Speed of light - why is it a constant?

[Shenstar]

What makes the speed of light a constant. I read the FAQ on special relativity but still don't understand why c (speed of light) exists as a constant.

It's like a rule like many others, why do they exist? Is there a part of space-time that limits this speed. Why are all the photons that ever existed limited by this speed?

 

[phyzguy]

Because that's the way the universe is. To quote Feynman, "You don't like it? Go somewhere else!"

 

[bcrowell]

FAQ: Why is the speed of light the same in all frames of reference?

The first thing to worry about here is that when you ask someone for a satisfying answer to a "why" question, you have to define what you think would be satisfying. If you ask Euclid why the Pythagorean theorem is true, he'll show you a proof based on his five postulates. But it's also possible to form a logically equivalent system by replacing his parallel postulate with one that asserts the Pythagorean theorem to be true; in this case, we would say that the reason the "parallel theorem" is true is that we can prove it based on the "Pythagorean postulate."

Einstein's original 1905 postulates for special relativity went like this:

P1 - "The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion."

P2 - "Any ray of light moves in the 'stationary' system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body."

From the modern point of view, it was a mistake for Einstein to single out light for special treatment, and we imagine that the mistake was made because in 1905 the electromagnetic field was the only known fundamental field. Really, relativity is about space and time, not light. We could therefore replace P2 with:

P2* - "There exists a velocity c such that when something has that velocity, all observers agree on it."

And finally, there are completely different systems of axioms that are logically equivalent to Einstein's, and that do not take the frame-independence of c as a postulate (Ignatowsky 1911, Rindler 1979, Pal 2003). These systems take the symmetry properties of spacetime as their basic assumptions.

For someone who likes axioms P1+P2, the frame-independence of the speed of light is a postulate, so it can't be proved. The reason we pick it as a postulate is that it appears to be true based on observations such as the Michelson-Morley experiment.

If we prefer P1+P2* instead, then we actually don't know whether the speed of light is frame-independent. What we do know is that the empirical upper bound on the mass of the photon is extremely small (Lakes 1998), and we can prove that massless particles must move at the universal velocity c.

In the symmetry-based systems, the existence of a universal velocity c is proved rather than assumed, and the behavior of photons is related empirically to c in the same way as for P1+P2*. We then have a satisfying answer to the "why" question, which is that existence of a universal speed c is a property of spacetime that must exist because spacetime has certain other properties.

W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

Palash B. Pal, "Nothing but Relativity," http://arxiv.org/abs/physics/0302045v1

R.S. Lakes, "Experimental limits on the photon mass and cosmic magnetic vector potential", Physical Review Letters 80 (1998) 1826, http://silver.neep.wisc.edu/~lakes/mu.html

 

[mjacobsca]

The answer to your question is that we don't know. My view is that SOMETHING limits a photon's speed, but we don't know what it is. Is it the fabric of space itself? If so, why? Until we can understand what space is like at the Planck scale we probably can't answer the question fully. Perhaps the photon is dragging on some sort of subspace foam, and even experiences friction? Perhaps it is working its way through 11 dimensions at once, and it takes time to go through them all? I think string theory and other TOE's have the chance to explain photon behavior at a deeper level and explain WHY the photon travels as fast as it does, but will they be able to explain why time is so intimately connected to it?

I do believe a TOE will one day explain all the constants in terms of a common quantum basis, but whether it will be too bizarre for us to grasp or test is a whole other matter.

 

[BruceW]

that link to "nothing but relativity" is amazing! Only assumptions about the anisotropy of space lead to the relativistic transformation laws and the fact that there is a particular speed which is the same when viewed in any frame!
Still, It doesn't explain why that special speed happens to be the speed of light in vacuum.

 

[Shenstar]

Thanks for those answers. They also made me think of another question. Why do atoms and those sub atomic particles get theIr energy from, why are electrons coupled with a nucleus. What keeps them their.

I think the answer will be something about weak force, strong force and maybe electromagnetic force. But what 'powers' these forces at the atomic quantum level?

 

[danR]

I think what is irksome fundamentally is the notion that a rate should be a constant, and time and space metrics, which millenia of mensuration tell us deeply should be more fixed, are subservient to this rate.

At any rate, I think P2 and P2* are both rooted in empiricism. P2* just pretends it doesn't know that.

 

[bcrowell]

Still, It doesn't explain why that special speed happens to be the speed of light in vacuum.

There are various ways of proving this. One is that Maxwell's equations have an invariant speed in them, and relativity says there's only one invariant speed, so they have to be the same. Another is to develop the properties of the energy-momentum four-vector, and you can show that a massless particle is accelerated to c by any infinitesimal perturbation.

Thanks for those answers. They also made me think of another question. Why do atoms and those sub atomic particles get theIr energy from, why are electrons coupled with a nucleus. What keeps them their.

I think the answer will be something about weak force, strong force and maybe electromagnetic force. But what 'powers' these forces at the atomic quantum level?

I'd suggest you start a separate thread for this in High Energy, Nuclear, Particle Physics.

 

[danR]

deleted. Someone answered it above.

 

[skbrant]

There are various ways of proving this. One is that Maxwell's equations have an invariant speed in them, and relativity says there's only one invariant speed, so they have to be the same. Another is to develop the properties of the energy-momentum four-vector, and you can show that a massless particle is accelerated to c by any infinitesimal perturbation.

But the Maxwell's equations, with its invariant speed c, could be just an incomplete model for the behaviour of light. In principle, the limit speed which appears in the Lorentz transformations could be something slightly greater than the observed speed of light. Even if the Maxwell equations are exactly correct, the observed photon speed is never exactly equal to c, because there are always, among other things, the interactions of the photons with the thermal background, or even virtual electron-positrons pairs of the vacuum. Perhaps when these quantum effects are taken into account the photon could be even slower than the neutrino? Of course these effects are very small and will not explain the anomalies recently observed in the OPERA experiment. But in principle, the speed of the photon could be smaler than the limit speed and perhaps some other particle could be faster than light (but slower than the limit speed, of course).

 

[Parlyne]

But the Maxwell's equations, with its invariant speed c, could be just an incomplete model for the behaviour of light. In principle, the limit speed which appears in the Lorentz transformations could be something slightly greater than the observed speed of light. Even if the Maxwell equations are exactly correct, the observed photon speed is never exactly equal to c, because there are always, among other things, the interactions of the photons with the thermal background, or even virtual electron-positrons pairs of the vacuum. Perhaps when these quantum effects are taken into account the photon could be even slower than the neutrino? Of course these effects are very small and will not explain the anomalies recently observed in the OPERA experiment. But in principle, the speed of the photon could be smaler than the limit speed and perhaps some other particle could be faster than light (but slower than the limit speed, of course).

