comes with a geodesic and just assumed it had to be included indirectly?  Is there some systematic relation between refraction bending and the geodesic?

More recently I did a Hamiltonian analysis showing that consideration of only time dilation is quite sufficient to give most GR results.  It is quite a bit simpler than doing geodesic calculations. 

Anyway, I want to identify what is another way of looking at the same thing, and what is different.  Perhaps it will help focus the discussion to post a figure I made for another QA chain, which shows how the 2x bending looks in an equivalence setup.  It looks like equivalence doesn't work and is blatantly wrong.  This is another example of the kind of thing I'm trying to address logically, not just wave under the rug. 

Robert ~

I have understood what you are trying to do.

In response to your remark "It seems unreasonable to adopt a theory which cannot be analytically compared to any other similar theory", I would answer that it is reasonable to adopt GR because (within the limits of its applicability) it is true! It is unfortunate that its complexity often forces us to resort to approximations of various kinds when calculating its consequences. We "adopt" theories because observations of nature compel us to. We can't expect nature to comply with our hope for mathematical simplicity.

I wonder if this is any use to you: for the trajectory of a photon, in the two-dimensional slice of the Schwartzschild spacetime, fc2dt2 = f--1dr2 where f = 1 - K/r. So the wordline of a photon (a null geodesic) is given parametrically by ct(r) = integral (dr/f).

Robert,

I would say that the quantum and classical calculation of electron time of flight give the same result because both calculations include time and space translation symmetry. QM, however, provides a more complete theory. Returning to your discussion, GR includes the equivalence principle and SR in a theory which connects the distribution of mass and energy to the space-time curvature. I suspect (expect) GR provides a more complete description of the phenomena optical and particle trajectories. However, like it is with all theories, the more complete ones are distinguished from less complete ones by yielding results for some observed phenomena which are at odds with the less complete ones.

@All, some interesting inputs that I will have to study.  Thanks.

@Shuler:  I only discovered this question yesterday, otherwise I would have posted some of my previous answers here.

"Perhaps it will help focus the discussion to post a figure I made for another QA chain, which shows how the 2x bending looks in an equivalence setup.  It looks like equivalence doesn't work and is blatantly wrong.  This is another example of the kind of thing I'm trying to address logically, not just wave under the rug. "

The figure does not display the planet situation correctly. The relevant formula for the deflection angle is

Theta = (2GM/b v^2)*(1+v^2/c^2)  = (2GM/b)*(1/v^2+1/c^2),

where M is the mass of the planet (or sun), b is the impact parameter, v the velocity of the infalling particle and c the speed of light. (G is of course Newton's gravitational constant).

This clearly shows that for given (closest) distance b from the center of mass, the deflection angle is *smallest* when v=c (then the factor in the last parentheses takes its minimum value 2/c^2), it is bigger for any smaller velocity v.  The deflection angle is therefore smaller for light than for any particle with a smaller velocity. But the picture cannot capture this anyway as we are talking here of the asymptotic directions of the particle ray, coming from infinity and going to infinity on an approximate hyperbolic track.

The factor two that you are talking about is not a factor between light and other particles but a factor between a Newtonian and the final Einsteinian prediction for light. (With Newtonian gravity the result of the calculation is simply  Theta = 2GM/b v^2.) I will discuss shortly, where this factor 2 comes from. Actually, this is quite well understood.

The second problem with your representation is that you cannot invoke the equivalence principle to compare the elevator situation and the planet situation here. In such a comparison, the equivalence principle is valid locally only, which means that you should not use a calculation that involves integration over an infinite distance (which is the case for the deflection angle calculation, the angle is between the asymptotes of the near-hyperbola). You might compare falling heights over a short distance. If you do so, there is no factor of 2. The elevator calculation and the planet calculation agree for short distances. [As an aside, since the elevator corresponds to a homogeneous gravitational field, the long-distance direction change of light and any other horizontally ejected particle will be just pi/2. In a homogeneous field, contrary to Newtonian physics, a horizontally thrown object will travel a maximum finite distance, even if its initial velocity is c, the trajectory is not a parabola but a curve approaching a downward vertical tangent.]

Now where does the factor of 2 for light come from? If you use only the equivalence principle, you'll get Theta = 2GM/(b c^2). This result would be correct in a flat space. It is the deviation from a spacelike geodesic corresponding to infinite speed, or, less dramatically, consisting of positions evaluated at the same global time. This is discussed nicely in [1], with elementary calculations.

