What is Special Relativity based upon?

[arcnets2]

Hi all,
at the risk of boring everybody, I have a question that has probably been asked before.
We all know that Special Relativity is based upon the concept of invariant light speed. Let some light travel a distance s in time t, then
(ct)^2 - s^2 = 0 should hold for any observer (in an inertial system).
Let anything else be at rest in some inertial system, then (ct)^2 - s^2 for this body should be the same for all observers.

OK. This leads to time dilatation and space contraction. But how about the relativistic mass formula? It seems to me that we need some extra assumptions to get that. Or do you disagree?

 

[robert Ihnot]

Max Born writing, Einstein's Theory of Relativity, mentions that Einstein had a Thought Experiment to show E=mc^2, on p283, and Born adds:

"Based on the fact that radiation exerts a pressure. From Maxwell's field equations supplemented by a theorem first deduced by Poynting (1884) it follows the momentum transferred to an absorbing surface by a short flash of light is equal to E/c." He adds, confirmed experimentally by Lebedew, 1890, and with greater accuracy by Hull, 1901.

Born tells us that this thought experiment "does not make use of mathematical formalism in the theory of relativity." But Born does use the equation: Mv=E/c.

Now, it is true that Einstein did not put his famous equation in the original paper on Special Relativity, but published it later the same year: Does the Inertia of a Body Depend upon its Energy Content?"

In this paper Einstein concludes: If a body gives off the energy L in the form of radiation, its mass diminishes by L/c^2. There doesn't seem any evidence that Einstein either depended upon actual experimental fact, nor did he mention any new assumptions not already presented in his earlier paper.

So the answer seems to be: No. No additional assumptions were found necessary.

 

[jtbell]

But how about the relativistic mass formula?

There was a long discussion about this just a couple of weeks ago:

Derivation of the Equation for Relativistic Mass

A related discussion, also from a couple of weeks ago:

Derivation of the Equation for Relativistic Momentum

 

[bernhard.rothenstein]

basis of SR

Hi all,
at the risk of boring everybody, I have a question that has probably been asked before.
We all know that Special Relativity is based upon the concept of invariant light speed. Let some light travel a distance s in time t, then
(ct)^2 - s^2 = 0 should hold for any observer (in an inertial system).
Let anything else be at rest in some inertial system, then (ct)^2 - s^2 for this body should be the same for all observers.

OK. This leads to time dilatation and space contraction. But how about the relativistic mass formula? It seems to me that we need some extra assumptions to get that. Or do you disagree?

IMHO the simplest answer is: SR is based on experiment and on the principle of relativity which leads to the equations which account for the different relativistic effects.
Experiments lead to m=g(V)m(0) (Bucherer, Kaufmann). Many disagree with the name given to m but aggree that m(0) is Newton's mass. Never mind. Multiplly both sides of it with cc call mcc=E energy and m(0)cc rest energy (in accordance with the physical dimensions) and nobody will blame you. Multiplying with c we obtain mc which has no physical suport (a tardyon never moves with c).
Special relativity becomes involved when the tardyon moves relative to both the invoved inertial reference frames and so observers from the two frames measure energy and not energy.
Is there more to say?:rofl:

 

[masudr]

Demanding that that [itex]c^2 t^2 - \vec{x}^2[/tex] is invariant in inertial reference frames actually leads to modelling spacetime as a 4-dimensional flat space, with the metric $\eta_{ab} = \mbox{diag}(1,-1,-1,-1)$, and defining Lorentz transformations so that this metric is unchanged by two copies of the transformation acting on it, see this post.

This is done so that the vector $[ct, x, y, z]$ transforms as it should. One then defines proper time as

$$d\tau = \frac{1}{c}\sqrt{g_{ab}dx^a dx^b}$$

and then four-momentum as that which is conserved in collisions comes out as

$$p^a=m_0 \frac{dx^a}{d\tau}$$

We recognise that the temporal component $\gamma m_0 c^2$ looks like $\frac{1}{2}m_0 v^2 + m_0 c^2$ and the spatial components $\gamma m_0 \vec{v}$ look like $m_0 \vec{v}$ at slow speeds (compared to c, of course). And then we say we can call $m\gamma$ relativistic mass, because it can make some formulae nicer.

That spacetime is a 4-dimensional flat space, with the above metric, and that Lorentz transformations (as defined in the above linked post) correspond to transformations between inertial frames appears to be the essential content of special relativity.

 

[coalquay404]

Is it not perhaps worth our while to place a few stickies at the top of the relativity forum so that questions like this don't get asked every day?

 

[lalbatros]

The additional assumption needed to get E=mc² is that the laws of mechanics are Lorentz-invariant. Or in other words, the laws of mechanics should have the same symmetrys as the laws of electromagnetism.

 

[arcnets2]

Wow, that helps a lot. And you are fast. THANKS!

 

[eigenglue]

OK. This leads to time dilatation and space contraction. But how about the relativistic mass formula? It seems to me that we need some extra assumptions to get that. Or do you disagree?

The extra assumption is that conservation of momentum holds true in all inertial reference frames. That leads to the to the relativistic mass formula.

 

[bernhard.rothenstein]

conservation of momentum

The extra assumption is that conservation of momentum holds true in all inertial reference frames. That leads to the to the relativistic mass formula.

Please extend why the extra assumption of conservation is compulsory. I think it is not. The concept of rest (Newtonian) and the addition law of velocities do the job!

 

[bernhard.rothenstein]

what to post and what not

Is it not perhaps worth our while to place a few stickies at the top of the relativity forum so that questions like this don't get asked every day?

I think that for a new born all the jokes (questions) are new! It is up to the participants to do not repeat given answers or to give illuminating ones.
I think that it would be better to put on the top of relativity forum the invitation to a polite styl of conversation!

 

[Hootenanny]

The extra assumption is that conservation of momentum holds true in all inertial reference frames. That leads to the to the relativistic mass formula.

Please extend why the extra assumption of conservation is compulsory. I think it is not. The concept of rest (Newtonian) and the addition law of velocities do the job!

I agree. Conservation of momentum in all inertia reference frames, isn't really an addditional assumption, but rather a consequence of the principle of SR;

"The laws of physics are the same in any inertia frame, irrespective of position of velocity. No frame is prefered."

 

[arcnets2]

OK. But I think an inertial system is defined by
a) actio = reactio holds
b) Newton's Law holds.
Isn't that equivalent to conservation of momentum?

 

[robert Ihnot]

labatros: The additional assumption needed to get E=mc² is that the laws of mechanics are Lorentz-invariant. Or in other words, the laws of mechanics should have the same symmetrys as the laws of electromagnetism.

Well, if we go back to Einstein's original paper on SR, he says on page 3, "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."

In short, is that not The Principal of Relativity?