You would be hard pressed to find a way to do this that wouldn't require a frequency dependent photon speed. The thing is, we use light over a range of frequencies that spans 20-some orders of magnitude. It would be hard to miss frequency dependence over that kind of range.

In particular, if you suggested that light simply had mass, you'd find that the speed of light looked like
$$v(\nu)=c\sqrt{1-\frac{m_\gamma^{\phantom{\gamma}2}c^4}{h^2\nu^2}}$$.

With this sort of behavior, there's no way it could look like all the light we use has anything close to the same speed, unless that speed was actually c.

I'll add that, from the standpoint of quantum field theory, interactions with vacuum fluctuations can only change the speed of a photon if the photon has mass in the first place.

 

[aleazk]

your question is somewhat vague. are you referring to the velocity of light in vacuum (a question which has an answer in the electromagnetic theory) or to the invariant speed limit that special relativity predicts or to the fact that this two velocities are equal (this last thing has to do with both theories, maxwell's equations and special relativity)?

 

[Spacie]

The question of the speed of light is best addressed in its proper historical context. It was Maxwell who ~150 years ago measured the speed of light in empty space, demonstrated its connection to magnetism and other forms of EM radiation, and described the laws that governed them. About 50 years later, Einstein postulated that for all observers, regardless of their frame of reference, the laws of nature should hold the same. That's how, historically, the speed of light became the "constant", or, rather, a "yardstick" with which all observers can measure stuff in their frame and through it all agree on their measurements, regardless of the particulars of their frames of reference.

Now, as for light moving in non-empty space, such as transparent matter, its speed can slow dramatically:
http://www.nytimes.com/1999/03/30/s...ergaard-hau-she-puts-the-brakes-on-light.html

In their paper, Dr. Hau, with Dr. Steve E. Harris of Stanford University and two of Dr. Hau's Harvard students, reported the results of their experiment in which a beam of laser light was slowed to the astonishingly low speed of 38 miles an hour. (By comparison, light in a vacuum travels about 186,000 miles per second.)

So, when people speak of "speed of light" what is meant is the speed of light in vacuum, i.o.w. empty space.

Logically speaking, the speed of light in vacuum is the fundamental property of space (being, in effect, the quantitative expression of its geometry).

...Perhaps when these quantum effects are taken into account the photon could be even slower than the neutrino?

But why are we comparing a photon with a neutrino? One is just a quanta of energy and another is a particle with mass, i.o.w. matter. Light is not matter and matter is not light. They cannot be treated as one and the same and expected to behave the same.

from the standpoint of quantum field theory, interactions with vacuum fluctuations can only change the speed of a photon if the photon has mass in the first place.

Indeed, interactions in vacuum are between the particles of matter with the vacuum. These fluctuations reflect the dynamic structure of space, through which light propagates.

A photon, however, cannot have mass, for then it would affect the curvature of space. Without mass, it only can only show the curvature that is already present, by tracing the structure of space as it propagates through it.

 

[Parlyne]

The question of the speed of light is best addressed in its proper historical context. It was Maxwell who ~150 years ago measured the speed of light in empty space, demonstrated its connection to magnetism and other forms of EM radiation, and described the laws that governed them. About 50 years later, Einstein postulated that for all observers, regardless of their frame of reference, the laws of nature should hold the same. That's how, historically, the speed of light became the "constant", or, rather, a "yardstick" with which all observers can measure stuff in their frame and through it all agree on their measurements, regardless of the particulars of their frames of reference.

Knowing the history of relativity is nice, of course; but, we don't need the foundational ideas that led to its discovery to be exactly correct for relativity to work. In particular, it is perfectly consistent to talk about the idea of massive photons, even though Einstein's original arguments all used light in a way that implied (in a blatantly a posteriori interpretation) massless light. The basic relativistic ideas are unchanged. The conclusions about light, however, do change; and, Maxwell's equations require some corrections. This leads to experimentally testable ideas (which, so far have been shown to be consistent with $m_\gamma = 0$.).

Now, as for light moving in non-empty space, such as transparent matter, its speed can slow dramatically:
http://www.nytimes.com/1999/03/30/s...ergaard-hau-she-puts-the-brakes-on-light.html


So, when people speak of "speed of light" what is meant is the speed of light in vacuum, i.o.w. empty space.

Logically speaking, the speed of light in vacuum is the fundamental property of space (being, in effect, the quantitative expression of its geometry).

No. Logically speaking there is simply a frame-invariant speed. That light travels at that speed (in vacuo) is an empirical fact, not a logically necessary one.

But why are we comparing a photon with a neutrino? One is just a quanta of energy and another is a particle with mass, i.o.w. matter. Light is not matter and matter is not light. They cannot be treated as one and the same and expected to behave the same.

We aren't. We're comparing the time of flight of a neutrino with what it would be if the neutrino were traveling at the speed of light. When it comes right down to it, relativity doesn't care whether you're talking about light, about a neutrino, or even about a flying cow. (Spherical of course.) All relativity really tells you is the relationships among its energy, momentum, mass, speed, etc.

Also, a photon is not "just a quanta of energy." First, "quanta" is plural. If you have just one its a "quantum." Second, photons carry energy, momentum, and angular momentum and have very specific interaction properties with all charged particles. It is not the case that they simply "are" energy.

Indeed, interactions in vacuum are between the particles of matter with the vacuum. These fluctuations reflect the dynamic structure of space, through which light propagates.

A photon, however, cannot have mass, for then it would affect the curvature of space. Without mass, it only can only show the curvature that is already present, by tracing the structure of space as it propagates through it.

First off, I was speaking in the terms of special relativity, where there is no curvature. Second, in general relativity, the curvature of spacetime responds to any source of energy or momentum; so, the fact that a photon has no mass only affect how, not whether, it curves spacetime. And, third, as pointed out above, nothing about relativity requires massless photons. In fact, nothing about relativity requires light to exist in the first place. I could use the ideas of relativity just as easily to describe a world consisting of nothing but quarks interacting under the strong force as I do to describe the world we actually live in.

 

[Spacie]

it is perfectly consistent to talk about the idea of massive photons, even though Einstein's original arguments all used light in a way that implied (in a blatantly a posteriori interpretation) massless light.

