h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Why?

Why does the equivalence of matter and energy just happen to be defined by the square of one of the universe's most fundamental constants? It seems far too neat to be coincidence.

Becauase of Human Mathematics.

The way we get to the Energy is the product of two human creations, (1) the Mass eliminated (M), measured using a human created scale of measurement, and (2) The speed of light (C), measured using another human created scale of measurement. If you were so inclined, you could probably equate E to any universal (as far as we can measure) constant.

Actually E=mc^2 comes from the relationship between Mass and Speed. This also gives a relationship between Kinetic Energy (=0.5mv^2) and Speed.

By looking at the total energy of a particle in mathematical terms we find an extra term that does not classically appear; the Rest Energy. This is the energy of a particle that is completely due to the particles mass. This term (Which is commonly denoted E, but should have a subscript R) is equal to mc^2

Re post 2 - I believe, though am willing to be corrected, that joules, metres and seconds were all already defined prior to Einstein. Otherwise yes, you could just define the unit of energy as mass times any convenient constant.

Indeed, the actual equation E=mc^2 is scale invariant. In its derivation (which I won't post here because all kinds of nasty symbols are involved, unless you want me to try ) no scales are used.

Only the actual values and our understanding of the links between those values is used. Hence when we say Mass (M) we don't state which unit is used, we only use our concept of mass.

Aries - I think you're hinting at the answer I'm looking for - can you expand? Not sure how to word this ... will try an example. I can visualise Ohm's law - I can easily see why resistance, current and voltage are related, and why each is dependent on the others. I can also see why a speeding mass has kinetic energy in proportion to its speed. But not why a mass at rest has energy in proportion to the speed of something else.

Apologies if this requires the entire derivation

No problem

Classical physics gives us the equation K.E. = (1/2)mv^2

Relativity tells us that as we increase the speed of an object it's mass increases. We therefore need to redefine what we mean by Mass; hence we need to redefine Kinetic Energy as well.

When we talk about K.E. classically, it is based on just mass and speed (as given above) but we now need to put in our altered form of mass. When we rearrange this altered equation we get the classically expected K.E. plus another term which doesn't rely on the particles speed (mc^2).

By looking at this on a graph, we see that even as an object slows down to v=0, the total energy does not reduce to 0. This mc^2 term actually tells us the energy that a particle has when it is not moving. i.e. it is the Rest Energy.

Is that better? I'll give the derivation if not

Just in case, I found the derivation on the internet. Unfortunately I can't reproduce the derivation in an easy to understand way because of the integral signs that are involved.

The method outlined in this article isn't complete because to form a complete derivation would mean having to go through the full headache of deriving Gamma and then a sequence of different physical values until we arrive at Kinetic Energy.

The value of c comes in when we correct the mass, the Lorentzian Gamma Factor (Or just Gamma Factor) includes the value of c:

Gamma = (1 - (v^2)/(c^2)) ^ (-1/2)

Here is the article:
http://www.btinternet.com/~j.doyle/SR/Emc2/Derive.htm

Thanks for that Aries, I'll have a read through it tomorrow.

This should be put into an entry y'know. One about 'why' it is as it is cos I remember asking the same question (and getting the same answer, pretty much) way back in the early days - although I can't find the thread.

Not surprising that it is a common question amoungst those who have got past the 'what is it' part and have some concept/understanding or the equation to then start wondering what the speed of light is doing there.

The difficulty in putting it into an entry is that it contains a lot of tricky symbols. Integration signs probably being the worst as there is no other way of expressing them.

h2g2 unfortunately has no facility for the creation of these symbols. Otherwise, it would be a good idea.

I might look into other ways in which it can be done, becuase I agree that it would make quite a good entry.

How about something along these lines

Let rest mass energy Eo = f(m) - an unspecified function of m

Generally Ev = f(m)/sqrt(1-v^2/c^2)

Perform Taylor expansion in v of first two terms

Ev ~ f(m)+ f(m)v^2/2c^2

For low v (Newtonian regime), Ev must ~ f(m)+mv^2/2

Hence f(m)/c^2 = m, hence f(m)=mc^2

It would work, but I'm not confident in the validity of it. I would prefer to use the full derivation from 'Physics 101 content'.

It is longer, but at least it is fully derived and proven.

It does work providing the Lorentz factor is taken as a given. (It's
an inductive - ie reverse logic - version of the standard textbook derivation):

http://www.phys.unsw.edu.au/einsteinlight/jw/module5_dynamics.htm

Unfortunately, neither approach answers Dave's original question (basically, where does the c^2 come from) because it comes from the Lorentz factor itself, which is already assumed. At this level it's an 'accident of algebra' that fits some data under special circumstances and therefore may hide a deeper truth.

The Lorentz Factor is applicable because of the requirement that the laws of physics are the same in all inertial frames of reference - 1st postulate of SR.

And the Lorentz Factor is derived purely from the requirement of the speed of light to be constant for all observers in inertial reference frames (ie nothing to do with mass, energy or momentum at all) - 2nd posulate of SR.

So where does the c^2 come from? It comes directly from the 1st and
2nd postulates of Special Relativity. Everything else is just algebra, isn't it?

Or am I missing something here?

Aries - thanks for the explanation, I now understand the maths and where the answer comes from.

Seth - yes, you have very eloquently just posted the question I was trying ask. So now I just need to figure out how the two relate

The c^2 in the Lorentz factor comes from its derivation, which is a lot easier to post on the forums fortunately.

Just a bit of free thinking here, Dave, but let's assume the precept that rest mass is a special form of energy. Furthermore, let's hypothesise that it is formed when energy is at such a density that it runs up against some universal limit. What might that limit be?

Now I can't do 'gamma' so let's take 'g' as the Lorentz Factor.

In the link in posting 14 there's a derivation for the increase of kinetic energy of a relativistic rocket as it accelerates:

dW = m.c^2.dg

Let's assume that the fuel for the rocket is 100% efficient conversion of its own mass to fuel (utilising E=mc^2!!)

dW = - c^2.dm

Hence dm/m = -dg

This integrates to

m = mi.exp(-g+1) where mi is the initial mass.

As m tends to zero, g tends to infinity, ie the rest mass has just enough energy to propel the last squillionth of the last particle of its mass to light speed.

>> It seems far too neat to be coincidence. <<

It does, doesn't it

Elegant, but I still don't like it.

I would still rather stick to the entire proof.

Maybe I'm re-stating what's already been said.

Special relativity redefined the definitions of mass and energy. A key component of special relativity is the lorentz factor, which is :

sqrt(1/[1 - (v/c)^2])

This factor shows up all the time in relativistic equations of motion. So that would be the short answer to why it's not a coincidence that c^2 shows up in the rest mass equation...

Yep, sorry, done that (Post 16)

The way in which the Lorentz Gamma Factor is derived involves the speed of light. I can post it's derivation but it is quite long. Anyone want it?