# Einstein's equation

1. Jan 29, 2004

### wiiz

Hello, this is my first post.
Some years back i read a book about einstein works. Among many things in the book, i read about his 'modified' newton's equation of motion (F=ma). Sadly, i had lost the book and can't recall the equation. All i can remember is it says that the einstein's equation should be used when a body is moving at close to or higher than speed of light. The F=ma is still correct for motions slower than that.

I hope someone can tell me what is the equation.
Thanks.

p/s: English is not my 1st language, so i hope everbody can understand what i'm trying to say here.

2. Jan 30, 2004

### Kalimaa23

Well, funny you would say that, because Newton's 2nd law is about the only one that does hold! Only use the correct form, dp/dt = F, instead of the (incorrect for changing mass) F = m.a

3. Jan 30, 2004

### DW

Mass does not change with speed. Both
$$F^\lambda = \frac{DP^\lambda}{d\tau}$$ and $$F^\lambda = mA^\lambda$$ work for relativity. The difference from Newtonian mechanics is primarily the time dilation encorporated in the proper time derivatives. These are the four vector relations. More than likely he is wanting to know the relation between ordinary force and coordinate acceleration for special relativity which is
$$\vec{f} = \gamma m[\vec{a} + \gamma ^{2}\frac{\vec{u}\cdot\vec{a}}{c^2}\vec{u}]$$. Even in this expression the mass m does not change with speed.

4. Feb 2, 2004

### franznietzsche

Well it does, because the as the force is applied and the mass accelerates its speed increases. As speed increases mass does, as according to the lorentz transformation. This is why it takes more and more force to accelerate an object at the same rate as the velocity approaches the speed of light. Its not the force equation thats wrong its the concept of mass being constant, its not. it increases with velocity.

5. Feb 2, 2004

### yogi

The answer given by Dimitri is quite correct - it shows Newton's genius - even in the 16th Century his concept of force as the rate of change of momentum is still true even after Special Relativity. But in one sense, there does not have to be an actual change in mass - one can arrive at the same result in consequence of time dilation - i.e., if m is considered constant .. the effective inertia will still be greater, that is, the rate of time passage between two systems in relative motion requires a greater force to bring about the same change in velocity in a short duration relative to the longer duration measured in the dilated time frame.

6. Feb 2, 2004

### DW

No it doesn't. The mass is invariant to the Lorentz transformation.

No it is not. The extra factors (plural) of $$\gamma$$ come from the time dilation relating the proper time in the time derivatives of the relativistic law of motion $$F^\lambda = \frac{DP^\lambda}{d\tau} = mA^\lambda$$ to the coordinate time in the time derivatives of the ordinary force equation $$f^i = \frac{dP^i}{dt} = \gamma m[a^i + \gamma ^{2}\frac{\vec{u}\cdot\vec{a}}{c^2}u^i]$$. The particle is obeying a four vector force law of motion and the result is a diminishing coordinate acceleration at constant ordinary force. THAT is why the coordinate acceleration diminishes with speed for constant ordinary force. It has nothing to do with a changing mass as the mass does not change with speed. The mass is invariant to the Lorentz transformation.

7. Feb 3, 2004

### Arcon

Yep. That is 100% correct so long as you're refering to relativistic mass. dw has the habit of refering to rest mass when others are discussing relativistic mass. Keep that in mind when you read his posts. His arguements have the nature of "Relativistic mass is not a function of speed because proper mass is invariant" since he will respond to someone discussing relativist mass with a statement which refers to the fact that proper mass is invariant. Thus his posts are always confusing on this issue.

These are the facts - they are undisputed in the relativity community
Fact #1) Relativistic mass is a function of speed.
Fact #2) Proper mass (aka rest mass) is not a function of speed.
Fact #3) When it comes to what "mass" means unqualified then different relativists mean different things.

8. Feb 3, 2004

### DW

No, he did not qualify it and call it relativistic mass and these days it is improper to use mass unqualified to refer to that. Mass unqualified used properly in todays terminology refers to that which is invariant. See
http://www.geocities.com/zcphysicsms/chap3.htm

No they don't. They have the nature of mass is invariant and relativistic mass is an obsolete concept.