However, general relativity tells us that spacetime about a mass is *curved*, and in the case at hand this means that also space (i.e. a spatial section of spacetime at fixed global time coordinate) is curved. For general relativity, it turns out that this additional curvature (which is absent in an elevator) yields another angle Theta = 2GM/(b c^2). For other theories of gravity, also briefly discussed in [1] (and in [2]), this additional angle could be different from the result of the equivalence principle alone, so there would be a factor different from 2. Refs. [3] and [4] give additional discussion and more details.

[1] R. P. Comer, J. D. Lathrop, Principle of equivalence and the deflection of light, Am. J. Phys. 46, 801 (1978)

[2] D. S. Koltun, Gravitational deflection of fast particles and of light, Am. J. Phys. 50, 527 (1982)

[3] J. Ehlers, W. Rindler, Local and Global Light Bending in Einstein's and other Gravitational Theories, Gen. Rel. Grav. 29, 519 (1997)

[4] S. Carlip, Kinetic Energy and the Equivalence Principle, arXiv:gr-qc/9909014v1 3 Sep 1999

I think the problem is this. In the famous "thought experiment", the interior of the constantly accelerated elevator experiences a uniform inertial field. The equivalence principle says this is indistinguishable from a uniform gravitational field. And so it is - in Newton's gravitational theory. But in GR there is no such thing as a 'uniform gravitational field'. From the Schwartzschild metric we can deduce the gravitational field in a local region near the surface of a large planet (a 'flat earth metric' *). But that is not a uniform gravitational field. The acceleration due to gravity is dependent on height. The acceleration of falling bodies in the elevator in Minkowski spacetime is not.

__________________________________

*    (1-k/z)c2dt2 - (1-k/z)-1dz2 - dx2 - dy2

Eric, actually, the acceleration of falling bodies in the elevator, using Special Relativity (I dislike the term Minkowski spacetime because it does not automatically apply to macro objects, so I'm going to quit using it), is in fact dependent on height.

Let the TOP of the elevator be in uniform in some arbitrary inertial frame O (for observer).  At some moment let O be co-moving with the TOP.  O releases a falling object FO into the elevator, at rest in the frame of O.

The BOTTOM of the elevator as we all know has to accelerate a little faster in the frame of O because the elevator is constantly becoming shorter in the frame of O.  So when the falling object FO passes the BOTTOM, it it observed to have this slightly greater acceleration.

So in fact even in an elevator using SR, the acceleration is dependent on height.

This is more than a curiosity. I found it 7 years ago and it was my first clue to an amazing coincidence regarding the exact magnitude of relativistic gravitational effects.  The effects of gravity do not diminish as relativistic mass and velocity increase.  (exactly as we concluded a Lorentz effect with greater limiting velocity than c would based on the analogy of the sonic world)

However, there is another oddity.  Acceleration must be expressed as d(mv)/dt.  As the mass increases (due to velocity), gravity has its unrelenting effect only on d(mv)/dt.  In propagating (integrating) this differential, one has to factor in an acceleration due to gravity and a slowing due to momentum conservation coupled with mass increase.  So gravity still cannot accelerate things beyond c.

The apparent simplicity of Einstein's elevator thought experiment is deceptive. We've probably all been taken in by it when learning relativity as students! I'd never given it a lot thought until this discussion came up, and now I'm bemused and intrigued. It seems to be valid only in Newtonian physics! Even in SR there is no such thing as a constant and uniform inertial field and I suspect that in GR the metric for a "uniform gravitational field" violates the empty-space field equations.

Nevertheless, the equivalence principle itself is not endangered. Inertial forces and gravitational forces are indistinguishable in GR in the sense that the Christoffel symbols describe them inextricably.

Charles ~ I said "Minkowski" spacetime because only SR is needed for the elevator experiment - the spacetime is flat everwhere, not just inside the elevator.

Robert ~ the mass of a test particle is irrelevant in GR - all test particles follow geodesics, irrespective of their mass. (The irrelevance of mass when a gravitational field acts on a test particle was discovered experimentally by Galileo...)

Charles ~

(1) It's usual to think of the observer in the elevator not looking out, but he may. His peculiar reference system works just the same outside. (The idea of not letting him look out is just to maintain his illusion that he's on the ground...)

(2) My terminology was careless. Yes, in a sense the Minkowski metric is a "uniform gravitational field". But that's the null case - absence of a gravitational field.  I didn't mean that. A metric in which freely falling test particles all have accelerations gz where g is a constant - something like that...

