Energy and Mass: What Happens to Energy in Relativity Theory?

[Dale]

Hi stevmg, perhaps it will help to consider the following questions:

1) Why is work defined as f.d in classical mechanics?
2) How can you prove that f.d is work?
3) Is the definition of work derived?
4) Is it handed down from God or otherwise have anything to do with the Bible etc.?

Once you have thought about those questions a bit, perhaps my statements and Frederik's will make more sense.

 

[stevmg]

It's been a long time since physics but I recall that work is f*d is reproducible by whatever combination of f or d gives the same result and will lift a weight to a given height - potential energy. Are you going to make me drag out my 50 year old physics textbook? I even have one from the 1920s (my father's.)

The "Bible" reference was an example of a tautology, not a reference to any derivation of any physics or mathematical formulae.

I restated that Fredrik's derivations were not tautological but that I did not understand them. My take now is that momentum is preserved even in spite of time dilation which slows things down in the moving time frame and thus the gamma factor is necessary to equate the moving time frame momentum and the resting gime frame momentum.

Again, Dr. Shankar's lectures really explain well, until you hit Lectures 14 and 15 simple relativity. He doesn't explain momentum conservation in ordinary terms but chooses to use the 4-space vector to do it. I am sure he is right but it is not intuitive to me.

 

[Dale]

I recall that work is f*d

That is correct, but how is that formula derived? How can you prove that f.d is equal to work?

 

[stevmg]

I would have to go back to my old college texts on physics for that. Today I have no access to them.

 

[Dale]

Well, let me give you the answer, which you can verify yourself later on if you want. There is no derivation, there is no proof, it is simply a definition.

I can define any term I like, say I want to use the term "glub" to refer to the cross product of acceleration and velocity (g = a x v). That is perfectly fine, glub is now a defined term and I can make experimental predictions of its value and carry out experiments to determine the accuracy of my predictions. I don't need to derive or prove that g = a x v because that is simply the definition of glub. Any attempt to prove it or derive it will necessarily be tautological, g = a x v because g is defined as a x v.

 

[stevmg]

If a particle gains "mass" by virtue of its velocity, does this mass have increased gravitational pull commensurate with the increase or is it just a virtual mass increase because of the conservation of momentum?

 

[Jonathan Scott]

If a particle gains "mass" by virtue of its velocity, does this mass have increased gravitational pull commensurate with the increase or is it just a virtual mass increase because of the conservation of momentum?

Basically, it's all consistent with relativity.

In a frame moving along with the particle, it is effectively at rest, so in that frame gravitational effects work in the usual way (so for example it doesn't get any nearer to being a black hole). If you then work out the gravitational effect in that frame on a particle which is at rest in the original frame (so it appears to be moving with the reverse velocity relative to the particle) that gives the acceleration in the frame of the moving particle, then if you transform the acceleration to the original frame, you get the gravitational effect of the particle as seen from the original frame.

It's fairly easy to do this using a simple weak field approximation to GR combined with special relativity transformations, but when you do this you discover that the effect of gravity is not just due to a scalar potential like an electric field. It also has components which relate to velocity (like magnetism) plus additional tensor terms which don't have any direct electromagnetic equivalent. These terms are vanishingly small for everyday gravitational effects, but ensure consistency under relativity transformations.

 

[starthaus]

The simplest derivations involve photons, but you don't like that.

Most books I've seen define momentum as p = γmv and/or energy as E = γmc² and go on to show that this definition reduces to the Newtonian values for small velocities and that these definitions transform correctly when you switch to a different frame, and then postulate that these quantities are conserved in collisions. The justification is that relativity built on these assumptions correctly predicts the results of experiments.

If you actually want to work out these formulas without knowing in advance what the answer is, it seems to be a difficult exercise. I had a go myself in the thread "Derivation of momentum & energy formulas".

