(E,p)(E,-p) What pushes on a scale?

[Phrak]

You hold in the palm of your hand a classical particle with momentum

$$P = (E,cp_{x},cp_{y}, cp_{up})$$ where $$cp_{up}=0$$ .

Placing it gently on a scale you ask, what part of

$$P = (E,cp_{x},cp_{y}, cp_{up})$$

is pushing down on the scale?

 

[lbrits]

Usually you weigh particles in their rest frame, otherwise they move around too much =) In that case, you are measuring the first component of $$P$$.

You might consider the following: A gas of such non-interacting particles, in a box. The average 4-momentum of the fluid is then $$\left\langle P \right\rangle = ( \left\langle E \right\rangle, 0, 0, 0)$$, so the mass if the gas is going to be $$\left\langle E\right\rangle$$.

 

[Phrak]

What if the momentum is non-zero?

 

[lbrits]

Then it won't be in the palm of your hand for very long.

 

[Phrak]

OK, so backing up a little, in everyday life, where things sit still on a scale.

$$\langle P \rangle ^2 = m^2 = E^2/c^4 - p^2/c^2$$, defining m after some units adjustment, and obtaining your equation when $$p=0$$ .

Under restricted conditions, m is what could be called gravitational mass.

But I think you misunderstand me; I really don't know what the general solution is.

I can guess halfway through it. Using the equivalence principle, the problem is equivalent to asking what resists acceleration a_z, by a force f_z, in the lab frame. So I might guess that it involves the 4-vectors of force and acceleration, whatever they are. Do you think so?

 

[lbrits]

I have some notes which might help:

http://www.physics.thetangentbundle.net/wiki/Special_relativity/force
http://www.physics.thetangentbundle.net/wiki/Special_relativity/motion_under_constant_force

The "3-force" is what the experimenter applies to the object. The measurement on the scale, if you will.

 

[Dale]

I would say that neither E nor p push on the scale. Instead, the scale pushes on the particle and measures the force with which it pushes.

 

[Phrak]

You've got a point Dale. If the scale acts with Fz in the lab frame, what dv/dt, in either frame, does the particle have?

 

[lbrits]

A quick perusal of the links I gave would show that the answer to that question is
$$\vec{f} = \frac{1}{\gamma} \frac{d^2 \vec{x} }{d\tau^2} = \frac{1}{\gamma} \frac{d}{d\tau} \left( \gamma \frac{d\vec{x}}{dt} \right) = \frac{d}{dt} \left( \gamma \frac{d\vec{x}}{dt} \right)$$

Cmon people... no need for spoon feeding.

 

[Phrak]

lol, lbrits. I need spoon feeding. To be honest, the equations in those links were pretty haphazardly done, but for review purposes. The deductive chain was obscure.

 

[lbrits]

The first link assumes basic familiarity with SR, but I've tried to make it more self contained. Thanks.

 

[Phrak]

Sorry, I didn't know you wrote it. I hope you can take it as constructive criticism from someone who hasn't delt with SR in a while, and didn't devote a great deal of time reading it. To be fair, I've been involved in another persuit that takes up what little brain power my wee brain can handle.

 

[Dale]

You've got a point Dale. If the scale acts with Fz in the lab frame, what dv/dt, in either frame, does the particle have?

Since you are already starting with the http://en.wikipedia.org/wiki/Four-momentum" is simply dp/dtau.