Charles this is the problem of GR you can't assume speed of light is absolutely constant, just locally can and you can't assume equivalence principle is absolutely applied, just locally can. This tends to complicate the equations and prevent unification with Q. Mec.

Eric, I don't understand why you thought it necessary to provide me with elementary education on the equivalence principle, on which I am probably a leading authority (owing to 2011 paper "Isotropy, equivalence and the laws of inertia" in regards theory, not experimental such as UW Eöt-Wash Group).  :D  If you would point out the comment that provoked that, I will endeavor to clear it up.

Charles, the complexity of GR is mostly not in its mathematical form, but in its physical assumptions. The hardest one is assuming speed of light variable in different gravitational potentials. This assumption simply made the unification with quantum mechanics a nice dream that mostly we can't get to if we stick with these assumptions. The locality of GR is a real problem that prevents the unification. If you look to the problem as a transformations between classical observers as its in GR you won't get anything. While if the equations are remade to be from the view of tiny particles or may be photons this might make it much easier and smoother to follow with Q.Mech. Who Knows?

So why they aren't unified yet???

its has been 85 years until now, I think its enough time to realize the difficulty!!!

Robert ~ "Eric, I don't understand why you thought it necessary to provide me with elementary education on the equivalence principle, on which I am the world's leading authority (regards theory, not experiment)"

What have I said?! What have I said?!! - I honestly don't remember. If you've taken offence at something, I apologize. Bear in mind that some of my posts are just my spontaneous thoughts, not directed at any particular person. Even if it was a direct response to one of your posts it would only have been my attempt to express my own thoughts on the matter. My intention in RG is not to "educate" anybody, but to have interesting conversations. I have taken great interest in your contributions - I admire them and have learnt from them.

@Eric, no problem.  : )  I get tired and cranky and sensitive sometimes.  You have made some of the most helpful comments by the way, and I enjoy your posts also.

@Sadeem, the variance of the speed of light may have been a theoretical prediction (along time time dilation) at the time GR was developed, but it is very much an empirical fact now, well verified (again, along with time dilation - they are intimately related).  This does make the writing of wave equations in QM ambiguous and possibly more complex depending on what the correct formulation is.  Charles by the way has some very good ideas.  But this has not been the show stopper in unifying QM and gravity.  The non-renormalizability (or whatever sophisticated term is technically correct) as I understand it has been the problem, according to most physicists.

But you can see from various posts on this and related threads, there is at least some suspicion that gravity may not be unifiable with the other 3 forces.  I have even proposed, based on 3 pieces of circumstantial evidence, that it may operate at a much higher Lorentz limiting velocity.

The bottom line is, that I think the underlying difficulty of the physics is the real reason.  Perhaps aggravated by the human insistence on pursuing unification as opposed to just describing what is there.  Einstein ended his career and became somewhat of an outcast by blindly pursuing unification of all forces under his gravity model.  Now it appears the rest of the physics community has done no better at unifying gravity under the QM model which is so successful at the other forces.

There is no mechanism in the QM formalism which can produce time dilation, or spatial curvature.  Coordinates are not variables in QM.  What you are effectively pointing out is that even if we get rid of spatial curvature, we still will have a problem with time dilation.  I have spent a lot of time thinking about this subject and if you care to read the result you can view my 2013 paper on quasi-measurement dynamics.  This is an attempt to construct a time dilation mechanism within QM.  It is very long and extremely difficult, so I don't recommend it to anyone else, but since you have an interest in that specific topic it may be worth your while.

BTW, there is a "free parameter" in the 2013 paper that I have since nailed down in the currently pending Hamiltonian paper, which I can also send if you like, but I am not posting it yet.

Article On Dynamics in a Quasi-Measurement Field

Robert ~

Please enlighten me - my knowledge of QM is sketchy. Relativistic quantum field theory is Lorentz- invariant. Time dilation effects are a consequence of Lorentz invariance. So it seems to me that time dilation effects must be already built into QM in some way. So I'm a bit confused when you say "There is no mechanism in the QM formalism which can produce time dilation..."

@Lord: "The apparent simplicity of Einstein's elevator thought experiment is deceptive."

I can second that.

"We've probably all been taken in by it when learning relativity as students! I'd never given it a lot thought until this discussion came up, and now I'm bemused and intrigued. It seems to be valid only in Newtonian physics! Even in SR there is no such thing as a constant and uniform inertial field and I suspect that in GR the metric for a "uniform gravitational field" violates the empty-space field equations."

