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

[stevmg]

Here is the tautology in that derivation posted - It is a very good derivation indeed, though.

\dot p=\frac{d}{d\tau}(\gamma m \dot x)=\dot\gamma m\dot x+\gamma m\ddot x=m\gamma^3\dot x^2\ddot x+\gamma m\ddot x=\gamma m\ddot x(\gamma^2\dot x^2+1)

=\gamma m\ddot x\left(\frac{\dot x^2}{1-\dot x^2}+1\right)=\gamma m\ddot x\left(\frac{\dot x^2+1-\dot x^2}{1-\dot x^2}\right)=\gamma^3 m\ddot x

Also, in Plakhotnik's article, equation 2 is derived eventually and NOT accepted as fact.

The loose part of his "intuitive article" is the approximation in equation (3) on page 105 as there is approximation he uses and then integrates THAT. That may not be so "kosher."

Since the equation (2) as cited is correct it will always work in any proof.

In my own history, for example, I remember a calculus test in which a cone was being filled up with water. Given was the angle of the cone, the current height of the water (so the area at the surface could be calculated), and the flow of water into the cone. The question was "what is the rate that the height was rising at that particular time"

My roommate answered the question and was marked wrong because he he made the assumption, which was correct, that the rising height rate was inversely proportional to the area of the surface. But, at that point in time, we had never proven that his assumption (and the assumption was correct) was true, so the instructor marked him wrong. Now, the instructor was a good guy and told him that had that one question made the difference between a B or a C or a D and an F he would have given him credit for it, but it did knock him down from A to a B, so the instructor felt comfortable "zinging" him to make his point. Mike (my roommate who is now a retired PhD professort at the University of Oregon in Biochemistry) was wrong, though.

P = mu is true but Plakhotnik doesn't accept it as true until shown
SQRT(1 - u^2/c^2) to be true

I am out of my league, now but attach another simpler proof from Karl Calculus Tutor.

I'm happy and I do know what a tautology is:

""The Bible says ...(thus and so)


(thus and so is true because it says so in the Bible"

I'm sure you've heard that before.

Stephen Garramone, M.D. (Col, Ret), USAF, MC

 

[Fredrik]

Here is the tautology in that derivation posted - It is a very good derivation indeed, though.

\dot p=\frac{d}{d\tau}(\gamma m \dot x)=\dot\gamma m\dot x+\gamma m\ddot x=m\gamma^3\dot x^2\ddot x+\gamma m\ddot x=\gamma m\ddot x(\gamma^2\dot x^2+1)

=\gamma m\ddot x\left(\frac{\dot x^2}{1-\dot x^2}+1\right)=\gamma m\ddot x\left(\frac{\dot x^2+1-\dot x^2}{1-\dot x^2}\right)=\gamma^3 m\ddot x

It's not a tautology. The first step just uses the definition of p (the same definition as in the article you posted), and some of the other steps use the definition of gamma.

Also, in Plakhotnik's article, equation 2 is derived eventually and NOT accepted as fact.

I noticed that they claimed to be doing that, but I don't even have to read the "derivation" to know that it's not really a derivation. You can't derive special relativistic stuff from the Newtonian/Galilean theory. So what they're doing is something else entirely. See my previous two posts for an explanation of what.

 

[stevmg]

To Fredrik -

I really do appreciate all the time you have spent trying to wrap my brain around this. I am not a physicist, nor do I work with physics, and my masters in math is also old . I took differential equations in the Spring of 1960, so we are going back a long way. Ooooh, do I forget that crap, although I can still do it if I have the texts in front of me. I am an M.D. and believe me, the only thing MDs and DO (osteopaths) do with math is ratio, at most for drug dosing.

Now, I forgot something about logic (I never had a course in logic and it was taught to me in a computer class much later) which your examples brutally reminded me of (yes, you can end a sentence in preposition as English is a Germanic language, not a Latin one.)

One CAN start off with a supposition such as momentum = gamma*mv

Then, taking that, and, by a series of logically EQUIVALENT statements one can work to something that has been previously proven or known to be true (but not on the basis of the original supposition:)

statement A: momentum = gamma*mv
A <--> B
B <--> C
C <--> D
.
.
.
Y <--> Z
Z "2 + 2 = 4" [for argument sake]

Then you have proven A. A is NOT a tautology in that case - something you have been tyrying to tell me for the past few days but, as stated, my old brain has had a hard time absorbing. My problem was that it has been so damn long since I did diff eq that I cannot follow what you did very well and I don't really need to. The two articles that I posted are of sufficient rigor that that's all I need to understand it at my level.

You do have to agree, though that this so-called relative mass equation (which really is not an increase in mass but an increase in the effect of the momentum at high velocity due to relativity - I use the word "velocity" as it is a vector that has to be turned) is not "proven" in physics texts but is assumed, while relativistic time, velocity and distance are mathematically proven in these same texts. I also know that particles with "increased mass" due to high velocity do NOT have increased gravitational effect commensurate with their "new" mass.

Muchas gracias, vielen Dank, whatever, for all your help!

Stephen M. Garramone, M.D,
Melbourne, FL

 

[stevmg]

Attached is a .pdf file from a lecture series which fully explains the derivation of p= gamma*u where u is the velocity in the moving time frame, m is the original mass at rest. It is lecture 4 that he goes into it.

The major contributor to this time dilation

Once momentum in relativity is explained, the so-called "relativistic mass" equations are explained:
m = m₀*gamma

Actually, when this "new" mass is created, it really has no true "mass" effects. Gravity is not affected nor is there more of a bending of light. This equation is to make the Newtonian equation p = m*v be more applicable.

 

[stevmg]

Great lectures on all aspects of simple relativity

http://www.youtube.com/watch?v=pHfFSQ6pLGU&feature=SeriesPlayList&p=FE3074A4CB751B2B

start with lecture 12 and go through lecture 15

 

[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