This is a beautiful derivation, DrGreg. As an alternative, less mathematical (maybe more physics related), I use the one based on the Euler-Lagrange formalism attached in my blog. (no.2)

 

[starthaus]

If a particle gains "mass" by virtue of its velocity, does this mass have increased gravitational pull commensurate with the increase

No, you can't stick relativistic mass into Newton's law.
Nevertheless, the total energy of the particle increases: TE=mc^2/sqrt(1-(v/c)^2) and that results into an increased strength of the gravitational field around the particle in discussion.

 

[Count Iblis]

I agree with DaleSpam and Fredrik. The real issue, i.m.o., is actually deriving the classical limit in a proper way. My opinion is that most textbooks don't give the correct derivation of the classical limit. What happens is that c = 1 units are not used and then the c to infinity limit or equivalenty letting v go to zero relative to c, becomes a rather trivial calculus exercise.

A meaningful derivation of the classical limit should start with special relativity formulated in c = 1 units. Then mass is identified with rest energy and is thus not an independent physical quantity from energy. The same is true for time vs. space and momentum vs. energy, of course.

The classical limit has to be derived from this without any ad hoc insertion of c. Of course, c will re-appear in the equations, but only as a dimensionless rescaling parameter appearing as a result of studying some non-trivial scaling limit of the theory.

Such a fully rigorous derivation in which new units for mass and time apart from energy and space, respectively, emerge in a scaling limit is not given in textbooks. But it seems to me that the whole point of the classical limit is exactly that in classical physics mass and energy are incompatible quantities while in relativity they are physically the same quantity. So, one has to derive exactly this fact.

 

[stevmg]

The note by Fredrik is beyond meaningless as the square root as presented comes from where? It comes from Maxwell et al electromagnetic theories which are tautological in this as they are downstream from the relativistic mass equation.

Citing the Taylor series expansion is actually an expansion of
E = m(c^2)[1 - (v^2)/(c^2)]^(-1/2)
which already assumes the veracity of the relativistic mass equation. Again, tautological reasoning.

You cannot prove something is true by assuming it is true and then basing the proof on that "fact." You can only use that sort of technique and prove that the negative or contradiction of a given assumption false reductio ad absurdum.

Did I say that? Well, that was in March and have I learned a lot from you folks since...

Humble apologies...

 

[PhilDSP]

Any derivation of E=mc² will be based on the SR definitions of some other terms, so the value of such "derivations" is questionable. You might as well start by defining energy to be

$$E=\sqrt{\vec p^2c^2+m^2c^4}$$

If we restore factors of c, this becomes

$$W=\gamma mc^2-mc^2$$

This "derivation" assumes that we have already accepted the SR definitions of four-velocity, four-momentum, force and work.

If I understand Fredrik correctly he admitted that this is an SR based derivation. So actually there is some merit to your March observation that it is a bit self fulfilling. To step further back and derive the energy-mass equation from more basic principles you can take the approach that J. J. Thompson took. If you care to take a plunge into that I'd be happy to help out but since you're the one who seems most interested in getting to the bottom of this at the moment you should be prepared to do a little work.

 

[stevmg]

If I understand Fredrik correctly he admitted that this is an SR based derivation. So actually there is some merit to your March observation that it is a bit self fulfilling. To step further back and derive the energy-mass equation from more basic principles you can take the approach that J. J. Thompson took. If you care to take a plunge into that I'd be happy to help out but since you're the one who seems most interested in getting to the bottom of this at the moment you should be prepared to do a little work.

1) There has been absolutely NO merit in anything I have had to say... DaleSpam, jtbell starthaus, DrGreg et al can all testify to that. Teaching me anything is like driving a piece of straw into a rock - eventually, success happens provided the straw is going fast enough and precisely at a right angle to the surface of the rock at impact. Again, humble apologies to the contributors listed (and those not listed) above.

2) I am interested in getting to the bottom of this so giving me the J.J. Thompson reference may be of good order.

stevmg

 

[PhilDSP]

Is it possible for you to acquire a copy of J. J. Thompson's monograph called "Beyond the Electron"? If not, PM me and we can work something out.

That text has both the full mathematics and layman's explanation of what is happening in his later explorations and thoughts.

 

[stevmg]

Ordered it from Amazon.com for $18 and change. Good used edition. 1928 vintage.

stevmg