Have a look at my post in the thread "Is it possible to derive the constant & uniform velocity of light ..."  There is a notion of uniform acceleration and uniform gravitational field in general relativity, even though the requirement to make it coordinate independent makes that notion more complex than in Newtonian physics. Here [1] is a reference explaining it all. The uniform gravitational field has a metric that gives a vanishing Riemann curvature tensor. (Essentially the Rindler metric.)

[1] G. Munoz, P. Jones, The equivalence principle, uniformly accelerated reference frames, and the uniform gravitational field, Am. J. Phys. 78, 377 (2010)

There is quite a lot of information related to the question in the Link given below.

http://star-www.st-and.ac.uk/~hz4/gr/HeavensGR.pdf

Eric, regarding your question:

@Homeier, thanks for the link to Heavens' GR lecture notes.  FYI to all interested parties, on another thread this morning someone posted a link to a similar set of notes by t'Hooft http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2013.pdf .  It can be useful to see the slightly varying perspectives of these different professors who have been sufficiently concerned with "explaining" GR to write up their notes nicely and publish them.  Of course our contributor Charles has some notes also to which he has often linked, but I'll repeat here http://rqgravity.net/HomePage , and which he often expands in response to various discussions on RG.

Charles ~

Thank you for your answer. I understand that QM (as we know it) can only be formulated in flat spacetime. The 'relativistic' in "relativistic QM" is strictly special relativitistic. QM is not only unable to handle time dilation in curved spacetime - it's unable to handle anything in curved spacetime! But since QM is covariant under the Poincaré group, I can't figure out why the  time dilation aspects of the Lorentz transformations cannot be taken over into QM. In other words, each inertial frame will have its own formulation of QM, and these formulations will be related through the Lorentz transformations...

@Shuler:

"Time is an independent variable in QM. "

So is space in relativistic quantum field theory. The operator describing position measurements becomes distinct from the variable describing space.

" It is not the outcome of any quantum process and cannot be influenced by QM processes.  In other words, QM has no way to make the 2nd clock of Charles' comment run more slowly. "

Nevertheless, quantum field theory is a relativistic theory, so time dilation and length contraction are built in. The Dirac equation is Lorentz invariant. Therefore, your statement does not sound very well-founded.

@Lord:  "QM is not only unable to handle time dilation in curved spacetime - it's unable to handle anything in curved spacetime! "

QM does not have any problem with curved spacetime as long as that is a fixed background. What QM does have problems with is a dynamic spacetime, i.e. one that may change according to a field equation. In QED, time and space are not treated as dynamic fields -- and such a treatment would indeed cause major problems. But QED in a spherically curved universe could be easily be devoloped via appropriate modification of the quantization rules. We know how to to QED on the surface of a sphere, so we should not be unable to do it at the interior of a curved 3 space.

Anyway, the interest to consider GR just an effective theory, describing a metric field on a fixed spacetime brackground can be motivated from the quantum point of view. It is conceptually much easier to quantize a field on a fixed background than to quantize space and time themselves. The technical difficulties are of course not removed by this reinterpretation.

K Kassner ~ I accept your point. My statement was hasty and careless! I'm aware that quantum theory can be done in principle on a fixed curved spacetime background. There is also quite an accumulation of literature on the quantization of "small" perturbations of the metric on a fixed curved background. That's all that "quantum gravity" has achieved till now ( I presume it's essentally the theory of a spin-2 quantum field...).

The unified theory that will satisfactorily include all known fundamental interactions, including gravity, is still nowhere in sight after almost a century. Whether modifications (or reformulations) of GR or of QM (or both) will be needed is an open question. I have a hunch that quantization of spacetime itself is not what is required.

It became clear in the late twentieth century that the electromagnetic, weak and strong interactions are all expressions of the gauge principle. The Yang-Mills idea seems to be basic and fundamental. Einstein's theory is not a gauge theory, but there are gauge theories of gravity. See http://www.amazon.com/Gauge-Theories-Gravitation-Commentaries-Classification/dp/1848167261

for a thorough review. (It's a very expensive tome, but amazon allows a view of the contents list, which includes journal references.)

The "Poincaré Gauge Theories" treat the Poincaré group as a Yang-Mills group. This was first done by T W B Kibble in 1960, and the idea has since been extensively investigated and refined by F W Hehl and his coworkers. The gauge field for the Lorentz subroup is curvature - the source is energy-momentum density; the gauge field for the translational subroup is torsion - the source is spin density.

