# Relativistic and rest mass

1. Feb 25, 2005

### Symbreak

relativistic mass is defined by the lorentz contraction equations (i.e time & legnth).

but why should mass increase only for speeds very close to c? is there any theory that explains this?

we know that rest mass is basically the resistance to acceleration and the higgs theory postulates that mass arises when a group of charges interacts with the higgs field.
Is it possible that rest mass and relativistic mass are actually the same phenomenon? perhaps the question is not what sets mass at a certain value, but what stops mass being infinite.

2. Feb 25, 2005

### jcsd

The relatvistic mass M is a function of speed given by:

$$M = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$$

So it increases whenever v (speed) increases, but it it is only signifcant when v is a signifcant fraction of c. Note that the value of a relatvistic mass depends on the relative velcoity of the observer, also when v = 0 $M = m_0$.

Rest mass is defined as the inertial mass (i.e. the m in F = ma) of an object in it's rest frame, howver it is not generally it's inertial mass (infact in special rleativty the general concept of inertial mass of an object is not easy to define as an accleration is not always parallel to the force producing it).

3. Feb 26, 2005

### pmb_phy

Why do you say that? Its not defined by the Lorentz factor. It just happens that the Lorentz factor appears in the derivation.
Mass is always a function of speed. Its just that at low speeds the Lorentz factor is close to one.
Relativistic mass is defined such that mv is conserved quantity. rest mass is identical to relativistic mass at low speeds and represents the bodies ability to change momentum.
They're both identically the same thing. Its just that rest mass = rel-mass(0)
That is incorrect. Inertial mass, as the term is used in relativity, is idenical with relativistic mass. It is not the quantity that you're thinking of, which is transverse and longitudinal force. Rel-mass represents a bodies ability to change momentum. Hence the reason you'll see in all derivations of $m = \gamma m_0$, i.e. that rel-mass is defined as that quantity m such that mv is conserved (i.e. m = p/v).

Pete

4. Feb 26, 2005

### jcsd

But for me at least the whole concept of inertial mass is directly rooted to F = ma, so I'm not sure it's particualrly good idea to call any quanitty in relativty the inertial mass (except the rest mass of an object in it's rest frmae which is unambigiously the inertial mass in that frame). Anyway I think the issue has been discussed enough

5. Feb 27, 2005

### pmb_phy

I don't know that Symbreak has discussed it enough. He/she may wish to know more.

F = ma is incorrect even in non-relativistic mechanics. F = ma is a relationship for when m is not a constant of motion. The correct relationship is F = dp/dt (as Feynman pointed out in his text). F = dp/dt was due to Newton and F = ma was due to Euler.

Pete

6. Feb 27, 2005

### Symbreak

Thanks for the replies.

One problem I have with the relativistic mass is why the increase of mass is near c and not any other velocity? There seems to be a conflict with GR. Every inertial velocity is equivalent (we cannot distinquish a constant velocity v from absolute rest, only from the perspective of the observer). Then why does mass increase for a v near c? Why should we discriminate a 'high velocity' from a 'low velocity' if they are both the same?
Another thing. If c always travels the same to all observers, no matter what their velocity, what about observers which travel near c? According to the 'time dilation' equation of Lorentz, time would slow down for these observers. If time does - then by definition, light (a wave which moves at a rate c in time) must move slower!

7. Feb 27, 2005

### jcsd

Yes F = dp/dt is prefereable to F = ma, but usually it is the equation F = ma that is used to define the inertial mass of an object.

8. Feb 27, 2005

### kevinalm

Something that is rarely pointed out directly is that the difference (Mrela - Mrest) is the mass equivalent of the kinetic energy. Mrela is the instantaneous inertial mass. I let you ponder the implications.

For a moving observer the universe appears contracted, thus preserving the observed value of c.

9. Feb 27, 2005

### pmb_phy

You can always follow your heart with definitions and use what works for you.

Pete

10. Feb 27, 2005

### pmb_phy

That's pretty well known so I don't understand your comment.

Pete

11. Feb 27, 2005

### kevinalm

Which one? The first?

Most entry level texts don't drive home the relationship Ekinetic = (Mrela-Mrest)c squared. At least in my experience. The poster was unclear as to an underlying reason for the Mrela tending to infinity as v goes to c, at least that was my read. As you accelerate a mass, you add kinetic energy. Which manifests as additional inertial mass. Which requires an ever increasing amount of energy to accelerate the mass. The limit of this is M goes to infinity as v goes to c. If you're reasonably skilled with differential equations you can derive the equation for Mrela from the postulate that Ekinetic=(Mrela-Mrest)c squared, and nothing else.

As to the second, I believe I misunderstood the poster. Upon rereading, I think he's jumping reference frames and confusing himself.