A quick note on Relativity

This post is intended to be a very concise rebuttal to one of the most popular misconceptions about the theory of Special Relativity.

—  1. Brief Historical Review  —

For the ones that really want to have a better grasp of this complicated issue you can go to:

• the wikipedia article
• The MacTutor History of Mathematics page
• the articles by Dr. SkySkull
• this very interesting article which I haven't actually read... 

But what I really want to say is that we can basically say that the catalyst of Relativity was the Electromagnetic Theory of Maxwell. After deriving the full set Maxwell's equations Maxwell was able to show that the Electromagnetic Field propagated itself like a wave whose speed was $ {c=1/\sqrt{\epsilon_0 \mu_0}} $.

This simple fact had at least two problems to it:

1. According to Galileo's transformations any speed was relative to the frame were it was being measured. But this value of $ {c} $ was an invariant.
2. According to classical Physical ideas if some phenomenon was described by a wave equation than something had to be waving.

In the case of the electromagnetic field that question of what exactly was waving was a very though nut to crack. Just remember the simple fact that we can see the Sun even though the space between us and the Sun is essentially void.

To solve these two problems with one go the Physicists proposed that there was a substance that existed all over the space that waved whenever the electromagnetic radiation was being carried. And the speed of light was indeed an invariant, but it was an invariant relatively to the aether...

But that of course begged the question of why didn't we find any variation in the speed of light since the Earth's movement relative to the aether was changing all the time?

The answer was that the methods weren't good enough to detect such a little variation. And the hunt was on to devise a method with enough sensibility to detect the more than obvious changes.

A lot of things were tried, all of them failed, a lot of partial answers were proposed to explain this failure but a coherent picture was missing!

—  2. Enter Einstein  —

This of course changed in 1905 when a young man named Einstein publish some of the best articles in the history of our business.

One of these articles was a major rethinking of the concepts of space and time and really is a beauty to be read for any self-respecting physicist. But I digress...

With just two simple (and seemingly contradictory axioms) Einstein was able to show why no variation in the value of the speed of light could be detected and he also predicted many more phenomena that initiated a whole new cultural Zeitgeist for the years to come.

His two principles were:

1. The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.
2. Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body 

—  3. The misconception  —

After this somewhat lengthy introduction is time for me to state what I consider to be the popular misconception: "The theory of relativity says that nothing can travel faster than light."

Well, we surely don't see that being said by Einstein in the previous quote! What he says is that the speed of light is an invariant (even though even in University textbooks we can read the first postulate as saying that the speed of light is the limit of all speeds in the Universe and that it is the same for all observers).

—  3.1. Moving Frames  —

Well if it isn't a postulate (if it were a postulate it would always be up for experimental scrutiny) maybe it is something that we can derive in Special Relativity?

Well, sort of...

Imagine that you have two inertial frames $ {S} $ and $ {S'} $ and you consider $ {S} $ to be a stationary frame and that $ {S'} $ has speed $ {v} $ relative to $ {S} $. If you know that the speed of a given body to be $ {u'_x} $ in $ {S'} $ than you can show that the velocity of the same body in the inertial frame $ {S} $, $ {u_x} $, to be:

$ \displaystyle u_x=\frac{u'_x+v}{1+\frac{u'_xv}{c^2}} $

Now if you admit that $ {u'_x < c} $ and that $ {v < c} $ you can show that $ {u_x < c} $. But if you don't admit the initial hypothesis than the value of $ {u_x} $ isn't bounded above by $ {c} $.

And now for a proof of the previous assertion:

If $ {u'x < c} $ and $ {v < c} $, that for some $ {\alpha, \beta > 0} $ it is $ {u'_x=c-\alpha} $ and $ {v =c-\beta} $.

Now $ {u_x} $ is

$ {\begin{aligned} u_x &= \dfrac{c-\alpha+c-\beta}{1+(c-\alpha)(c-\beta)/c^2} \\ &= \dfrac{2c-(\alpha+\beta)}{1+(c^2-(\alpha+\beta)+\alpha\beta)/c^2} \\ &= \dfrac{2c-(\alpha+\beta)}{2c^2-(\alpha+\beta)c+\alpha\beta}c^2 \\ &< \dfrac{2c-(\alpha+\beta)}{2c^2-(\alpha+\beta)c}c^2 \\ &= \dfrac{2c-(\alpha+\beta)}{2c-(\alpha+\beta)}c \\ &= c \end{aligned}} $

Which is $ {u_x < c} $ just like we said it would be.