These are classical theories, not quantum theories. However, since the Yang-Mills fields for internal symmetry groups (photons, Z and W bosons and gluons) are quantizable (though this poses immense problems  - see, eg., http://psroc.phys.ntu.edu.tw/cjp/v30/987.pdf). The question is then "is a quantum Poincaré gauge theory possible?"

A different approach would be to look again at the work of S N Gupta (Proc Phys Soc London 1952 A65 608). He took the standard linear theory of a massless spin-2 field (in flat spacetime), inserted energy-momentum of all other fields as a source term, then modified to include energy-momentum of the spin-2 field itself. Continued iteration of this procedure converges to Einstein's gravitational equations(!) A similar trick can be done to derive Poincaré gauge theory (http://ericlord.webring.com/ericsfiles/pdf/40.pdf). The starting-point for this kind of procedure is quantizable, so I wonder if a full quantum theory could be built up iteratively...

These are just some random thoughts I'm throwing out there. Or, as they say, "that's my two cents" (-:

@Kassner, the ability to "produce" spacetime dynamics, as GR does, was what I was referring to as lacking because space and time are independent variables in QM.  I'm aware there is a relativistic version of QM based on Dirac et. al.  At that point in the discussion we were in a hypothetical as regards if one didn't have to produce spatial dynamics one would still have a problem with time.  This is what I have been investigating for 8 years and which I call inertial theory (loosely).  It is beyond the pale of what most people consider, and roughly equivalent to a background-independent approach to quantum gravity.

@Lord: " Einstein's theory is not a gauge theory, but there are gauge theories of gravity."

Why do you say that? Aren't coordinate transformations gauge transformations? I presume you would then say classical electrodynamics is not a gauge theory either? So when does a theory that has gauge invariance qualifiy as a gauge theory and when does it not? Is it the necessity to identify some physical field with the gauge transformations? Then classical electrodynamics would not qualify, QED would, but general relativity presumably would, too. After all, you can obtain general relativity from special relativty by first requiring general covariance from the latter and then "reifying" the Riemann curvature tensor, i.e. making a dynamical object out of the left-hand side of a compatibility condition in special relativity (saying that the Riemann tensor must vanish). The condition is abandoned and replaced by appropriate dynamical equations for the new dynamical object (obtained by requiring limiting cases to become Newtonian).

K Kassner ~

“Aren't coordinate transformations gauge transformations?”

In a sense, yes. What I said was that Einstein’s theory [by which I mean the theory that comes from the Hilbert Lagrangian √(-g)R] is not a gauge theory. The group of general coordinate transformations has only four spacetime-dependent parameters. The gauge potentials therefore constitute a tetrad eai and the gauge fields turn out to be the torsion Faij (skewsymmetric part of the connection), not the Riemann curvature tensor. So the resulting gauge theory is not Einstein’s theory (it is in fact the gauge theory of the translational part of the Poincaré group). One can contrive to get Einstein’s theory from it by a suitable choice of Lagrangian for torsion (which, as we all know, was tacitly assumed to be zero by Einstein).

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“So when does a theory that has gauge invariance qualifiy as a gauge theory and when does it not? Is it the necessity to identify some physical field with the gauge transformations?”

Yes, that’s what I mean by a gauge theory – there is a physical field whose existence is necessary to render a symmetry spacetime-dependent.

“Then classical electrodynamics would not qualify”

Why do you say that? – isn’t electromagnetism a “physical field”?? Electromagnetism is necessitated to allow the phase of a complex field to be spacetime-dependent

Robert ~

I’ve been having trouble understanding your concerns about the nature of time in QM. The Schrödinger equation (ih/2π)∂/∂t = H is a component of the Lorentz-invariant 4-vector equation (ih/2πc)∂/∂xμ = pμ. So why aren’t all the characteristics of time in classical SR taken over into QM?

Mozafar ~

Perhaps it’s a matter of what one expects of the structure of a theory if it's to be called a gauge theory. The structure of GR looks superficially like a gauge theory with the connection as gauge potential and the curvature tensor as gauge field. However, the group of general coordinate transformations has only four independent spacetime-dependent components, not sixteen. So the connection has too many components to be its gauge potential. The potential should look like a set of four vectors - ie., a tetrad. That’s the Poincaré gauge theory approach. The tetrad is treated as the gauge potential for the translational subgroup of the Poincaré group (which of course turns into general coordinate transformations when "gauged"). The connection is the gauge potential for the Lorentz rotations of the tetrad rather than a gauge potential associated with the coordinate transformations. Kibble (1960) chose √ (-g)R as the Lagrangian, but the curvature tensor here is not the Riemann tensor (the connection is not symmetric). So the theory reduces to Einstein’s theory only if torsion is taken to vanish.

