# Relativistic mass equation

1. Mar 25, 2014

I know the reasons why the relativistic mass equation (M= gamma Mo) is out of favour with what seems to be a majority of theoretical physicists but I am unsure about one particular thing:

Suppose we knew the mass and velocity of a relativistic particle and wanted to calculate its KE.
Isn't use of the relativistic mass equation, along with (M-M0)c squared, the quickest and easiest way to get to the answer? In other words although some may consider the equation to be a historical curiosity, does it still have its uses?

I have considerd this question before but am still not sure about the answer.

Thanks

2. Mar 25, 2014

### PAllen

A few people still like it, but your example is one of the reasons most don't like it. The original value was to replace the m in p=mv. For KE one would want, by the same philosophy, to replace the m in 1/2 mv^2, but that doesn't work with relativistic mass. If you you are going to use a relativistic formula anyway (yours is s relativistic formula, not Newtonian), why not consider KE = E - E0 = mc^2(γ-1).

For me, it all comes back to the 4-vector notion, in which gamma doesn't go with the mass at all, it is part of the 4-velocity. 4-Momentum is then mU, and energy it the timelike component of this in the basis of the measuring instrument.

3. Mar 25, 2014

### Bill_K

Kinetic energy itself is a leftover concept from Newtonian mechanics. In relativity the quantities you deal with are the covariant quantities, and there's hardly ever an occasion when you'd want or need to calculate KE.

4. Mar 25, 2014

Thank you PAllen but the equation you presented is the same as the one I presented. The only difference is I used Mo instead of M. I know that the concept of rest mass is out of favour and I know why but I used Mo in reference to the original equation. I'm quite happy to use M or m in place of Mo but when I use your equation I am just using the original equation in a different format and without the confusion of definitions of mass.

Last edited: Mar 25, 2014
5. Mar 25, 2014

"Hardly ever" is not the same as never. There are occasions when I might want to calculate KE and that's why I asked the question.

6. Mar 25, 2014

### Staff: Mentor

There are some problems for which some people find relativistic mass to be useful. For example, many years ago I had to calculate the trajectory of relativistic electrons passing between two parallel charged plates. Applying $F=m_{r}a$ seemed as easy as anything else, although I wouldn't argue with anyone who prefers to do the calculation some other way.

However, problems of this sort are few and far between, and the overwhelming majority of people studying relativity do so not because they need to solve a very specific class of problems, but because they want to understand relativity. If understanding relativity is the goal, than spending a lot of time learning about relativistic mass, then even more time learning about its limitations, is counterproductive.

Last edited: Mar 25, 2014
7. Mar 25, 2014

### PAllen

In mine, only one mass is used. There is no concept or relativistic mass in my formula. The gamma has nothing to do with the mass.

8. Mar 25, 2014

### pervect

Staff Emeritus
You can just as easily use the energy equation, and subtract the rest energy.

9. Mar 25, 2014

Yes I agree but in the original equation,the one I presented, there is only one mass namely Mo. The only difference is one of symbols. I know that generally speaking relativistic mass is considered to be outdated and misleading and I am familiar with the reasons for that. I like to think of M (as in the original M=gamma Mo) as being equal to the mass plus the kinetic energy expressed in mass units.

10. Mar 25, 2014

Yes you can express it mass units or energy units.

11. Mar 25, 2014

Thanks but the problems are probably not as rare as you think. The old mass variation equation is still taught in A level physics courses in UK schools (eg AQA physics A "turning points in physics" topic). Until and if the examiners modify the specification teachers still have to go along with it. Admittedly the relativity taught is at a very basic level.

12. Mar 25, 2014

Apart from a few things that need clarifying with regards to things like symbols used I think I am in broad agreement with everybody here.But there are still a couple of problems:

!.What should A level students be told?
2.The second point is about terminology. There are three terms in the equation when expressed in energy units KE = E-Eo
What should each of these three terms be called? KE is called kinetic energy but what do we call E and Eo. It might be tempting to call E total energy and Eo rest energy but those terms imply the existence of total mass and rest mass.