-?! What is mass in your definition? According to relativity, mass is the force that curves space around it. The light only follows that curvature.

If you refer to the equivalence of energy and mass, you have to agree that mass is a particular expression of energy.

The conclusions about light, however, do change; and, Maxwell's equations require some corrections. This leads to experimentally testable ideas (which, so far have been shown to be consistent with $m_\gamma = 0$.).

thankgod for that

That light travels at that speed (in vacuo) is an empirical fact, not a logically necessary one.

Yes, you're right the speed of light in vacuum is an empirical fact = the law of nature = the inherent property of space.

We're comparing the time of flight of a neutrino with what it would be if the neutrino were traveling at the speed of light.

Here you are making an assumption, the basis of which is not clear. You apparently assume that a neutrino, like a photon, also has its "constant speed" of flight. But from where does it follow? Is it possible that its speed depends merely on the force with which it was propelled? We know that this is not true of a photon: SR states that its speed through vacuum will always be the same, regardless of the speed of matter that emitted it (or observed, lol). A neutrino, having a mass, should have its speed dependent on the speed of the object that emitted it. That's the key difference between matter and light according to relativity.

When it comes right down to it, relativity doesn't care whether you're talking about light, about a neutrino, or even about a flying cow. (Spherical of course.) All relativity really tells you is the relationships among its energy, momentum, mass, speed, etc.

That's right. There is difference in how energy, momentum, mass, and charge relate. All are expressions of energy that have different effects on geometry of space.

Also, a photon is not "just a quanta of energy." ...Second, photons carry energy, momentum, and angular momentum and have very specific interaction properties with all charged particles. It is not the case that they simply "are" energy.

Nothing is "simply" energy but each particular expression of energy is manifested differently in space. You're right to point out that photons interact with charged particles. Something that cannot be said about a neutrino. That and the fact that a neutrino has mass, while photon does not, may explain the difference in how they interact with vacuum.

First off, I was speaking in the terms of special relativity, where there is no curvature.

Oh yes, there is.

Second, in general relativity, the curvature of spacetime responds to any source of energy or momentum; so, the fact that a photon has no mass only affect how, not whether, it curves spacetime. And, third, as pointed out above, nothing about relativity requires massless photons. In fact, nothing about relativity requires light to exist in the first place. I could use the ideas of relativity just as easily to describe a world consisting of nothing but quarks interacting under the strong force as I do to describe the world we actually live in.

You're mistaken, sorry.

 

[WannabeNewton]

Oh yes, there is.

No there isn't. SR assumes the flat background metric $\eta _{\mu \nu }$. It is trivial to show that $R^{\alpha }_{\beta \gamma \delta } = 0$ identically for this flat background space- time.

 

[PeterDonis]

that link to "nothing but relativity" is amazing! Only assumptions about the anisotropy of space lead to the relativistic transformation laws and the fact that there is a particular speed which is the same when viewed in any frame!

I second this comment. In fact, I think a link to that paper, plus a brief summary of its argument, should be in the FAQ for this forum. It really does a good job of explaining why it isn't *light* that's special; it's very general properties of spacetime that lead to an invariant speed. Then it's just a matter of giving physical reasons (as bcrowell did) why massless particles have to travel at the invariant speed.

 

[Spacie]

No there isn't. SR assumes the flat background metric $\eta _{\mu \nu }$. It is trivial to show that $R^{\alpha }_{\beta \gamma \delta } = 0$ identically for this flat background space- time.

The background metric is flat, but Lorentz transformation curves it (resulting in constant speed of light for all observers, right?) Introduction of Minkowski's spacetime is merely an improved method of calculating Lorentz transformation dynamically.

 

[PeterDonis]

The background metric is flat, but Lorenz transformation curves it (resulting in constant speed of light for all observers, right?) Introduction of Minkowski's spacetime is merely an improved method of calculating Lorenz transformation dynamically.

No, WannabeNewton is correct. The Riemann tensor is identically zero for Minkowski spacetime, and SR assumes Minkowski spacetime. (If you're trying to talk about SR without talking about spacetime at all, just using the Lorentz transformations, then there's no point in talking about "curvature" or a "metric" at all; those concepts assume you are talking about spacetime.)

I'm not sure what you mean by "Lorentz transformation curves it". The existence of an invariant speed is a property of spacetime, and holds whether spacetime is flat or curved; and the Lorentz transformation is just a way of expressing coordinates in one frame in terms of coordinates in another frame, using the invariant speed.

 

[WannabeNewton]

The background metric is flat, but Lorentz transformation curves it (resulting in constant speed of light for all observers, right?) Introduction of Minkowski's spacetime is merely an improved method of calculating Lorentz transformation dynamically.

Just to add on to Peter's already complete answer, a lorentz transformation is an element of the Poincare group which consists of all the isometries of Minkowski space - time. These isometries are diffeomorphisms $\phi :M \mapsto N$ such that the pullback $\phi^{*}g_{\mu \nu } = g_{\mu \nu }$. Therefore, a lorentz transformation will not cause space - time to curve. The structure of the space - time remains the same.

 

[aleazk]

it's very general properties of spacetime that lead to an invariant speed. Then it's just a matter of giving physical reasons (as bcrowell did) why massless particles have to travel at the invariant speed.

maybe this could help:

this was already asked:

At the risk of sounding silly, I would like to ask: How certain are we that the c in Einstein's equations is actually the speed of light? As far as I understand it, all of relativity etc. still holds based on the speed limit "c", and Einstein derived his equations by assuming that light traveled at c. Do we know this? In other words, is it possible that something could travel faster than light but not faster than c?

This is a very good question, it's not silly at all. You see, if the metric of spacetime is flat, then you can show that, in an inertial system, the null geodesics have the equation dx/dt=c, where c is some constant (all this is geometry, we don't know the value of c yet). Now, if you take Maxwell's equations, you can prove that electromagnetic waves always travel on null geodesics (this is called "the geometrical optics aproximation"). Then, c is equal to the speed of light in an inertial system. For the detailed calculation, see "General Relativity"- R.Wald.

New entry: on the other hand, acording to General Relativity, the deflection of light by the Sun is 4GM/Rc^2, here c is the geometrical c. Since we know the experimental value of this deflection, we can use the above formula to obtain c, wich, of course, result to have the same value than that of the speed of light.

