# Relativistic Kinetic Energy

1. Jan 24, 2009

### RobSoko315

I am currently reading Einstein's book "Relativity: The Special and General Theory", and I came across I point I don't quite understand.

Einstein says:
In accordance with the theory of relativity the kinetic energy of a material point of mass m is no longer given by

$$\frac{1}{2}$$mv2

but by the expression

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

This doesn't make sense to me. According Wikipedia, its

$$\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$$ - mc2

My question is, who's right? Perhaps its a typo in the version of the book I have. But even if it is a typo, it seems future equations he explains build on the one he gives (above). Also, the calculations I did of Wikipedia's equations are much closer to the classical equation when v is slow.

2. Jan 25, 2009

### malawi_glenn

Wiki is correct, the TOTAL energy is $\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} = \gamma m c^2 = E_{kin} + E_{mass}$

What happens in Eisteins book, I have no clue since i dont have it myself.

3. Jan 25, 2009

### JesseM

You can see that section online here. It's probably just a case of the terminology not having been settled on at the time Einstein was writing--maybe it wasn't common back then to talk about "rest mass energy" and "kinetic energy" as distinct things, and they just used "kinetic energy" to mean any energy that wasn't potential energy. But in the modern way of speaking, yes, wikipedia is correct.

4. Jan 25, 2009

### malawi_glenn

By reading that page, I have to agree with JesseM: it is just a matter of convention. As Einsteis wrote, the term mc^2 is not associated with velocity, it sets the zero of the Energy scale.

5. Jan 25, 2009

If you look at Einsteins original paper entitled Dynamics of the Slowly Accelerated Electron he derives the correct ie the Wikipedia equation.I am assuming there is a misprint in the book referred to above.

6. Jan 25, 2009

### robphy

I think you mean that
"Dynamics of the Slowly Accelerated Electron" is the title of a section in
"On the Electrodynamics of Moving Bodies", which is the title of Einstein's 1905 paper.

7. Jan 25, 2009

Yes I do mean that.Thank you.

8. Jan 25, 2009

### Staff: Mentor

Hi RobSoko315, welcome to PF.

It is pretty common knowledge that wikipedia is often wrong, so please don't hesitate to ask whenever you see something there that is unclear. In this instance it is correct, but that is by no means a general endorsement of its correctness elsewhere.

9. Jan 25, 2009

### Naty1

In the section where the "wiki formula" is derived
http://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies

Wiki sez:

and
which is I believe what Malawi notes above.

So it sounds like a convention as noted in posts above ...yet a bit further in that same section Wiki correctly notes that the taylor expansion of the "wiki formula" for low speeds
approximates KE as 1/2mv^2....so maybe that's what underlies the convention...

10. Jan 25, 2009

### RobSoko315

Thank you DaleSpam for the encouragement

And thanks everyone else for an answer.

-Rob

11. Jan 29, 2009

### Naty1

Another good discussion can be found in SIX EASY PIECES by Richard Feynmann on pages 85 to 91...he concludes we "shift the origin" as stated in post #4...but his comments and insights are as always interesting.

12. Oct 20, 2009

### Sam 4325

I have a few questions posed to me 2 of which are addressed here, while these questions are not entirely physics related; I hope to get your input.

1. If we ever are going to explore our galaxy or the universe the implications of E=mc^2 is the limiting factor that won't let us. Why?

2. Does light have mass?

3. Does thought have energy?

4. In what form and what implications would that have on the concept of a soul?

Thanks!

13. Oct 20, 2009

### Staff: Mentor

Do a search for "relativistic rocket", I believe Baez has a good page on the subject.
An individual photon does not have any invariant mass, but it is possible for a system of photons to have mass when considered together as a whole.
These are completely inappropriate questions for this forum. Please re-read the rules that you agreed to follow when you registered.

14. Oct 21, 2009

### Rasalhague

What are the necessary and sufficient conditions for a system of photons to have mass. Is it a necessary condition that there be more than one photon? Does it matter what direction the photons are travelling in relative to each other?

15. Oct 21, 2009

### Staff: Mentor

That their momentum not be in the same direction in space.
Yes and yes.

16. Oct 21, 2009

### Rasalhague

Excellent, thanks DaleSpam! At last it's starting to make sense...

17. Oct 21, 2009

### Bob_for_short

Isn't mc2 a potential energy?

18. Oct 21, 2009

### JesseM

No, it's not considered that in current terminology--"potential energy" is always understood in terms of the potential associated with some force field. mc^2 is usually called something like "rest mass energy".

19. Oct 21, 2009

### Bob_for_short

And how about calculation of the classical electron radius from mc2? It is equal to the potential energy, isn't it?

20. Oct 21, 2009

### JesseM

True, but according to the wikipedia article, in that case they are starting from the (incorrect) assumption that the electron's mass is entirely due to electrostatic potential. I guess I should have qualified my statement about rest mass energy though--for a single pointlike particle rest mass is unrelated to potential energy, but for a bound system composed of multiple parts, the "rest mass" (and thus the rest mass energy) of the system as a whole can be greater than the sum of the rest masses of the parts, it will include both potential energy (so that a hydrogen atom has a smaller rest mass than the sum of the rest masses of a free proton and free electron) and kinetic energy (so a gas-filled box would have a greater rest mass after it was heated up). In the classical electron radius calculation, I think they are modeling the electron as something like a continuous charged fluid, with each point in the fluid having zero rest mass on its own, and presumably also with zero kinetic energy, so the entire rest mass of this bound composite system would be due to the potential energy. I haven't actually studied the classical electron radius calculation though, so if I'm mistaken someone please correct me.