General questions about relativistic derivations and deductions

[SeventhSigma]

My questions here, in order:

1. How do Maxwell's equations demand that c is constant for everyone? People say that his equations "derive" c but I always thought that the permeability variables were experimentally derived. I fully understand the derivation of gamma from the time dilation light clock -- that's all good and well. That derives from the concept of c being the same for all observers, but I want to know the logical steps made in determining this as a necessary condition to begin with.

2. How do we know where else gamma applies? How do we derive relativistic mass, relativistic length, relativistic momentum and mass-energy equivalence? I've searched through so many threads on this forum and all I really see are heated disagreements and handwaving. I want to know the logical steps that led us to go "We need to use gamma here and based on this we derive such and such." I figure all relativity needs to assume is that c is held constant and therefore everything else changes to adjust given sufficiently high speeds -- but based on this I want to know how everything since this is derived.

3. If we define energy as an ability to do work, what exactly is the underlying "fuel" here? If we know E=mc^2, this tells us the maximum amount of energy a particular mass can leverage to perform work, correct? What is it about mass that determines its ability to do work?

 

[Bill_K]

SeventhSigma,

You can't really learn it here, you should sit down with a good book on special relativity. We can only address specific questions that might come up after you've gone through a coherent development of the subject.

How do Maxwell's equations demand that c is constant for everyone?

You can't derive Special Relativity from Maxwell's Equations. Maxwell's Equations are invariant under Lorentz Transformations, and in that sense they *allow* Special Relativity. But the interpretation of the Lorentz Transformations as something having real physical significance goes beyond Maxwell.

How do we know where else gamma applies? How do we derive relativistic mass, relativistic length, relativistic momentum and mass-energy equivalence?

By writing down a set of equations for relativistic mechanics that is consistent and covariant under Lorentz transformations. Starting with the fact that p and E/c must be the components of a 4-vector.

If we define energy as an ability to do work...

We define energy as the quantity which is conserved and reduces to the familiar expression for energy in the nonrelativistic limit.

 

[Mike_Fontenot]

[...]
I figure all relativity needs to assume is that c is held constant and therefore everything else changes to adjust given sufficiently high speeds -- but based on this I want to know how everything since this is derived.
[...]

I don't think c being constant for all inertial observers IS derived ... it's just what experiments seem to keep telling us. And those experimental results were hard to swallow for a long time, because they are inconsistent with Newtonian physics.

If you ASSUME that the speed of any given light pulse is always measured to be the same by all inertial observers, and if you also assume that there is no preferred inertial frame, then the Lorentz equations follow. (See Einstein's "Relativity, the Special and General Theory", by Crown publishers, for a simple derivation (in one of the appendices)). Everything else in special relativity then follows, once you have the Lorentz equations.

Mike Fontenot

 

[SeventhSigma]

According to http://en.wikipedia.org/wiki/Four-vector A Four-vector is different from a vector that can be modified with a Lorentz transformation. (EDIT: Sorry I totally misread that. Four-vectors CAN be transformed. But why? How is this different from a normal vector? A normal vector has a direction and magnitude. I presume this is a "three vector" in the sense that it lacks a time component?)

What do Lorentz equations do? What's the point of the transformation? What is invariance? That the equations don't change anywhere?

 

[Vanadium 50]

I don't think c being constant for all inertial observers IS derived ...

Of course it is. Do the derivation. Where does the speed of the observer enter into it?

 

[bcrowell]

Here's a FAQ entry on this topic, with an additional note appended.

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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

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Basically what you're assuming is a different set of axioms than any of the ones discussed above: you want P1+Maxwell. Maxwell's equations are not consistent with Galilean relativity, because they change their form under a Galilean transformation. So if Maxwell's equations hold and we want to have a principle like P1, then we have to have a transformation between frames that is not the Galilean transformation. By P1 we want the laws of physics to have the same form in all frames, and Maxwell's equations are a law of physics, so they must have the same form in all frames. But they predict a definite value for the speed of light, so P1+Maxwell implies P1+P2. From P1+P2 you can prove the Lorentz transformation, and then to check for self-consistency you need to verify that Maxwell's equations retain their form under a Lorentz transformation.

IMO P1+Maxwell is not a desirable axiomatization, because it gives a special role to electromagnetic waves. In 1905, the electromagnetic field was the only fundamental field known, so it made sense to give it a special status. Today we know that there are lots of fundamental fields, so it's silly to base all of relativity on something having to do with just one of them.

 

[Mike_Fontenot]

Of course it is.

Some people DO go to great lengths to avoid making that assumption. Even Einstein, in his Crown book, DID make a point of avoiding it, and instead showed that he could instead start with an arbitrary definition of simultaneity. But if you read essentially all of his other writings on special relativity, he usually just states that all inertial observers will measure any given light pulse to have exactly the same speed, and then just gets going with the stuff that really matters.

I think attempts to avoid starting with the speed-of-light assumption, while possible, result in a more arbitrary-seeming approach. Much more straightforward, in my opinion, is to start with the speed-of-light assumption, based purely on the experimental evidence, and then go from there. In practice, that's usually what's done, even if it's not talked about much.

It's the kind of thing that can be debated and argued about endlessly, but which doesn't amount to a hill-of-beans in practice.