Again all of that this shows is that one can't boost a body to have a speed bigger than $ {c} $ by using an inertial frame whose speed is smaller than $ {c} $ if the speed of the body in $ {S'} $ is smaller than $ {c} $ too.

—  3.2. Accelerating bodies  —

But we still have another chance, though. What if we don't use boosts? What if we accelerate a particle during a sufficient long time interval? Doesn't a speed bigger than $ {c} $ happens eventually?

Let's see.

First we have to define the linear momentum of a particle in Special Relativity. If we want the linear momentum to be conserved in relativistic collisions the definition that makes sense is

$ \displaystyle \vec{p}=\gamma m \vec{u} $

Where $ {\gamma} $ has its usual meaning and $ {\vec{u}} $ is the speed of the body.

The definition of force still is $ {\vec{F}=\dfrac{d\vec{p}}{dt}} $. Thus $ {\vec{F}=\dfrac{d}{dt}(\gamma m \vec{u})} $.

It is possible to show that for a body of constant mass (which is always the case if something doesn't get aggregated to the body or if the body doesn't get broken up - summing up: we aren't falling for the "relativistic mass" erroneous idea) one has

$ \displaystyle  a=\frac{F}{m}(1-u^2/c^2)^{3/2}  $

Now imagine that you are pushing on a body and its speed is getting bigger and bigger. Than what we have is $ {u \rightarrow c} $ but in this case it is $ {a \rightarrow 0} $. Thus the closer the speed of the body gets to $ {c} $ the less it is accelerating until that finally its acceleration is $ {0} $.

The conclusion is that if a body starts with $ {u < c} $ than it would take an infinite amount of time to get to a speed of $ {c} $ and from then on its speed would be $ {c} $ for ever.

—  4. Conclusion  —

Contrary to what is normally said the fact that the speed of light is limit of all speeds on the real world isn't a postulate.

It does hold with the caveat that the bodies have to have speeds less than $ {c} $ to begin with, but nothing in Special Relativity forces the bodies to have speeds less than $ {c} $ when they start moving.

—  5. Appendix  —

I think that in order to finish this post I just have to mention that it is possible to derive the theory of Special Relativity with just the first postulate and one doesn't need to make any mention to the speed of light.

The idea is the following one: imagine that you have three inertial frames $ {S_1} $, $ {S_2} $ and $ {S_3} $ whose axes are parallel to each other.

Then you boost $ {S_2} $ relatively to $ {S_1} $ with a velocity $ {v_{21}} $.

You also boost $ {S_3} $ relatively to $ {S_2} $ with a velocity $ {v_{32}} $.

The question now is: Are the axes of $ {S_1} $ and $ {S_3} $ still parallel?

If you answered the previous question with a resounding yes, than what you're doing is saying that we always live an Euclidean timespace and that the dynamics that rules us are the classic ones.

If on the other hand you just said: Gee, I don't really know... than you're saying that is none of our business to tell the spacetime continuum how it ought to behave and you are willing to study it and only then conclude what its true nature is.

This is just what Feigenbaum did in this great article: The Theory of Relativity - Galileo's Child whose reading I highly reccomend and whose abstract I just have to quote (no bold in the original):

  We determine the Lorentz transformations and the kinematic content and dynamical framework of special relativity as purely an extension of Galileo's thoughts. No reference to light is ever required: The theories of relativity are logically independent of any properties of light. The thoughts of Galileo are fully realized in a system of Lorentz transformations with a parameter $ {1/c^2} $, some undetermined, universal constant of nature; and are realizable in no other. Isotropy of space plays a deep and pivotal role in all of this, since here three-dimensional space appears at first blush, and persists until the conclusion: Relativity can never correctly be fully developed in just one spatial dimension.