9. Feb 3, 2004

### Arcon

When someone doesn't qualify the term mass then the meaning must be distilled from the context in which it is used. It is quite clear from the context that the person means relativistic mass and not proper mass. If you're unsure about what the person means then the appropriate thing to do is to ask them. Not assume they means something else and then attempt to correct.

You should have known from the content of my post that this was the point I was making clear for him.
That is incorrect. That is merely your personal opinion. Yes. It is a popular personal opinion. But one which is not universally adhered to. In fact Alan Guth himself told me personally that he finds the concept useful. He even uses it in his lecture notes in his MIT course The Early Universe.
That is incorrect. Counter examples abound in the modern relativity literature. For the correct meaning of relativistic mass see http://www.geocities.com/physics_world/sr/inertial_mass.htm. If you have any questions feel free to ask.
That is incorrect. This is a topic that has been debated for many decades. However he concept of mass as "invariant mass" cannot be given a general meaning. It is of limited use.

Plenty of recently published physics literature use the concept and use it quite succesfull. For example:

Cosmological Principles, John A. Peacock, Cambridge University Press, (1999)
Relativity: Special, General and Cosmological, Rindler, Oxford Univ., Press, (2001)
Basic Relativity, Richard A. Mould, Springer Verlag, (1994)
[/b]Introducing Einstein's Relativity,[/b] Ray D'Inverno, Oxford Univ. Press, (1992)

There was even an article published just recently in the American Journal of Physics on the concept of relativistic mass. There were a few others in the same journal too.

Apparatus to measure relativistic mass increase, John W. Luetzelschwab, Am. J. Phys. 71(9), 878, Sept. (2003).
Relativistic mass increase at slow speeds, Gerald Gabrielse, Am. J. Phys. 63(6), 568 (1995).
In defense of relativistic mass, T. R. Sandin, Am. J. Phys. 59(11) 1032 (1991).

Numerous references to examples contrary of dw's claim may be found at --
http://www.geocities.com/physics_world/relativistic_mass.htm

However if you can prove that the terminology used in all these texts and journal articles is "obsolete" then please do so. Also define "obsolete" since you're not using it in the original meaning since, by definition (and in this context), something is only obsolete when it is never used in modern relativity literature. That is clearly incorrect as I've demonstrated to you on many occasions in the past.

10. Feb 3, 2004

### Arcon

Actually in that expression the quantity

$$\gamma m$$

is
the mass being discussed. m is proper mass.

As long as people try to use a term before making crystal clear how that term is defined this type discussion will go on forever. The only question in this thread is what is the definition of the term "mass"? of which there are two answers (1) Relativistic Mass and (2) Proper Mass.

11. Feb 3, 2004

### DW

No, $$\gamma m$$ is the energy. m is the mass being discussed.

Actually m is the mass according to every frame, not just the proper frame.

That wasn't the question posed for this thread, but anyway only one is correct in the context of modern relativistic physics terminology and that is the definition of mass as an invariant. And, it is the same value according to every frame, not just the value for the proper frame.

The context is modern relativistic terminology.

If it is the popular opinion then it is the modern relativistic terminology as I state and is not incorrect nor just my personal opinion. I am a bit surprised that you come right out and say here that the popular opinion (modern relativistic terminology) is incorrect.

How unfortunate for his students.

That is your own site, not a referrence, and is wrong. Exceptions do not prevent the modern relativistic terminology from being what it is.

Just because you want everything I say to be wrong and without meaning does not make it so. It can and has been given a general meaning and its use is greater than that of relativistic mass which is what makes that an obsolete concept.