(remember that if, e.g., the metric is flat then the null geodesics from event p form the boundary of the region in spacetime that can be reached by timelike curves also diverging from p, first theorem in Wald's causality chapter)

new entry: another way of see it, and maybe more simple, is the following: if you put a flat lorentz metric in spacetime, then you can show that the transformation formulas between inertial systems are the lorentz transformations, where c is some constant with units of velocity. now, this transformation formulas you just derive will leave maxwell's equations unchanged only if you take c equal to the speed of light in vacuum

 

[Spacie]

No, WannabeNewton is correct. The Riemann tensor is identically zero for Minkowski spacetime, and SR assumes Minkowski spacetime. (If you're trying to talk about SR without talking about spacetime at all, just using the Lorentz transformations, then there's no point in talking about "curvature" or a "metric" at all; those concepts assume you are talking about spacetime.)

SR was not set in Minkowski spacetime. In fact, Einstein did not like Minkowski spacetime at first. It is the improvement in calculations that convinced him to use it.

I'm not sure what you mean by "Lorentz transformation curves it". The existence of an invariant speed is a property of spacetime, and holds whether spacetime is flat or curved; and the Lorentz transformation is just a way of expressing coordinates in one frame in terms of coordinates in another frame, using the invariant speed.

That's right, and in doing so they describe the curvature of space (including when this curvature is 0), how it differs and is transposed (projected or mapped -? not sure what the right term would be) from one coordinate system to another. I see it from geometric point of view.

edit: From the geometric point of view, if you map the event from one sys of coordinates to another, this event will appear "curved" or deformed in comparison to how it looked in its own coordinates. That's what I meant by "Lorentz transformation curves it".

 

[PeterDonis]

SR was not set in Minkowski spacetime. In fact, Einstein did not like Minkowski spacetime at first. It is the improvement in calculations that convinced him to use it.

Historically, that may be true. So what? The fact remains that if you are talking about a metric and curvature, you are talking about spacetime. If you go back and read Einstein's original 1905 papers on relativity, you will see that he never used the terms "metric" and "curvature". He only started using them after he became convinced that a spacetime viewpoint was useful (and that was as much because it was necessary to generalize his theory to include gravitation as for calculational convenience).

That's right, and in doing so they describe the curvature of space (including when this curvature is 0),

No, the Lorentz transformation does not describe curvature. The Riemann tensor describes curvature.

how it differs and is transposed (projected or mapped -? not sure what the right term would be) from one coordinate system to another.

In a flat spacetime, a Lorentz transformation can be global, but in a curved spacetime, it can only be local, so its description of how quantities change when changing coordinate systems is also local. Also, in a curved spacetime, a Lorentz transformation only works between certain special kinds of coordinate systems, the ones that make the spacetime locally look like a piece of flat Minkowski spacetime, and in those coordinate systems, which can only cover a small patch of spacetime around a given event, the curvature of the spacetime does not appear. So a Lorentz transformation can't describe how the curvature of a spacetime gets transposed when you change coordinate systems (i.e., how the components of the Riemann tensor change when you change coordinate systems).

 

[Spacie]

In a flat spacetime, a Lorentz transformation can be global, but in a curved spacetime, it can only be local, so its description of how quantities change when changing coordinate systems is also local. Also, in a curved spacetime, a Lorentz transformation only works between certain special kinds of coordinate systems, the ones that make the spacetime locally look like a piece of flat Minkowski spacetime, and in those coordinate systems, which can only cover a small patch of spacetime around a given event, the curvature of the spacetime does not appear. So a Lorentz transformation can't describe how the curvature of a spacetime gets transposed when you change coordinate systems (i.e., how the components of the Riemann tensor change when you change coordinate systems).

Thank you for your very good explanation. I'd like to point out that Lorentz transformation is the central feature of both SR and GR and that it was proposed by Lorentz way before SR and the contribution of Minkowski. In terms of geometry, if you take an event described in one sys of coordinates, the same event will appear deformed when transposed to another sys of coordinates. Perhaps you're right in saying that curvature should be used only in reference to spacetime. I should have said deformation (of the appearance of event), which, however, does not change the geometrical gist of the matter.

 

[PeterDonis]

Thank you for your very good explanation. I'd like to point out that Lorentz transformation is the central feature of both SR and GR and that it was proposed by Lorentz way before SR and the contribution of Minkowski.

This is true, but it doesn't change the fact that the Lorentz transformation, as a specific kind of coordinate transformation, has only a very particular role in GR; it does *not* play a general role as a transformation between arbitrary coordinate systems.

In terms of geometry, if you take an event described in one sys of coordinates, the same event will appear deformed when transposed to another sys of coordinates.

An event is a single point. How can it be "deformed"?

Perhaps you meant to say an "object" can appear deformed? If so, that is true, but a Lorentz transformation can only lead to very particular types of "deformation" (length contraction and time dilation, if we consider the latter to be a "deformation" in the time dimension). These "deformations" can occur in flat spacetime, which is why a Lorentz transformation can describe them. In a curved spacetime, much more general types of "deformation" are possible, which a Lorentz transformation can't describe.

Perhaps you're right in saying that curvature should be used only in reference to spacetime. I should have said deformation (of the appearance of event), which, however, does not change the geometrical gist of the matter.

You are correct that it's all about geometry, but a Lorentz transformation can only be used in a very particular type of geometry, flat Minkowski spacetime (or a small, local patch of a curved spacetime that looks like a patch of Minkowski spacetime to a good enough approximation). There are much more general types of geometries and coordinate transformations that are used in GR.

 

[Spacie]

I'm glad I e-spoke to you about this. Thank to you now I have a much better understanding of geometry in GR.

An event is a single point. How can it be "deformed"?

An event is not a single point. An event has at least a length. An event is an action.

... There are much more general types of geometries and coordinate transformations that are used in GR.

Ah, but all those different geometries and coordinate transformations... what would they be without Lorentz' idea of how to explain the null result in MM experiment? Could SR or GR be even conceived without Lorentz transformation?

 

[PeterDonis]

I'm glad I e-spoke to you about this. Thank to you now I have a much better understanding of geometry in GR.

Thanks!

An event is not a single point. An event has at least a length. An event is an action.

Not as the term "event" is standardly used in relativity (both SR and GR). If you were using it differently, then I understand why you said what you said; but in the standard terminology, an "event" is a point in spacetime: a specific location in space at a specific time. Mathematically, in a given coordinate system, an event is described by a single set of coordinate values (x0, x1, x2, x3).