That’s why I prefer to say that “Einstein’s theory is not a gauge theory”. But perhaps our disagreement is more a matter of terminology rather than disageement about the physics.

@Lord:

'“Then classical electrodynamics would not qualify”

Why do you say that? – isn’t electromagnetism a “physical field”?? Electromagnetism is necessitated to allow the phase of a complex field to be spacetime-dependent'

I said that because the gauge invariance of *classical* electrodynamics does not produce a new field. I also said in my post that QED would qualify. If you require gauge invariance of quantum mechanics, you get the photon as a new field. But if you look at classical electrodynamics alone (i.e. without coupling to quantum mechanics), the gauge invariance of the electromagnetic fields (with respect to a gauge transformation of the four potential) does not produce new physics. So classical electrodynamics is a gauge-invariant theory but not a gauge theory. QED is gauge invariant and a gauge theory.

My question was more or less why GR is not both.  I used to consider it the gauge theory that comes out on requiring SR to be gauge invariant with respect to arbitrary coordinate transfomations.I mean if you formulate SR for arbitrary coordinate systems what you get is what many people would consider GR (formula-wise). But it would be GR with a zero Riemann tensor, so it is still SR indeed. If you then require the Riemann tensor to become a dynamical object and write down the simplest action permitted by symmetries to obtain its equations of motion, you have GR up to the constant that appears in the Hilbert-Einstein action, which you can get from the Newtonian limit.

Therefore, my feeling was that SR and GR are similarly related to each other as quantum mechanics without electrodynamics and QED. In the first case, the gauge field would be - roughly - the Riemann tensor, in the second the electromagnetic field tensor. Apparently, the analogy is incomplete (apart from the obvious fact that in the first example, no quantum mechanics is involved), but then I am by no means an expert on gauge theories.

Kassner ~

When a field theory posses a symmetry, additional fields (gauge potentials  and gauge fields) have to be present for the symmetry to be spacetime dependent. The sources of the gauge field are contributed by the original theory. Whether we are dealing with classical or quantum physics makes no difference.

If the equations of a complex field ψ are invariant under ψ′ = eiαψ with constant α, the invariance can be maintained under spacetime dependent α only if there is a supplementary field Ai with transformation law Ai′ = Ai + ∂iα. That is the electromagnetic gauge potential. QED is not needed to deduce that. What you are saying, essentially, is that a gauge-invariant field without sources is not a gauge theory. That’s a matter of terminology. Maxwell’s theory does includes sources (charges and currents). The gauge principle simply allows us to see how these sources arise from other fields, which may be regarded as classical fields prior to quantization.

Incidentally, the term “gauge invariance” (Eichinvarianz) was coined in 1927 by Hermann Weyl, who attempted to obtain electromagnetism from general relativity through invariance under gij′ = eαgij

Your third paragraph is a neat summary of the path that led Einstein from SR to GR. It sounds simple put like that, but it took him ten years!

Yes, there is an intriguing analogy between GR and gauge theory, though it's not an exact correspondence.

@Eric, you asked about my "concerns" about time and stated all the classical characteristics of "time" in SR carry over to QM.  These are two different statements or questions.

First the latter.  I have no concern about SR and QM.  As far as I know that is well understood.  Time is an independent variable in a particular reference frame in either.  I was talking about GR, where the curvature tensor, which includes time , is a dependent variable and changes with other parameters dynamically.  It is not sufficient just to "use" the time and space coordinates of GR in QM (though that is certainly a worthwhile intermediate goal, and Charles has done a lot of work on it), but QM (or something) should explain how the mass-energy tensor produces the curvature tensor.  Essentially, QM only explains motions (of Schrodinger-Dirac waves) guided by the curvature tensor (i.e. spacetime).

Einstein was insistent that space and time were not just background coordinates, but could be acted on.  In the course of the conversation, some questions arose about whether it was necessary to include "space" (in which case it wouldn't strictly be GR anymore, though it might be a closely related theory with the same Hamiltonian ... the discussion was whether the spatial coordinates contributed to the Hamiltonian and whether they could be transformed away by a suitable substitution).  That still left time which must both act on and be acted on. 

