# Einstein's:Mass increase resulting from Acceleration increase

1. Jun 29, 2005

### Spin_Network

I am looking for the ultimate reference to the notion by Einstein, that increase in Acceleration will result in increase in Mass?

Any links or handwaving would be much appreiciated, as I intend to place a fundemental number of questions, in order to expand my knowledge on LQG, thanks.

2. Jun 30, 2005

### Aer

I may be totally off base here (and I am sure others here will correct me if that is the case), but (I have been told) most physists have abandoned the idea of increasing mass with acceleration. Instead an objects mass is considered constant even though the objects total energy increases with velocity, that is e=f(m,c,v) (I do not know the exact equation). Of course this equation reduces to e=mc^2 for v<<c.

3. Jun 30, 2005

### Chronos

Confusion sets in when you don't subtract the energy it takes to boost the relativistic mass from the local inertial reference frame.

4. Jun 30, 2005

### pervect

Staff Emeritus
Nope, you're on base, see for example the link:

Does mass change with velocity?

Relativistic mass changes with velocity, but the usefulness of the concept is questionable. Mass (without anyh qualifiers) is taken to be invariant mass, which does not change with velocity.

Acceleration changing mass is "right out".

5. Jun 30, 2005

### Garth

It depends how you measure it, whether from a frame dependent observer's point of view or from the frame independent space-time 4D point of view.

From the POV of the observer with a definite preferred frame of reference, their own, foliated into 3D space + time, a moving mass appears to increase with velocity:
$$m=m_0\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
where $$m_0$$ is the object's rest mass measured in its 'co-moving' rest frame.
The 'faster it goes the harder it is to push'. The force necessary to accelerate the object:
$$F=m\frac{d^2x}{dt^2}$$
tends to infinity as v tends to c.

However from the perspective of 4D space-time the four force is given by;
$$F^\mu=m_0\frac{d^2x^\mu}{d\tau^2}$$ where
$$d\tau^2=dt^2 -\frac{1}{c^2}(dx^2+dy^2+dz^2)$$ in a SR Minkowski space-time.
and the mass remains invariant $$m_0$$. The effect of the 'increase' in mass this time is accounted for by the definition of time; from the 4D perspective the invariant proper time $$\tau$$ is used not the observer dependent t.

It may be considered more 'pure' in relativity to use frame independent invariant mass and invariant time, however as real observers we are locked into a preferred time, our own, and we have to look out into the universe from that frame of reference. To any observer the masses of other moving objects appears to increase with velocity as above.

The choice of whether to use an observer dependent 3D + time or a frame independent 4D perspective is closely connected with how we want to account for the frame dependent concept of energy, as the observer dependent 'relativistic' mass subsumes the classical concept of kinetic energy as $$E = mc^2$$.

I hope this helps.

Garth

Last edited: Jun 30, 2005
6. Jun 30, 2005

### Dr.Brain

Mass of a moving object deviates from the rest mass $m_o$ as:

$m= \frac {m_o}{\sqrt 1- \frac {v^2}{c^2} }$
As velocity increases , mass increases and with increasing velocity, it increases more rapidly.

BJ

Last edited: Jun 30, 2005
7. Jun 30, 2005

### Garth

I think I have just said that....
But again note: only from the observer's frame dependent POV.

Garth

8. Jun 30, 2005

### pmb_phy

It is incorrect to make this assertion since not all authors use the term in that way. MTW is one example. There are places in their text in which "mass" means relativistic mass. Its only in certain fields where it is easier to mean "rest mass" when someone uses the term "mass." However most people grasp what the term means either from the author stating it outright or when its obvious from context. This is true of other terms in relativity too. Take time dilation as an example; When people say that "the neutron has a half life of 15 minutes" it is assumed that they are referring to the proper life when they use that term. However nobody who knows relativity would say that time does not altered with speed.

Its quite easy to make errors if one does not completely understand the definitions correctly. In fact I know of one SR text which makes a mistake because of his usage of the term "mass" as proper mass.

Let me give you an example: Let S be an inertial frame of referanc in flat spacetime. If a magnetic field has a value of B and the electric field has a value of E = 0 then what is the mass density of the field in frame S? I'll get back next week to follow up with the answer.

Pete

Last edited: Jun 30, 2005
9. Jun 30, 2005

### pmb_phy

I can't see how that could be determined unless you took a poll. The usage of the term can be found in online university lecture notes, in new relativity texts and in the physics literature.

The term "mass" cannot be used as "invariant mass" in all generality since it can only be used under certain circumstances, namely for point particles or bodies wich are under stress. E.g. consider an electric dipole in an E field. Since the net charge on an electric dipole is zero the particle will have a rest frame. Rel-mass is defined as m = p/v. Some people claim that relativistic mass is defined as "m = E/c^2". However in this dipole example E/c^2 does not equal p/v so the terms cannot be defined to be equal/same since the definitions are different and since they are not always equal.

Pete

10. Jun 30, 2005

### pmb_phy

You must be refering to velocity, not acceleration, right? If so then the article was published in either 1906 (or 1907??). Sorry that I don't recall the name of the paper or the name of the journal. In the first section of this article Einstein shows how energy carries mass with it. In the second section he uses the term to speak of the mass of radiation. He also uses this notion on page 101 in his text The Meaning of Relativity when he's addressing the changing of a bodies mass with a change in the gravitational field in which it sits.

Pete

11. Jun 30, 2005

### pervect

Staff Emeritus
As I recall, I did not agree with this assertion the last time you made it, though I don't recall offhand which sections of MTW you interpreted as making this statement. You can quote them if you like, but I'm pretty sure they will not be clear-cut, and that I will disagree with your intepretation of what MTW actually said. If you feel like re-hashing this old point, though, go ahead.