13. Mar 25, 2014

### PAllen

Energy is energy (a frame variant quantity); mass is mass, an invariant. The full equation to focus on is:

E^2 = (mc^2)^2 + p^2 c^2

Then you are not tempted to convert total energy to total mass, making a frame dependent notion of mass.

14. Mar 25, 2014

That's good for me but it's probably too heavy going for A level students.

15. Mar 25, 2014

### Bill_K

AQA Turning Points in Physics also tells us that "time runs slower when you are moving." I guess in comparison, their use of relativistic mass is a minor point.

16. Mar 25, 2014

### Staff: Mentor

No one disputes that it's still taught, and therefore that you may need it just to pass some tests. The dispute is over whether it's good pedagogy to teach it - that is, SHOULD it still be taught, or is there a better way of using limited classroom time to communicate the basic principles of SR. Note that relativistic mass is NOT one of those basic principles; it's a very specific application of them in a specific class of problems.

In some ways this discussion reminds me of discussions of whether Latin should be taught in American high schools. There's no doubt that most native English speakers will derive some non-zero benefit from a year or so of diligent Latin study; but enough to justify displacing a year or so of English literature?

Last edited: Mar 25, 2014
17. Mar 25, 2014

### atyy

But is there a conceptual connection to GR, since the principle of equivalence in Newtonian gravity is that inertial mass is gravitational mass? Since the inertial mass is energy in special relativity, that suggests that the energy should be the source of gravity in relativistic gravity. Then requiring that the equations be tensorial, the source should be either the stress-energy tensor or something like its trace.

18. Mar 25, 2014

### PAllen

Hmm. In my mind, the only form of inertial mass in SR is invariant mass, which includes KE only to the extent it becomes part of the invariant mass of the system.

19. Mar 25, 2014

I'm not sure where you are getting that from. The specification is very brief referring to proper time,time dilation the equation and muon decay.

20. Mar 25, 2014

### Bill_K

That is literally what it says on Page 21:
It then goes on to discuss the usual train example, but there is no hint given that the effect is symmetrical, which I think is the whole point of relativity.

Are we looking at the same document?

Last edited: Mar 25, 2014
21. Mar 25, 2014

As the syllabus stands at present it has to be taught. The problem is how do teachers do the extra that is necessary without taking up too much extra time and without jeopardising students chances of exam success.
Syllabus changes have to take into account the likely future educational paths of the student population and only a very small fraction of this population will study physics beyond A level.

22. Mar 25, 2014

We're looking at different things. I was referring to the AQA 2014 specifications. Page 33 for syllabus A and page 26 for syllabus B.

23. Mar 25, 2014

### DrStupid

The original value was NOT to replace the m in p=mv. In his paper from 1905 Einstein replaced Galilei transformation by Lorentz transformation only. Everything else remained unchanged. In the result the former invariant m became frame dependent. Many physicist (including Einstein himself) continued to use it in SR. Later it has been antiquated in favor of the rest mass in order to get a frame independent property again. This is what the term "mass" is generally used for today.

24. Mar 25, 2014

### PAllen

Einstein abandoned it, and wrote about doing so, calling it a bad idea. Mass remains an invariant scalar in SR. For a system of particles, the invariant mass is the norm of the total 4-momentum. Einstein, in 1905, derived that momentum was mγv, from which he factored mγ as a mass factor. This was before any notion of 4-vector treatment was developed. Even before adopting 4-vectors, Einstein was writing that m versus m0 was a bad idea.

4-vector treatments leads to a different factoring: gamma comes form derivative of position by proper time. It is part of the tangent vector, nothing to do with the mass.

25. Mar 26, 2014

### DrStupid

Mass as defined by Newton (by def. 2 and lex 2 & 3) does not remain invariant in SR. It needs to be refined in order to make it invariant again.