Instead of digging up every author you can find who still uses outdated terminology get up to speed with the modern terminology. See instead-
http://www.geocities.com/zcphysicsms/chap3.htm
http://www.weburbia.demon.co.uk/physics/mass.html
"Space-Time Physics" by Taylor and Wheeler, 2nd edition, Freeman Press (1992).
"Does mass really depend on velocity, dad?" by Carl E Adler, American Journal of Physics 55, 739 (1987)
"The Principle of Relativity" by Einstein
"Concepts of mass" by Max jammer
"Einstein's Revolution" by Elie Zahar.
etc etc

Last edited: Feb 3, 2004
12. Feb 4, 2004

### Arcon

That is incorrect. $$\gamma m$$ is the mass of the particle whose proper mass is m. The mass-energy, E, is $$E = \gamma mc^{2}$$. The mass being discussed is the velocity dependant mass as they've already told you - more than once I might add.
The question posed was the correct expression for force. In case you're unaware of this fact, people discuss many things in a thread other than the subject posted. i.e. they discuss things which pertain to the question.

And as I've demonstrated many times, several thousand times as I recall, you're claim that there is only one correct usage of the term mass is incorrect. The mass a particle is the time component of the particle's 4-momentum 4-vector. The energy of a particle is proportional to the time component of the particle's 4-momentum 2-form. I.e. m = P0/c, E = cP0.

Not only wrong but also illogical.

Then you've just proved what I've been saying all these years - You're not paying attention to what I've been teaching you. Because something is used more it doesn't mean that it's the correct usage. It only means that a particular group of individuals use it more. I.e. it's probably the case that more particle physicists use proper mass since it is the subject of their research. However it's also probably that more cosmologists/GRists use relativisitc mass since that is what they use more since it is relativistic mass which plays a role of source in general relativity that is analogous to the role charge plays as source in EM. One need only crack open a cosmology text to see that fact.
Very typical of you waite. Belittle those who you don't agree with. Anyone would take Guth over you any day of the week.

That is a lie. If you claim there is an error then state your case. Don't simply post a lie like this.

An illogical conclusion and an illogical statement.

Wrong.

13. Feb 4, 2004

### DW

I already corrected you on that above.

As I said, it was the energy.

No, as I've already told you mass does not change with speed.

The one I posted was, yes I know.

And you have been wrong thousands of times. The mass is not the time component. That is the energy. The mass is the center of momentum frame energy and for a free particle is the "length" so to speak of the momentum four vector. The mass is the m given by $$(mc)^2 = g_{\mu} _{\nu} P^{\mu} P^{\nu}$$.

No P0 is the energy.

No this is only the case for special spacetime coorinates. Don't confuse the energy with the energy parameter for which even that is also not always P0.

(snipped a flame)

Where it comes to jargon yes it does.

No a true relativity guy is more interested in what quantities are frame invariant as relativity is really a theory of invariance. Besides, only in a sufficiently low speed and linearized limit does that analogy hold valid in which case the energy in this analogy reduces only to the mass anyway.

So you say.

(snipped some flames)

So you say.

(snipped some flames)

So you say.

Yes and yet the terminology is modern, unlike some of the more recent inappropriate uses of "relativistic mass" in a few obscure sources you probably took several days to dig up.

So you say.

No that was the fault of Plank Tolman and Lewis which was a concept for a time used by Einstein and then latter argued that it should be done away with by Einstein.

(snipped a flam)

This is your mistaken website, not a referrence.

And latter argued against its use as I said.

Neither book is anchient, but the "relativistic mass" terminology is obsolete no matter the order of publication.

(snipped a flame)

And he is wrong. It is the energy, just not the energy parameter.

(snipped flames and speculations about what I think)

Arguement by analogy is not logical.

Yes it is. You are confusing particle energy which by definition is that element just as by definition energy density of a distribution is $$T^{00}$$ with the energy parameter which is not always that element.

Last edited: Feb 4, 2004
14. Feb 4, 2004

### Arcon

I know what you wrote. I know what you claimed. What you wrote and what you claimed are both wrong. First off it should be obvious to anyone that E and m = m_o/sqrt[1-(v/c)^2] have different dimensions and they are defined differerently
Proper mass is not change with speed. Relativistic mass changes with speed. Please explain what part of that fact you're having trouble understanding.

