# Mass increases with acceleration ?

1. Jan 28, 2007

### becsy

"Mass increases with acceleration"?

Hey! I have a Question. I was reading through some of the forums and something popped into my head. "Mass increases with acceleration" so does that mean that is decreases with decceleration? i would think it would but im not to sure. If i got in an elevator would my mass change? and if it does how would it change. its all confusing.

2. Jan 28, 2007

### Chris Hillman

What changes when a body undergoes acceleration?

No wonder you are confused! It is completely incorrect to say that "mass increases with acceleration". What you probably read somewhere (on this forum? I hope not!) was the claim that "relativistic mass increases with velocity". This should be restated as "relativistic kinetic energy increases with velocity".

The mass of an object is a kinematically invariant property; that is, it does not change simply due to the motion of the object.

Specifically, the expression given by Einstein for the energy (mass plus kinetic energy) of an object in terms of its mass and its velocity is:
$$E = m \, \cosh \, \operatorname{arctanh} \, v = \frac{m}{\sqrt{1-v^2}} = m \, \left( 1 + \frac{v^2}{2} + \frac{3 \, v^4}{8} + \dots \right)$$
where $m$ is the mass (sometimes misleadingly called "rest mass"), $m \, v^2/2$ is the Newtonian kinetic energy, and the remaining terms are relativistic corrections to the Newtonian kinetic energy. The effect of these additional terms is to ensure that as the velocity approaches unity (the speed of light in relativistic units in which we measure both time and distance in meters), the kinetic energy of the object diverges.

Having said this, I should probably add that the corresponding expression for the magnitude of the relativistic momentum is
$$\| \vec{p} \| = m \, \sinh \, \operatorname{arctanh} \, v = \frac{m \, v}{\sqrt{1-v^2}} = m \, \left( v + \frac{v^3}{2} + \frac{3 \, v^5}{8} + \dots \right)$$
where $m \, v$ is the Newtonian expression for the momentum, and the remaining terms are relativistic corrections, which again have the effect of ensuring that as velocity approaches unity, the magnitude of the momentum diverges. Note that while mass is a kinematicaly invariant quantity, the kinetic energy and momentum are both observer-dependent quantities.

(The qualifier "kinematical" refers to the fact that, in general relativity, if you heat an object, you are adding energy to the system, so that its effective gravitational mass increases very slightly. This effect is much to small to affect any cooking in your kitchen, however!)

Unfortunately, I should warn students that several current threads on this board concern claims by a dissident which are, to say the least, highly idiosyncratic, and in my view, quite unhelpful to students trying to grasp the fundamentals of relativistic physics. A book I highly recommend is Taylor and Wheeler, Spacetime Physics, First Edition (the second edition unfortunately dropped discussion of an invaluable concept, the "rapidity" of a "boost", which is the analog of the angle of a rotation). This widely used and excellent textbook represents mainstream pedagogy in teaching the basic ideas of relativistic physics, and there are many good reasons for studying such a textbook rather than a dissident approach.

Last edited: Jan 28, 2007
3. Jan 28, 2007

### disregardthat

Shouldn't v^2 be divided by c^2 in the equation to find the relative mass? That is what has been said everywhere else at this forum.

4. Jan 28, 2007

### bernhard.rothenstein

mass

In special relativity not?

Last edited by a moderator: Jan 28, 2007
5. Jan 28, 2007

### Staff: Mentor

Note that Chris is using units in which c = 1 and v is between 0 and 1. (See his reference to "relativistic units".)

6. Jan 30, 2007

### cesiumfrog

In similar spirit, should it be noted that rapidities (and the previous equations in hyperbolic trig form) are now particularly idiosyncratic? Rather than representing mainstream pedagogy, you're saying they've been purged from modern textbooks. Granted, it was a cute concept (suggesting insight into the absoluteness of rotation, like acceleration rather than velocity), perhaps it's just considered harder to build further upon?

7. Jan 30, 2007

### Chris Hillman

I said no such thing: I said that in the second edition of a particular book, the second author chose to drop a topic, a decision which observers other than myself regard as a serious mistake.

Why not follow up on some of the reading in Lie theory which I suggested in some recent posts? We are talking about exponentiation and an affine parameter in one dimensional subgroups. That's why your assertions, quite frankly, are nonsense

8. Jan 30, 2007

### cesiumfrog

Sure, if you think it's relevent, in which threads have you suggested reading in Lie theory?

To clarify, are you saying that "exponentiation and an affine parameter in one dimensional subgroups" is the reason my assertion (that rapidities have become an idiosyncratic, rather than mainstream, approach to SR) is nonsensical? How?

Also, could you specify the dissident posts you mentioned earlier?