Ah, but all those different geometries and coordinate transformations... what would they be without Lorentz' idea of how to explain the null result in MM experiment? Could SR or GR be even conceived without Lorentz transformation?

Historically speaking, that's an interesting question. If I remember correctly, Einstein did not cite the MM experiment in his initial 1905 papers, but he did draw on Lorentz's previous work, which was prompted (I believe) by that experiment.

Physically speaking, as the paper linked to by bcrowell earlier in this thread makes clear, the principle of relativity plus the homogeneity and isotropy of space are enough to narrow down the allowable local transformations to two: Galilean and Lorentz. I agree you would still need some other piece of information to choose between them, such as the null result in the MM experiment (the result would not be null if spacetime were locally Galilean instead of locally Lorentzian). But you could certainly develop the theory of both types of transformations, mathematically, without knowing for sure which one was the "right" one, physically.

As for GR, obviously local Lorentz invariance is a key component, but there is also a curved spacetime version of Newtonian gravity, which has local Galilean invariance. It was developed by Cartan in the 1920's. See here:

http://en.wikipedia.org/wiki/Newton–Cartan_theory

So you could also develop the theory of gravity under both types of transformations, mathematically, without knowing for sure which one was right, physically. In the Lorentz invariant version, obviously you would need a way to handle local Lorentz transformations, but they would be derived from the theory, not fed into it as prior information.

 

[Parlyne]

-?! What is mass in your definition? According to relativity, mass is the force that curves space around it. The light only follows that curvature.

If you refer to the equivalence of energy and mass, you have to agree that mass is a particular expression of energy.

Mass and force are separate things. And, in GR, it's the stress/energy tensor (which includes energy, momentum, pressure, mass, etc.) which sources gravity. Not just mass.

What I'm talking about is inertial mass. You know, the kind of mass that shows up in $\vec{F}=m\vec{a}$. Or, in SR, shows up in $E=\sqrt{m^2 c^4 + |\vec{p}|^2 c^2}$.

Yes, you're right the speed of light in vacuum is an empirical fact = the law of nature = the inherent property of space.

You're missing my point. The fact that the mathematical structure we call special relativity has an invariant speed can be mathematically derived from the spacetime geometry. That fact that the physical phenomenon we call light travels at that speed is empirically derived and, as such, contingent. It's always possible that increasingly precise measurements will find that the speed of light actually has a bit of energy dependence, indicating that light actually has mass and, as such, travels slower than the invariant speed. (But, as I pointed out above, no such evidence exists at this time.) The fact that a property of light itself matters here means that the speed at which light travels is not just an inherent property of space.

Here you are making an assumption, the basis of which is not clear. You apparently assume that a neutrino, like a photon, also has its "constant speed" of flight. But from where does it follow? Is it possible that its speed depends merely on the force with which it was propelled? We know that this is not true of a photon: SR states that its speed through vacuum will always be the same, regardless of the speed of matter that emitted it (or observed, lol). A neutrino, having a mass, should have its speed dependent on the speed of the object that emitted it. That's the key difference between matter and light according to relativity.

There's no assumption here that the speed at which the neutrinos in the experiment traveled is invariant. Your argument is akin to saying that I can't measure how fast I was driving my car because my car doesn't travel at an invariant speed. It's a non-sequitur. The distance from source to detector is well defined, as is the time of flight. That let's you calculate the speed. Nothing strange. Nothing necessarily relativistic either (except the procedures necessary to keep the clocks synchronized).

Nothing is "simply" energy but each particular expression of energy is manifested differently in space. You're right to point out that photons interact with charged particles. Something that cannot be said about a neutrino. That and the fact that a neutrino has mass, while photon does not, may explain the difference in how they interact with vacuum.

This doesn't change that the corrections to a particle's mass due to interactions with vacuum fluctuations are (unless the particle is a scalar; but, that's another issue) proportional to the value the mass would take without those interactions. If one is 0, both are 0.

 

[Spacie]

...in the standard terminology, an "event" is a point in spacetime: a specific location in space at a specific time. Mathematically, in a given coordinate system, an event is described by a single set of coordinate values (x0, x1, x2, x3).

Ah! that explains why my incorrect terminology led me to require an additional spatial dimension in order to visualize action in GR. Even though, from a purely geometrical standpoint, a point in space can only designate a location. An action implies some sort of change (in my head at least), which implies the need for some length, be it in time or space (which in GR are conveniently merged together. I would have to chew on this. Thanks!)

If I remember correctly, Einstein did not cite the MM experiment in his initial 1905 papers, but he did draw on Lorentz's previous work, which was prompted (I believe) by that experiment.

You remember absolutely correctly. I have just reviewed the history, so it's still fresh in my head.

Physically speaking, as the paper linked to by bcrowell earlier in this thread makes clear, the principle of relativity plus the homogeneity and isotropy of space are enough to narrow down the allowable local transformations to two: Galilean and Lorentz. I agree you would still need some other piece of information to choose between them, such as the null result in the MM experiment (the result would not be null if spacetime were locally Galilean instead of locally Lorentzian). But you could certainly develop the theory of both types of transformations, mathematically, without knowing for sure which one was the "right" one, physically.

As for GR, obviously local Lorentz invariance is a key component, but there is also a curved spacetime version of Newtonian gravity, which has local Galilean invariance. It was developed by Cartan in the 1920's. ...

So you could also develop the theory of gravity under both types of transformations, mathematically, without knowing for sure which one was right, physically. In the Lorentz invariant version, obviously you would need a way to handle local Lorentz transformations, but they would be derived from the theory, not fed into it as prior information.

All this is very interesting. Thanks. I'll go study this.

 

[Spacie]

Mass and force are separate things. And, in GR, it's the stress/energy tensor (which includes energy, momentum, pressure, mass, etc.) which sources gravity. Not just mass.

So, according to you, mass and force are separate things, but energy and mass are the same thing. Right? Then you would have to define each of these 3 things for me. My understanding is simple: everything is energy. Forces are geometrical representations of energies (= energies "geometrized"). Mass is a particular geometrization of energy (or energies, if you will) that results in curvature of spacetime we call gravity.

As for stress/energy tensor being the source of gravity, I confess that I have trouble with this "metaphysical" interpretation. I consider myself a pragmatist and realist. So in my layman view, saying that stress/energy tensor, which is a mathematical abstraction, is the source of gravity, which to me is a very real force that keeps me securely on Earth and satellites from falling, is not far removed from saying that Atlas is holding the sky on his shoulders, or like in medieval times they said that heavens were held up by the decree of God.