I *really* do think that it is not too much to ask that people who mean "relativistic mass" should actually SAY "relativistic mass", not "mass".

It's quite simple - mass does not increase with velocity, relativistic mass does increase with velocity (I have no problem with that statement).

I'm willing to give older textbooks a "grandfather clause", on this point, for that matter, some of them may be written in such a confusing way. (MTW is pretty old, but it is modern enough not to suffer from this sort of confusion).

12. Jun 30, 2005

### Integral

Staff Emeritus
I am going out on a limb here.

Perhaps the best way to prevent a recurrence of an old and ongoing quibble.

Let us define, for conversations in these forum, the meaning of the word "mass" to be REST MASS vs Relativistic mass.

Can we now direct the conversation toward the meaning and interpretation of the relativistic mass, rather then pointless argument over which is the correct viewpoint.

My question:
If applied energy is not increasing the mass, what is it increasing?

13. Jun 30, 2005

### Aer

I cannot answer your question without risk being banned from physicsforums. As the answer (unfounded, no experimental data to prove or disprove it that I have found anyway) would imply one of the postulates of Special Relativity to be in slight error.

14. Jun 30, 2005

### Integral

Staff Emeritus
I am not looking for, nor do I want an explantion which lies outside of the bounds of General Relativity.

15. Jun 30, 2005

### pervect

Staff Emeritus
If you apply a force to a single particle to accelerate it, you increase it's total energy (as measured in some particular inertial coordinate system). It's rest energy (rest mass) would remain the same, being an invariant. I would suggest that we say that the energy "went into" the kinetic energy of the object.

At least that's what we'd say if we were doing special relativity. If we were doing GR, we'd start asking about the presence of timelike Killing vectors, or the asymptotic flatness of space-time, and lose at least 90% of our audience in the process :-)

Applying energy to a system of particles to increase the temperature of said system yields a different result. Here, the energy of the system increases, and the system is still at rest, so we say that the rest mass (rest energy) of the system increases. This is true with any definition of energy or mass that I'm aware of.

People tend to think that if we have a system with a "relativistic mass" of mr, we can apply Newton's law to it simply by replacing mass with relativistic mass. This is false for a moving system (the only case when relativistic mass is different from invariant mass). The acceleration of a moving system will depend on the direction of the force as well as the "relativistic mass". The acceleration is NOT in general equal to a = F/mr. We can get around this by talking about the "transverse mass" of the system. In the past some authors did in fact do this (I believe Einstein has done this). But now we have _three_ kinds of mass to worry about. (And you thought things were confused with only two :-)).

People also tend to think that the gravitational field of a system with a "relativistic mass" of mr is -G mr /r^2 and that it's uniform in all directions. This is another all-too-common false notion that is caused by people assuning that relativistic dynamics is just a matter of substituting "relativistic mass" wherever "mass" occurs.

16. Jun 30, 2005

### Aer

Nor do I want to give you such an explanation, although I said nothing of general relativity - special relativity is a "special" case of general relativity and it is only this special case that that the idea would challenge. Anyway, I don't understand your question. Isn't "relativistic" mass a fundamental part of the current understanding of relativity?

17. Jun 30, 2005

### pervect

Staff Emeritus
"relativistic mass" is not particularly fundamental. Some people like it, some people don't. Read the sci.physics.faq

Does mass change with velocity?

18. Jul 1, 2005

### pmb_phy

I'm simply stating a fact. I'm not attempting more that that but its quite clear that MTW is doing what I said. I know relativity far too well to make such a silly mistake that you assumed I did.

However all one has to do is open the book and read it and that's all that is required. If someone has MTW and wants the reference then I'll post it.

Pete

19. Jul 1, 2005

### pmb_phy

Who are these "people"? Are they students? Teachers?

As with any subject, if a student makes a mistake like this then they will have to study more.
What is mr"? This is false for a moving system (the only case when relativistic mass is different from invariant mass). The acceleration of a moving system will depend on the direction of the force as well as the "relativistic mass". The acceleration is NOT in general equal to a = F/mr.

Last edited: Jul 1, 2005
20. Jul 1, 2005

### pervect

Staff Emeritus
After a few zillion times (I exaggerate slightly, perhaps) of trying to tell people that, it is wrong to subsitute mr (relativisitc mass) for m in Newton's laws (either the F=ma variety, or the F=gmM/r^2 variety), I get just a little bit tired of repeating myself.

So by people, I mean people on physics forums, many of which are not particularly knowledgable about relativity.

Some people don't seem to listen, either. (Not much that can be done in that case, I suppose, but it doesn't help the frustration level). I could point to a recent example or two on the general physics forum of this phenomenon, BTW, but I think it would be better if I kept things less personal and didn't single out any particular individuals in this manner.

The sci.physics.faq also comments on the large number of people who get confused, rather than enlightened, by relativistic mass.

In spite of this, I'm not on any campaign to stamp out the concept. If people like the idea of relativistic mass, that's fine with me, as long as they use it correctly. Part of correct usage is to identify that one is in fact talking about relativistic mass, rather than invariant mass. Making this distinction manifest would avoid the age-old confuison about the fact that a photon has no invariant mass (rest mass), though it does have energy (and so has "relativistic mass").

What I object to is when people use "mass" with no qualifiers to mean "relativistic mass". This is not done in modern textbooks, and I certainly am not aware of it being done anywhere in MTW's "Gravitation".