Since both yogi and Dimitri are talking about a change in mass with speed they are talking about relativistic mass. If you don't believe me then simply ask then and they'll tell you. If you're uncertain what a person means then you should always ask them.

And each of those thousand times you were wrong.
Wrong. That is only proportional to energy in some cases. I've proven that fact to you several times.
Now you're chosing to create new terminology for some reason. There is no such thing as an "energy parameter" - There is only energy. Please learn what energy is.
Please do not start trouble yet again. Learn how to take criticism

15. Feb 4, 2004

### Arcon

yogi and Dimitri - When you used the term "mass" above, i.e. when you said it changes with speed, did you mean proper mass (aka "rest mass"), m0 or did you mean relativistic mass, m = m0/sqrt{1-(v/c)2}?

16. Feb 4, 2004

### DW

That depends on your choice of units alone and so is irrelevent.

And the mass which you call proper is the mass according to every frame, not just the proper frame. It is invariant.

(snipped flame)

I don't care if they meant "relativistic mass". They just said "mass" and as I corrected, that does not change with speed.

(snipped flame)

No it is the energy by definition. You are confusing energy with energy parameter again.

Just because you haven't learned GRs terminology doesn't mean that I am making it up. See-
http://tinyurl.com/2ne2b

(snipped flame)

So you say.

No their relativistic mass concept was energy which includes light.

Yes he did.

As I said, he did. Now you just PROVED my point. He argued against the use of the concept in his later years as demonstrated by your own quote which is a part of why the modern terminology does not encorporate it.

(snipped flames)

We are not talking about time or any other coordinate. The relativity of spacetime coordinates is not analogous to the invariance of physics.

So you say.

Last edited: Feb 4, 2004
17. Feb 4, 2004

### Arcon

Incorrect. Once more you're confusing the distinction between an equality with that of a definition. If what you claim is true then the frequency of a photon would would have the same physical meaning as the energy of the photon since E = hf for a photon. But that is quite untrue. In fact nothing could be further from the truth. Just because two quantities are proportional it does not mean that they have the same physical meaning. Arguing otherwise is like saying that momentum density is not really momentum density because it's proportional to energy flux. Please learn the difference between equality and definition.

If you're confident in your claim then prove that m_o*c*dt/dT is the energy of a particle under all circumstances.
Depends on how mass is defined.
That is incorrect. If you don't care what they meant then you're being rather rude. The purpose of this forum is discussion and learning. Why don't you want to know what they mean? Besides - it has never been the case that just because someone uses the term "mass" it means "proper mass." That you keep assuming that is why your posts are always confusing to people.
Incorrect. An assertion you've never been able to back up - one that you've only been able to constantly repeat.
Both incorrect as well as irrelevant. I was explaining to you that it was Einstein, not Tolman who proved that radiation has mass.

And as I've explained - Einstein never argued that one should not say that light has mass. When you repeat "Yes he did." all that means to me is that you can only repeat your statement. Try to prove your claim rather than repeat your claim.

Incorrect. I though that I've already explained this to you. Einstein argued against M = m_0/sqrt[1-(v/c)^2]. He did not argue against M = m_o*dt/dT and it was that which he held to be the mass of a slowly moving particle, not m_o. And it is m_o*dt/dT which is the correct/modern definition of relativistic mass. In fact this is very important in GR where dt/dT does not always equal 1/sqrt[1-(v/c)^2]. It is for this reason that Einstein said that mass increases when its near other masses. You were the one who brought up Einstein. Are you now going to abandon him now that I've shown you that you didn't know what you were talking about?

By the way. I asked you a question. Why didn't you answer my question? I asked you why you refered to Jammer's book when Jammer argues against your claim?

Also, it you feel capable of it, prove that the magnitude of the energy-momentum 4-vector is rest mass and not rest energy. Claims that they are proportional are invalid according to you so no arguement based on proportionality will be valid.

18. Feb 5, 2004

### Phobos

Staff Emeritus
out of control again

under review