There got to be a more pragmatic interpretation of the source of gravity in GR. Like, according to Newton, it's just what goes on between two masses, which are "real" things (even though they act somewhat mysteriously at a distance, but still, this is a more "realistic" view). So, my head is satisfied that the tensor accurately describes the math involved. What is lacking is a pragmatic know-how acceptable for the benefit of my "gut feeling" that just keeps on rebelling.

What I'm talking about is inertial mass. You know, the kind of mass that shows up in $\vec{F}=m\vec{a}$. Or, in SR, shows up in $E=\sqrt{m^2 c^4 + |\vec{p}|^2 c^2}$.

In both cases, I visualize a sail inflated by wind. Which is only a 2D plane curved in 3D. There seems to me that to distinguish between inertial and invariant masses, from the geometrical standpoint, some additional dimensions are in order. What do you think?

The fact that the mathematical structure we call special relativity has an invariant speed [for light] can be mathematically derived from the spacetime geometry.

This is an example of circular reasoning. The mathematics of SR are based on the assumption that speed of light, as "a law of nature", is invariant for all observers. You have to remove that initial assumption and prove that it works out the same in the end in order to claim that this as a logical outcome of the theory. (And the situation with GR in this regard is even worse, since it employs Minkowski's spacetime, which, in turn, stands on the axiom from which it follows a priori that nothing can ever possibly move faster than light in vacuum.)

That fact that the physical phenomenon we call light travels at that speed is empirically derived and, as such, contingent. It's always possible that increasingly precise measurements will find that the speed of light actually has a bit of energy dependence, indicating that light actually has mass and, as such, travels slower than the invariant speed. (But, as I pointed out above, no such evidence exists at this time.) The fact that a property of light itself matters here means that the speed at which light travels is not just an inherent property of space.

Once you start messing with the concept of a mass and make it loose like this, IMO this makes an entirely different theory than SR, let alone GR, which, essentially, is based on 3 concepts: light, matter and space, with mass being a property of matter but not light. The consequent equivalence of mass and energy does not invalidate the underlying geometry, from which it follows. It can't. For, otherwise it would be equivalent of pulling the carpet from under your own feet, or hacking off the brunch on which you sit. That's where, in my view, you're making a logical error in reasoning. Yes, everything is energy, but geometrically speaking, each expression of energy is distinct and has distinct "geometrical consequences".

This doesn't change that the corrections to a particle's mass due to interactions with vacuum fluctuations are (unless the particle is a scalar; but, that's another issue) proportional to the value the mass would take without those interactions. If one is 0, both are 0.

See, for me, the whole trouble with QM is in the fact that it replaced geometry of space with abstract properties of point-like particles of matter, which are also forces. This makes it very difficult, if not impossible, to visualize, and, consequently, to understand.

 

[Parlyne]

So, according to you, mass and force are separate things, but energy and mass are the same thing. Right? Then you would have to define each of these 3 things for me. My understanding is simple: everything is energy. Forces are geometrical representations of energies (= energies "geometrized"). Mass is a particular geometrization of energy (or energies, if you will) that results in curvature of spacetime we call gravity.

Energy and mass are emphatically not the same thing, no matter how many bad popularizations claim otherwise. The content of $E=mc^2$ is to say that mass is a type of energy, not to say that mass and energy are totally equivalent. To take an extreme case, a (presumably massless) photon can carry energy.

Forces can't really be thought of as a representation of energy, particularly since it's not too hard to construct situations where the action of a non-zero force does not change the energy of any part of a system. (All you need here is a force which is always perpendicular to the motion of the object it acts on.) Operationally, you can define a force as anything which changes the momentum of an object. Mass represents an object's inertial resistance to its motion being changed. (This, of course, can be made into a more mathematically precise statement.) There's no need to go to any sort of geometrical construction. These definitions haven't changed conceptually since Newton. All that's changed is the math necessary to describe them in a manner consistent with the present understanding of space and time.

As for stress/energy tensor being the source of gravity, I confess that I have trouble with this "metaphysical" interpretation. I consider myself a pragmatist and realist. So in my layman view, saying that stress/energy tensor, which is a mathematical abstraction, is the source of gravity, which to me is a very real force that keeps me securely on Earth and satellites from falling, is not far removed from saying that Atlas is holding the sky on his shoulders, or like in medieval times they said that heavens were held up by the decree of God.

There got to be a more pragmatic interpretation of the source of gravity in GR. Like, according to Newton, it's just what goes on between two masses, which are "real" things (even though they act somewhat mysteriously at a distance, but still, this is a more "realistic" view). So, my head is satisfied that the tensor accurately describes the math involved. What is lacking is a pragmatic know-how acceptable for the benefit of my "gut feeling" that just keeps on rebelling.

I wasn't making some grand metaphysical statement. It seems like you're interpreting what I said as meaning that "the stress-energy tensor is why there is such a thing as gravity," when I actually meant that "the geometry of spacetime is determined by the stress-energy tensor of all the stuff in the spacetime in much the same way that the Newtonian gravitational field is determined by the mass of all the stuff in space."

In both cases, I visualize a sail inflated by wind. Which is only a 2D plane curved in 3D. There seems to me that to distinguish between inertial and invariant masses, from the geometrical standpoint, some additional dimensions are in order. What do you think?

You've lost me here. I don't really see a need to try to make mass a geometrical quantity; so, I don't really see your point.

This is an example of circular reasoning. The mathematics of SR are based on the assumption that speed of light, as "a law of nature", is invariant for all observers. You have to remove that initial assumption and prove that it works out the same in the end in order to claim that this as a logical outcome of the theory. (And the situation with GR in this regard is even worse, since it employs Minkowski's spacetime, which, in turn, stands on the axiom from which it follows a priori that nothing can ever possibly move faster than light in vacuum.)

It's only circular because you changed what I wrote. What I said was that the mathematics of SR requires that there be some speed which is invariant under boosts to any inertial reference frame. This has nothing to do with light or anything else, even though Einstein first came up with the mathematical structure by thinking of light. To put it more concretely, Einstein's original axioms are mathematically equivalent to:

1) The spacetime interval between to events (that is, points in spacetime), given by $s=\sqrt{c^2\Delta t^2-|\Delta \vec{x}|^2}$, is independent of the inertial frame in which the positions and times are measured.

and

2) Any path traveled by light in a vacuum will have spacetime interval of 0.

