Is Kinetic Energy Different in Relativity?

[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.

 

[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 don't have it myself.

 

[JesseM]

What happens in Eisteins book, I have no clue since i don't have it myself.

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.

 

[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.

 

[Dadface]

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.

 

[robphy]

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.

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.

 

[Dadface]

Yes I do mean that.Thank you.

 

[Dale]

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.

 

[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:

For higher speeds, the formula for the relativistic kinetic energy[5] is derived by simply subtracting the rest mass energy from the total energy:

and

The mathematical by-product of this calculation is the mass-energy equivalence formula—the body at rest must have energy content equal to:

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...

 

[RobSoko315]

Thank you DaleSpam for the encouragement

And thanks everyone else for an answer.


-Rob

 

[Naty1]

Another good discussion can be found in SIX EASY PIECES by Richard Feynman 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.

 

[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!

 

[Dale]

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?

Do a search for "relativistic rocket", I believe Baez has a good page on the subject.

2. Does light have mass?

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.

3. Does thought have energy?

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

These are completely inappropriate questions for this forum. Please re-read the rules that you agreed to follow when you registered.

 

[Rasalhague]

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.

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 traveling in relative to each other?

 

[Dale]

What are the necessary and sufficient conditions for a system of photons to have mass.

That their momentum not be in the same direction in space.

Is it a necessary condition that there be more than one photon? Does it matter what direction the photons are traveling in relative to each other?

Yes and yes.

 

[Rasalhague]

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

 

[Bob_for_short]

...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.

Isn't mc² a potential energy?

 

[JesseM]

Isn't mc² a potential energy?

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".

 

[Bob_for_short]

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

 

[JesseM]

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

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.

 

[Naty1]

Does thought have energy?

If you accept that thinking (and controlling the body's life functions) consists of information processing, (I guess via chemical and electrical reactions) and ask if information processing in general consumes energy, then yes. In fact, the human brain seems to use three or four times the energy relative to its mass as does the rest of the body which means it's been allocated a large share of resources...nothing metaphysical about all that...

 

[Rasalhague]

If you accept that thinking (and controlling the body's life functions) consists of information processing, (I guess via chemical and electrical reactions) and ask if information processing in general consumes energy, then yes. In fact, the human brain seems to use three or four times the energy relative to its mass as does the rest of the body which means it's been allocated a large share of resources...nothing metaphysical about all that...

Also, according to this paper, a conscious brain consumes more energy than the deeply anaesthetised (and the authors presume unconscious) brain.

http://www.ncbi.nlm.nih.gov/pubmed/19549837?ordinalpos=1&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_DiscoveryPanel.Pubmed_Discovery_RA&linkpos=4&log$=relatedarticles&logdbfrom=pubmed