So I don't know if you call it a "concern."  I would call it an "interest."  I started out studying inertia and realized that it essentially defines time.  So acting on inertia, if one acts on the inertia of all things (a kind of reverse equivalence), this changes the time scale over the region of action.  The classical expression of such an action was well developed in 1912 when Einstein derived an expression for it.  Essentially it is a Machian expression with gravitational potential mediating the action.  This idea resurfaced several times (Sciama, Wheeler & Ciufolini, Ghosh, etc.) and though I was working on a different formulation of inertia, when I encountered this one it seemed right and I immediately adopted it.  That doesn't say anything about "how" it comes about, though in 2013 I did publish a speculative paper on the subject, which is rather dense, but you can find it on my profile.

Robert ~

Sorry, I misunderstood (I was misled when you said "There is no mechanism in the QM formalism which can produce time dilation" - I was thinking SR). Of course, the relation between GR and QM is a whole other kettle of fish! You are a brave man to be grappling with it   (-:

@Francis: "In fact quantum theory does have a very substantial problem on a fixed curved space time. The probability interpretation requires the conditions satisfied by Stone's theorem, from which follows a (relativistic) Schrodinger equation in flat spacetime."

I've had a brief look at your paper -- at those parts of it at least that you refer to. First, I do not think that Stone's theorem has anything to do with flat spacetime. It gives you the unitarity of time evolution but says nothing about the second coordinate x in your wavefunction. Second, it seems to me that your flatness requirement comes from assuming that free particles correspond to plane waves. But that assumption need not be made.

Now, I know of one example at least of a Schrödinger or Dirac equation that is not solved in flat spacetime. This is the angular momentum part of the hydrogen problem. There you solve a Dirac equation on the surface of a sphere which is not a flat space and does not become Minkowski space by adding time. Since there is no major problem interpreting spherical harmonics as valid wave functions, I also do not quite buy your statement that the probabilistic interpretation requires flat spacetime. I am quite willing to consider that this may be important when you want a dynamic spacetime but even then I believe that it does not matter in principle whether your background is flat or has constant positive or negative curvature, as long as it remains fixed. In that case, also the ontological problem of whether spacetime is a substantive manifold or not, loses its importance. If it is fixed you can *pretend* it to be a substantive manifold without incurring a danger; if it is not fixed, the difference between a pretense and reality may show up.

I do not claim that it is possible without problem to quantize fields in *any* stationary spacetime -- in fact, I doubt that it can be easily done for a Goedel universe. But I think for simple highly symmetric static spacetimes, there is a clear prescription of how to proceed. First, embed the spacetime in a flat higher dimensional space. Second, quantize the theory in the higher dimensional space. There it is known how to construct the commuators of the basic solutions of the field equation you want to quantize. Then consider a thin sheet about the curved hypersurface on which you want to construct your theory (corresponding to a fourdimensional manifold). Expand your field solutions in terms of eigenfunctions that factor between the "small" directions and the dimensions of our spacetime (here symmetry will help but is probably not required). As boundary conditions in the small directions, one may use periodic ones, then we are not far from what is actually done in string theories, where the idea of our spacetime being a compactified reduction of some higher-dimensional object prevails. (The compact dimensions are "rolled up" in circles, hence periodic b.c.)

After the expansion and after integrating out the compactified dimensions, you have field operators on your curved spacetime, their commutators, and a field equation for them. The latter is usually interpreted within the Heisenberg picture, not the Schrödinger picture. So the Hilbert kets and bras are only used to calculate expectation values and matrix elements tgey have no dynamics of their own.

The main problem in quantizing a classical field theory on a curved manifold is to obtain the right commutators, because curvilinear coordinates multiply the ambiguities in their construction. This is largely avoided by the detour through a higher dimension (which would not be helpful in the case of dynamically varying curvature). Of course, any such field theory would have to be checked against experiment in order to verify one's choices.

"Kassner, I think you have not looked very carefully at the solution of a wave equation. "

You think wrongly.

"The Schrodinger equation does follow from Stone's theorem,and its solution invokes Minkowski metric."

But not necessarily. It was artificially added by yourself. Everything goes through without the second coordinate. It is a Hilbert space argument essentially, where you do not need position representation.

"The theory of the Dirac equation assumes Poincare invariance as well as Lorentz invariance. The surface of a sphere is embedded in flat spacetime, so that is hardly a counterexample."