All of the usual SR discussion about time dilation, fast rockets, the twin paradox, and even the Lorentz transformation and Minkowski geometry follow from (1). (2) only comes into play when talking about light specifically. This is why I say that the massless nature of light is a separate issue from the structure of SR.

Oh, and to be clear, Minkowski geometry is a property of SR. The geometry is GR, while always having a Minkowski structure on a sufficiently local scale, is quite a bit more complicated, in general, as it is a dynamical quantity.

Once you start messing with the concept of a mass and make it loose like this, IMO this makes an entirely different theory than SR, let alone GR, which, essentially, is based on 3 concepts: light, matter and space, with mass being a property of matter but not light. The consequent equivalence of mass and energy does not invalidate the underlying geometry, from which it follows. It can't. For, otherwise it would be equivalent of pulling the carpet from under your own feet, or hacking off the brunch on which you sit. That's where, in my view, you're making a logical error in reasoning. Yes, everything is energy, but geometrically speaking, each expression of energy is distinct and has distinct "geometrical consequences".

I'm not making the concept of mass at all loose. I'm just pointing out that every observation which addresses the properties of light has finite precision and that, with added precision, we could always find something unexpected. An object with extraordinarily small mass and energy large enough to measure will be moving at speed very close to c. For sufficiently small mass, we would not yet have been able to measure the deviation, even if it is there.

The point that I'm trying to make is that relativity has nothing at all to do with light or with matter. SR is strictly about the geometry of spacetime, which has the effect of specifying what the kinematics of objects in the spacetime look like. GR adds the way that the geometry of the spacetime responds to the stress-energy tensor of the stuff in the spacetime; but, it does so with no reference to any specific properties of that stuff.

See, for me, the whole trouble with QM is in the fact that it replaced geometry of space with abstract properties of point-like particles of matter, which are also forces. This makes it very difficult, if not impossible, to visualize, and, consequently, to understand.

Quantum field theory (QFT) states that forces can be thought of (approximately) as resulting from the emission and absorption of certain kinds of particles. I don't know that this part is all that strange, since the emitted and absorbed particles are simple a way of moving energy and momentum from one particle to another. But, it's probably worthwhile to keep in mind that the geometry of SR is actually encoded into the structure of the QFTs we use to describe the world.

 

[kmarinas86]

What makes the speed of light a constant. I read the FAQ on special relativity but still don't understand why c (speed of light) exists as a constant.

It's like a rule like many others, why do they exist? Is there a part of space-time that limits this speed. Why are all the photons that ever existed limited by this speed?

Let's take an equation:

$$v^2 + u^2 = c^2$$

Given that all mass can be created from pure energy (light), we could say that there is energy inside of us that propagates and reflects at the speed of light $c$ relative to some arbitrary frame (later on we can show how this can appear to be the case in other frames). Make sense yet?

Now look at this page titled The mirror problem a thought experiment on time dilation:

http://www.schoolphysics.co.uk/age16-19/Relativity/text/Time_dilation/index.html

[PLAIN]http://www.schoolphysics.co.uk/age16-19/Relativity/text/Time_dilation/images/1.gif

So imagine that we moved at $v$ and light within us bounces within us at $c$. We would experience a time dilation equal to:

$$c/u = 1/sqrt(1-v^2/c^2)$$

This can be proven to be mathematically equivalent to the equation above:

$$u/c = sqrt(1-v^2/c^2)$$
$$u^2/c^2 = 1-v^2/c^2$$
$$u^2 = c^2-v^2$$
$$v^2 + u^2 = c^2$$

So while energy in an object does not move in a straight line, light that passes through the object can (if it is not disturbed) can. Two details to consider here:

1) Light that passes through an object transparent to it passes ahead of it at a relative velocity of c-v (if going in the same direction) or c+v (if going in the opposite direction).
2) The peak-to-peak time period between wave crests adjusts accordingly. If going in the same direction, this period increases by a factor * c / (c-v). If going in the opposite direction, this period decreases by a factor * c / (c+v).

As far as the object itself is concerned, it is time dilated by a factor of 1/sqrt(1-v^2/c^2), for whom (or 'which' if it is a thing and not a person) events of external origin occur in time intervals shorter by a factor of * sqrt(1-v^2/c^2). So for an object moving in the same direction as the light, the time period between peaks is observed to change by a factor of * [sqrt(1-v^2/c^2)] * [c / (c-v)], and so the frequency is seen to change by a factor of * [1/sqrt(1-v^2/c^2)] * [(c-v)/c]. These factors can be simplified to * $\sqrt{\frac{\left(1+v/c\right)}{\left(1-v/c\right)}}$ and * $\sqrt{\frac{\left(1-v/c\right)}{\left(1+v/c\right)}}$, respectively.

The wavelength observed by an 'internal', rather than 'external', observer must in this case increase by a factor of * $\sqrt{\frac{\left(1+v/c\right)}{\left(1-v/c\right)}}$, as would be predicted by the relativistic doppler effect. This again breaks into two terms which need explaining:

1) * sqrt(1-v^2/c^2), this is caused by length contraction of the observer relative to the wavelength of the incoming wave.
2) * c / (c-v), this is caused by the parallel motion of both the light and the observer, having a slight "scrunching" effect on the waveform inside the object.

If it weren't for the length contraction of the observer, then the wavelength of the light would actually be seen to shortened. This would becomes clear if we decided to project a standing interference pattern inside the object (let's say it's a box) by shining light through two small slit openings; the faster the object moved in the same direction as the light, the more scrunched this standing interference pattern would, if it were not for the length contraction, appear. So by definition, the object must contract further lengthwise relative to external electromagnetic waves when accelerating to properly account for the wavelength change that is actually observed of light. The factor by which they are contracted and the factor by which they are time dilated together produce the result that the speed of light $c$ that an 'external' observer sees is also the same speed of light $c$ that an observer 'internal' to that object would also see. The origin of the length contraction hypothesis predates Einstein's Special Theory of Relativity: http://philsci-archive.pitt.edu/987/1/Michelson.pdf. The same is true for the idea of time dilation: http://en.wikipedia.org/wiki/Relativity_priority_dispute#Harvey_R._Brown_.282005.29.

The 'external' electromagnetic waves, which do not interfere with the object, are completely neutral to whether or not the object is accelerating, moving, or not at all. They are independent and do not change their fundamental nature, that is to say they do not without becoming 'internal' respond with any particular action at a distance, whatever the observers and objects involved in such action.

