# Relativistic mass and energy

1. May 9, 2005

### nwall

well, i'm very new to this, but i'm interested in learning. recently i followed a link from these forums that lead me here http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html which states:

well, i'm a little confused now, because if energy and relativistic mass are the same thing then why do we have the equation E = mc^2? in the equation, E is the energy and m is the relativistic mass, right? that equation seems to say to me that there is a propper use for the term "relativistic mass" because while it may be related to energy by a constant, it's not the same thing. could anyone shed some light on this for me?

thanks

2. May 9, 2005

### dextercioby

c=1 can be a cause.

Daniel.

3. May 9, 2005

### JesseM

If m is the relativistic mass, then E = mc^2 will be valid for any object regardless of the velocity. But if m is just the rest mass, then E=mc^2 only works for an object at rest, and for an object with nonzero velocity you'd use this equation:

$$E^2 = m^2 c^4 + p^2 c^2$$

Where p is the relativistic momentum $$p = m v / \sqrt{1 - v^2/c^2}$$

4. May 10, 2005

### pmb_phy

You are correct. It is quite wrong to say that they are the same thing. If one quantity is proportional to another quantity that doesn't mean that they're the same thing. E.g. E = hf where E is the energy of a photon, h is Planck's constant and f is the frequency of the photon. E is therefore proportional to f. However this obviously doesn't mean that E an f are the same thing. E and m have a different definition and in some, but not all, cases they are proportional.

For an example where E does not equal to mc^2 please see

http://www.geocities.com/physics_world/sr/inertial_energy_vs_mass.htm

Pete

5. May 10, 2005

### Meir Achuz

An object at rest has energy given by E_{rest}=mc^2.
This relation is sometimes considered as an "equivalence" of mass and energy,
but it is not a true equivalence. The energy and mass of an object are closely related, but not equivalent. To within factors of c, the energy is the time-like component of the momentum four-vector, while the mass is the invariant length of the four-vector. The energy in the rest system equals mc^2, but while in motion, the energy varies with velocity.

In early expositions of special relativity, and still in some popularizations,
the mass is considered to vary as m=m_0/\sqrt{1-v^2/c^2} with m_0 called the "rest mass". Then E=mc^2 even in motion. However this interpretation goes against the principle that the mass, as an intrinsic property of an object, should be the same in all Lorentz frames. In the modern interpretation, m is the "invariant mass" of an object, and the equation E=mc^2 holds only in the rest system.

6. May 10, 2005

### pmb_phy

That only applies when the object has no forces acting on it.

Pete

7. May 10, 2005

### nwall

thanks pmb_phy and meir_achuz. that was a very helpful explanation. so i guess while the article was wrong in saying that relativistic mass and energy are the same, it was correct in saying that relativistic mass is an outdated concept because now everything is based on invariant mass? that makes much more sense. thanks again.

8. May 10, 2005

### sangra

part of the answer is that in standard units used by relativistic physicists c = 1
so E =m and m = m(rest mass)/sqrt(1-v^2/c^2).

9. May 10, 2005

### pmb_phy

No. Not everything is based on invariant mass. And relativistic mass is not outdated. The author is speaing about point particles. Since these particles don't have any internal structure then the time component of the 4-momentum will be mc.

So why is it you like energy rather than mass?

Pete

Last edited: May 10, 2005
10. May 10, 2005

### pervect

Staff Emeritus
Monty Python and the Relaitivistic Mass Question:

It's not outdated MUCH, said the crowd. "Maybe a bit", said sir Bevedere.
"A bit! A bit! A bit!" chanted the crowd.

"This new learning amazes me, Sir Bevedere. Explain to me once again
how ram's bladders may be employed to prevent earthquakes", aksed King Arthur.

11. May 10, 2005

### JesseM

In the case of E=hf, we have independent experimental techniques for measuring the frequency and the energy of a particle. But do we have independent experimental techniques for measuring the rest mass of an object and measuring its energy at rest? How do you measure rest energy?

12. May 10, 2005

### pmb_phy

That's not right. They're are different methods to measure E and m.

Pete

13. May 10, 2005

### JesseM

I didn't say there weren't, I just asked what they were.

14. May 10, 2005

### LindaGarrette

Can you give an example (in numbers) of the value of E changing as v approaches c? I just need some illustration to get my mind around it.

15. May 10, 2005

Staff Emeritus
A handy value to learn is that at 86.6% the speed of light*, the dilation factor

$$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

takes the value .5. And energy at a relative velocity of v is rest energy $$mc^2$$ over $$\gamma$$. So at 86.6% of c, the energy is doubled over the rest energy.

*Actually it's $$\frac{v}{c} = \frac{\sqrt{3}}{2} \approx .866$$. Work it out.

16. May 10, 2005

### pmb_phy

Ah. Sorry my good Sir. I applogize.

Let me get back to you in the morning.

Pete

17. May 10, 2005

### pmb_phy

The concept of rest energy is pre-relativity. Relativity did not prove that a body at rest has energy. Its a postulate on which to derive the relation between E and m. You can actually chose what the value of the rest energy is since energy is defined up to an arbitrary constant. Choose the constant to make life easy. The details of the rest will depend on specifics.

Pete

18. May 10, 2005

### JesseM

I suppose one method of measuring a particle's rest energy would be to collide it with the corresponding antiparticle and measure the energy released...

19. May 11, 2005

### pmb_phy

I've been thinking about Tachyons lately. The theory of tachyons does not assume $E^2 - (pc)^2 = m_0^2 c^4$ to be valid. So if you get a measured value for which thid relation does not hold then how were E and p measured and what does that say for the definitions of E and m0 ??

Pete

20. May 19, 2005

### Honorable_Death

I am almost positive E=mc^2 stated that relativistic mass DOES equal energy but rest mass doesnt. since the speed of light squared is constant, does that have to meen that energy does equal relativistic mass?