Are you telling me that the surface of a sphere is not curved? I can solve the Dirac equation for an electron confined to a spherical surface. This is a standard exercise. Hence I have the solution of the Dirac equation in a curved three-dimensional spacetime. The full Lorentz invariance is broken by the restriction to the sphere. But what is most important is that this is a simple counterexample to your statement that the probability interpretation requires flat spacetime. The solution on the sphere can be interpreted as easily as the solution in flat spacetime. I mean your paper is nice to read in general, but before making sweeping statements you should check whether what you say is verified by simple examples. The surface of a sphere is a perfect arena for the solution of the Schrödinger or the Dirac equation. So is the surface of a hypersphere, i.e. a three-dimensional space of constant curvature. No interpretation problem whatsoever.

Finally, my proposition of how to proceed in doing quantization on a curved four-dimensional spacetime was just to use a higher-dimensional flat spacetime. This means that I have the geneeralization of Poincaré and Lorentz invariance to higher dimensions, solve the equation there and restrict back to lower dimension. So your result can hardly be a counterexample since the curved spacetime is embedded in a flat superspace...

@Lord: "If the equations of a complex field ψ are invariant under ψ′ = eiαψ with constant α, the invariance can be maintained under spacetime dependent α only if there is a supplementary field Ai with transformation law Ai′ = Ai + ∂iα. That is the electromagnetic gauge potential. QED is not needed to deduce that."

Right. So cancel QED. It was meant as an example only. What I wanted to say is that you need fields outside of electrodynamics to obtain electromagnetic fields as a gauge field. Often this is considered a way of *deriving* electrodynamics. I.e., one starts from a *different* field theory. If you start from electrodynamics, you will not get anything directly out of its gauge invariance. There is no field ψ  to begin with. So electrodynamics alone, without any other fields, would be gauge invariant but not a gauge theory in your nomenclature.

@Francis:  I think I understand now in what sense you believe there is a problem with curvature. It is more or less the old communication problem between mathematicians and physicists.

Of course, one can solve the Dirac equation or, if we are not interested in relativistic effects, the Schrödinger equation reduced to the surface of a sphere to determine the probability distribution of a free electron in that curved two-dimensional space. But this is not the same equation, in a mathematical sense, as the equation in flat (two-dimensional) space. The operator in the Schrödinger equation on the sphere is the Laplace-Beltrami operator, not just the Laplacian; the gradients in the Dirac equation are different operators on the sphere than on the plane. For a physicist, these equations are essentially the same, whether formulated on the plane or on the sphere, both are versions of a generalized form involving covariant derivatives. Of course, one has to find the forms of operators on the curved manifold in some way. One way is to go via the tangent space and define the operators in curved space via appropriate projections. A pure mathematician might insist that one even does the solution in tangent space and projects then, but since the Laplace-Beltrami operator was used well before tangent spaces were invented, that distinction might appear moot to researchers that just want to solve the equations. For all practical purposes, the differential equation may be solved directly on the curved manifold.

Now, what you are suggesting on the basis of an interpretation of what measurements mean, is slightly different. You want to keep the time evolution in tangent space until a measurement is made and project only then. This is your teleconnection. I presume this will not, in general, give the same probability distributions than what you would observe, if projection was continuous, which would however be equivalent to solving differential equations directly on the curved manifold.

So I think you have a potential prediction there that could be tested experimentally. Nowadays, experimentalists can produce very good approximations to two-dimensional systems, and the surface of a sphere should not pose a major problem. Inject electrons on one side of the sphere and measure their position distribution on the other side of the equator at different times. Your teleconnection theory will presumably give a different result from the direct solution of the Schrödinger equation using the Laplace Beltrami operator. (If it does not, due to the high symmetry of the problem, put obstacles on the sphere, corresponding to simple potentials. Probably your theory can be extended to treat this case.) Let experiment decide.  (My guess is that *if* there is a difference in predictions, then the local evolution of the equation on the sphere will win over the teleconnection result. This does not mean that in the case of a dynamic spacetime the teleconnection approach does not have its merits, but I doubt that it is applicable without modification.)

I like your discussion of the cosmological implications, even though I think it is highly speculative. If you were right, you would revert the increase of the size of the universe by a factor of two that happened in the fifties of the last century when Walter Baade discovered an error in the luminosity - distance relationships assumed for cepheids... Then the old size estimates would have been closer to reality than new ones, even though they were based on a wrong premise.

Periodic relativity proposes an alternative to Schwarzschild metric which satisfies Einstein's field equations but does not depend on geodesic trajectories or weak filed approximation. The theory gives primary importance to two body systems and the multibody systems are of secondary importance.

Article Periodic relativity: Basic framework of the theory

Article Supplement to periodic relativity