Mike Fontenot

 

[Cleonis]

[...]
How do Maxwell's equations demand that c is constant for everyone? People say that his equations "derive" c but I always thought that the permeability variables were experimentally derived.
[...]

You are contrasting the two, but that is unfounded. The c that arises from Maxwell's equations is numerically the same as the velocity c that is assumed in the axioms of special relativity, but the interpretation is different.

The conceptual environment in which Maxwell's equations were developed was that electric and magnetic field are states of a medium, the socalled 'ether'.

Maxwell's equations do not imply that the speed of light is the same for all observers in inertial motion. They just imply a particular speed of propagation of electromagnetic waves in the luminiferous ether

In Maxwell's equations rate of change of electric field is correlated with strength of the induced magnetic field, and also rate of change of the magnetic field is correlated with the strength of the induced electric field.
These correlations imply the existence of electromagnetic waves, and they imply a particular speed of propagation of these waves. Big question: how come Maxwell's equations allow special relavity, with such a perfect fit that we can say that Maxwell's equations anticipated special relativity?
The answer ties in with wave propagation. A natural wave is a sine wave, which is continuous, and the mathematical description of a continuous propagating wave is a function of both space coordinates and time coordinates. Wave propagation in a medium gives rise to Doppler effects. The Doppler effects are characterized by symmetries that follow the Lorentz transformations. Noteworthy is that these Doppler effect symmetries apply for any wave propagation, not just propagation of electromagnetic waves. The first to undertake theoretical examination of Doppler effects in general was Woldemar Voigt, and in fact it is sometimes argued that Voigt's work anticipated special relativity.In terms of relativistic physics there is a single electromagnetic field, and magnetic effects can be seen as a relativistic side effects of charge carriers in relative motion.
In terms of relativistic physics the permeability constants are a consequence of the fundamental velocity c.There is a profound restructuring in going from ether based concepts to special relativity concepts.
- In terms of ether based concepts the electric field and the magnetic field are distinct entities, albeit with strong mutual couplings. Thus the speed of propagation of electromagnetic waves in the luminiferous ether arises from the permeability properties.
- In terms of relativistic physics there is a single electromagnetic field, and also a fundamental velocity c. The permeability properties then arise from that.

 

[Cleonis]

I think attempts to avoid starting with the speed-of-light assumption, while possible, result in a more arbitrary-seeming approach.

As Benjamin Crowell has pointed out in post #6 of this thread: logical systems have the property that there is a lot of freedom to interchange axioms and theorems without changing the content of the system as a whole.

I favor an approach that minimizes arbitrariness. Also, I feel that a minimal arbitrariness approach will minimize opportunity for a novice to misinterpret the introduction.

We have that the introduction of special relativity changed both the concept of light propagation, and the concept of mechanics of matter in motion. To focus on one and arrive only later at the other introduces arbitrariness.


Transition
It seems to me that the transition from pre-relativistic to relativistic was a replacement of the metric of spacetime. In pre-relativistic physics space was assumed to be described with the Euclidean metric, and time was assumed to flow uniformly.

To adopt the Minkowski metric is to change the concept of light propagation and the concept of mechanics (such as the velocity addition rule) in one move.


We feel that the transition of Newtonian to special relativity, and the transition of special relativity to general relativity must be related to each other in a deep way, but what is that relation?
In standard expositions SR is introduced by way of the light postulate, and GR is introduced by way of the principle of equivalence. Disadvantage: there is no hint of any relation between the two.

But there is such a strong relation! The relation is that in both transitions there was a fundamental rethinking of the metric of spacetime. SR gave us the Minkowski metric, and GR gave us the dynamic (+++-)-signature metric of the Einstein Field Equations.


The 2 fundamental assumptions of special relativity:
- The principle of relativity of inertial motion
- The Minkowski metric


(Many authors, among them Palash B. Pal, have demonstrated that the principle of relativity of inertial motion is very powerful: it constrains things down to just two options: galilean relativity and special relativity.
Palash B. Pal, "Nothing but Relativity," http://arxiv.org/abs/physics/0302045v1 )

 

[JesseM]

What do Lorentz equations do? What's the point of the transformation?

They transform from the coordinates of one inertial reference frame to another moving at a specific velocity relative to the first frame.

What is invariance? That the equations don't change anywhere?

The equations representing the laws of physics in terms of the coordinates of one inertial frame are the same as the equations that express these laws in the coordinates of a different inertial frame. Note that this wouldn't work for non-inertial frames, the equations of the laws of physics would look different in the coordinates of a non-inertial frame.

 

[SeventhSigma]

What does that mean? "This is what I see at this speed vs. what you see at your speed?"

 

[JesseM]

What does that mean? "This is what I see at this speed vs. what you see at your speed?"

No, an inertial frame doesn't refer to what you see visually, your own inertial rest frame is a specially-constructed coordinate system in which you are at rest, and which ideally is constructed out of rulers and clocks at rest relative to you, with the clocks synchronized according to the Einstein synchronization convention. It's important to understand the idea of inertial reference frames since pretty much all of SR is based on them, I suggest you might take a look at some of the intros to SR on this thread (for example take a look at this section of the "Physics virtual bookshelf" link, or this section of the relativity wikibook). You could also take a look at this thread of mine where I illustrated two ruler-clock systems representing different inertial frames, moving alongside one another.