There is a frame in which the speed of internal "bouncing" of light $u$ in an object is maximized such that $u=c$. This is called the rest frame of the object. If this rest frame were the same as that for all others, this would be called the Lorentzian Ether Frame. In Special Relativity, this frame would be unspecified, and even perhaps non-existent, though technically there can still be a frame in which the internal "bouncing" of light in mass $u$, or even that of any mass possessing degrees of freedom, is at an (unobservable) maximum $u=c$. In either case, the velocity of light changes apparent direction with respect to the directional movement of the observer, rather than speed, during frame changes of the observer, an effect which produces the aberration of light. The difference with the notion of a Lorentzian Ether Frame is that the length contraction has a component independent of any external observer's motion, and also, the time dilation has a physical component not relative to any external observer's motion. What do remain relative to the observer's relative motion are the frequencies and wavelengths of signals received from the object that notify the existence of the object to the observer, as well as the "observed" length contraction and time dilation of that object.

 

[PeterDonis]

The mathematics of SR are based on the assumption that speed of light, as "a law of nature", is invariant for all observers. You have to remove that initial assumption and prove that it works out the same in the end in order to claim that this as a logical outcome of the theory.

And the point of the paper that bcrowell linked to early in this thread is that it does exactly this: it removes the assumption that the speed of light is the same for all observers, and instead *derives* the theorem that there must be some invariant speed from the principle of relativity and the homogeneity and isotropy of space. The only way light comes into it is that we believe, based on our best current knowledge, that photons, particles of light, have zero rest mass, and any particle that has zero rest mass has to move at the invariant speed.

(And the situation with GR in this regard is even worse, since it employs Minkowski's spacetime, which, in turn, stands on the axiom from which it follows a priori that nothing can ever possibly move faster than light in vacuum.)

Minkowski spacetime is only one particular solution of the Einstein Field Equation, in which the stress-energy tensor is identically zero, there is zero cosmological constant, and the curvature is identically zero everywhere. There are lots of other spacetimes that are also solutions to the EFE but are very different from Minkowski spacetime.

GR does use the principle of *local* Lorentz invariance, i.e., a very small patch of any spacetime looks, locally, like a very small patch of flat Minkowski spacetime. But the logical basis for that works the same as I outlined above.

 

[Spacie]

Energy and mass are emphatically not the same thing...
a (presumably massless) photon...

I don't understand you. Sorry. To me you seem to contradict yourself, claiming that a photon may have mass, presumably because photon's energy may be interpreted as mass.. and then "emphatically" denying that. Are you simply messing with me, 'cause I'm a newb?

Forces can't really be thought of as a representation of energy...

All the time. Think of vectors.

Mass represents an object's inertial resistance to its motion being changed.

That's a good reminder for me, thanks. I am a strongly visual type, so I tend to "geometrize" everything. This view of mass says a lot about the properties of space.

There's no need to go to any sort of geometrical construction. These definitions haven't changed conceptually since Newton. All that's changed is the math necessary to describe them in a manner consistent with the present understanding of space and time.

I believe there is no understanding of space at all. There is no even theory of space, like, for example, what forces make up its dimensions, why 3, how space is formed and how it interacts with matter, etc, etc. And space and geometry go together.

I wasn't making some grand metaphysical statement. It seems like you're interpreting what I said as meaning that "the stress-energy tensor is why there is such a thing as gravity," when I actually meant that "the geometry of spacetime is determined by the stress-energy tensor of all the stuff in the spacetime in much the same way that the Newtonian gravitational field is determined by the mass of all the stuff in space."

My words were not directed at you, sorry if it seemed so. That's how it is said everywhere nowadays. Last time I consulted wiki, it said there too, that the stress-energy tensor is the source of gravity. A mathematical abstraction as a source of a very, very real force is... I am lost for words. So, no, my criticism was not directed at you but at this... incongruity.

The point that I'm trying to make is that relativity has nothing at all to do with light or with matter. SR is strictly about the geometry of spacetime, which has the effect of specifying what the kinematics of objects in the spacetime look like. GR adds the way that the geometry of the spacetime responds to the stress-energy tensor of the stuff in the spacetime; but, it does so with no reference to any specific properties of that stuff.

This makes no sense to me. The way I see it, these "advanced" interpretations of relativity only make people confused. The fundamental difference between light and matter is removed (everything is a point-like particle with properties in empty space), and this invites people to start comparing light with matter and treat them on par. This does not lead to clarity but has the opposite effect.

Quantum field theory (QFT) states that forces can be thought of (approximately) as resulting from the emission and absorption of certain kinds of particles. I don't know that this part is all that strange, since the emitted and absorbed particles are simple a way of moving energy and momentum from one particle to another. But, it's probably worthwhile to keep in mind that the geometry of SR is actually encoded into the structure of the QFTs we use to describe the world.

Really? I start thinking that you're simply messing with me.

 

[Spacie]

And the point of the paper that bcrowell linked to early in this thread is that it does exactly this: it removes the assumption that the speed of light is the same for all observers, and instead *derives* the theorem that there must be some invariant speed from the principle of relativity and the homogeneity and isotropy of space. The only way light comes into it is that we believe, based on our best current knowledge, that photons, particles of light, have zero rest mass, and any particle that has zero rest mass has to move at the invariant speed.

Yes, thank you very much, I am looking into this paper now. I have not done it before, because bcrowell's terminology seemed a bit over my head, when I first heard it (mainly because I have not used those words in years), so all I could do was duck, lol. That paper is exactly what I needed. Not because I doubt the constancy of the speed of light in empty space for all observers --quite to the contrary!-- but I have doubts about the conclusion that nothing can move faster than c. To me it seems that chargeless massive particles should be able to move >c and so I am trying to find the loophole in the theory that would allow that.

Minkowski spacetime is only one particular solution of the Einstein Field Equation, in which the stress-energy tensor is identically zero, there is zero cosmological constant, and the curvature is identically zero everywhere. There are lots of other spacetimes that are also solutions to the EFE but are very different from Minkowski spacetime.

GR does use the principle of *local* Lorentz invariance, i.e., a very small patch of any spacetime looks, locally, like a very small patch of flat Minkowski spacetime. But the logical basis for that works the same as I outlined above.

Thank you for all that. That's just what I needed. Minkowski spacetime is not good for my purposes, because, as I said, I found that it has the ceiling on the speed already "built in".

Thank you very much! :)