# Mass in terms of special relativity

According to special relativity mass doesnt add up like we think it does. That is, a system of 2 protons might not necessarily have a system mass of 2*(mass of one proton). If the protons are travelling their system mass can be more than the sum of individual masses.

I understand this in theory completely. But I'm looking for an experiment that has actually weighed two protons in motion and found the weight to be different. (need not be just protons or any other tiny particle)

I have heard about the water tumbler experiment in which someone tried to heat it up and then measure the change in weight (mass of heat) but was unsuccessful. I understand that this is not a straightforward question-answer question. But I had nowhere else to go.

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Edits that invalidate existing answers are generally invalid edits even when they come from the OP. Take the time to compose the questions you mean so that such edits are not necessary. Anything else s unfair to people trying to help you by posting answers. – dmckee Jan 23 '14 at 21:28
Try to ask a separate question instead of rewriting this one. – Manishearth Jan 24 '14 at 9:23

The LHC does this every time it collides two protons.

The protons in the LHC travel at very nearly the speed of light. So let's calculate their kinetic energy assuming the speed is $c$ and the mass is the rest mass:

$$KE = \frac{1}{2}m_0c^2 \approx 7.5 \times 10^{-11} J \text{ or } 0.47GeV$$

But we know that the proton energy in the last run of the LHC was 4,000GeV per proton and in the next run will rise to 7,000GeV per proton. The protons can't move faster than $c$, so the only explanation is that the protons are behaving as if their mass has increased.

Note that I'm being careful how I phrase this, and in particular I'm not claiming their mass has actually increased. That's because the kinetic energy of a relativistic particle is not simply given by $1/2mv^2$, but rather by:

$$KE = \sqrt{p^2c^2 + m^2c^4} - mc^2$$

where the mass in this equation is just the rest mass (also known as the invarient mass). You can interpret the increase in KE as being due to the proton mass increasing, but we don't normally view things this way because it causes confusion. For example if the mass really did increase you could argue that a clump of protons (i.e. matter) moving fast enough would get heavy enough to form a black hole, and this isn't the case.

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That's an indirect way to ascertain that the mass is actually increasing --- "the only valid explanation". What I'm looking for is this --- take a stationary tub of water (zero momemtum) and then heat it (increasing energy while the momemtum remains constant at zero). Now if you weigh this tub, theoretically the system mass should increase? – Black Dagger Jan 23 '14 at 11:35

If you consider a system with momentum/energy $p$, the definition of the (inertial) mass of the system is (in units $c=1$)

$m^2= p^2 = E^2 - \vec p^2$.

This is a Lorentz-invariant quantity, which does not depends on the choice of the inertial frame.

So, if we consider a system with one particle, the mass of the particle does not "increase" with speed, only the kinetic energy $E-m$ increases with speed, here the speed means the speed of the particle relatively to the inertial frame.

A (non-interacting) $2$-particle system has a "mass":

$M^2 = P^2 = (p_1+p_2)^2 = (E_1+E_2)^2 - (\vec p_1 + \vec p_2)^2 \\= (m_1+m_2)^2 + 2(E_1E_2 - \vec p_1.\vec p_2 - m_1m_2)$

Once again, $M$ is a Lorentz-invariant quantity, and does not depend on your choice of initial frame.

Now, if you check individually, the mass of the particle $1$, or the mass of the particle $2$, in a experiment, you always find $m_1$, and $m_2$. The individual mass of the particles $1$ and $2$ does not change, they are Lorentz-invariant quantities.

Finally, one subtelty, you are speaking of "weight", so you are speaking about the gravitational mass, but thanks to the equivalence principle, the gravitational mass is the same that the inertial mass.

[EDIT]

(due to the reframe of the OP question)

The above formula may be extended to many particles, with $M^2 = (\sum\limits_i m_i)^2+ \sum\limits_{i \neq j} (E_iE_j - \vec p_i. \vec p_j- m_i m_j)$. One may imagine, in a thermal point of view, that the particles get an increasing speed due to the energy supplied externally, and so $\sum\limits_{i \neq j} E_iE_j$ is increasing, while due to the thermal hypothesis, we have always $\sum\limits_{i \neq j} \vec p_i. \vec p_j = 0$. So, finally, $M^2$ is increasing.

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I have reframed my question. What you have answered is that you are just changing free-float frames, but your system is isolated and hence the mass stays constant. In one free-float frame the protons are moving while in the other they are at rest. Energy is changing, momentum is changing, but the mass is invariant. I get all that. My question is for an isolated system in which energy is being supplied externally. Let me know if its clear. – Black Dagger Jan 23 '14 at 12:57
I updated the answer, but, please, in the future, do not reframe your question too much. – Trimok Jan 23 '14 at 13:15
I made a correction to my edit. – Trimok Jan 23 '14 at 13:49

Well, I would like to point out that only the relativistic mass increases and it is just an observation. The real mass remains the same, no matter what happens. So in the above case, the actual mass of the protons will remain the same. So if there is any experiment that is conducted, the statistics will not be useful because only the original mass takes part. The increase is only relative and does not change anything.

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But I'm looking for an experiment that has actually weighed two protons in motion and found the weight to be different. (need not be just protons or any other tiny particle)

Practically all high energy physics experimental data are based on the formula given in Trimok's answer.

Take the now famous discovery of the Higgs. How do you think it was found? By looking at invariant masses of decay products.

Di-photon (γγ) invariant mass distribution for the CMS data of 2011 and 2012 (black points with error bars). The data are weighted by the signal to background ratio for each sub-category of events. The solid red line shows the fit result for signal plus background; the dashed red line shows only the background.

Note the little bump? It is the signal that a resonance of two photons with invariant mass at 125GeV with a width less than 10 GeV is there and a strong candidate for the Higgs boson. Note the two particles whose rest mass individually is 0, whereas the rest mass of the Higgs is 125 GeV.

The experiment has detected two photons in its electromagnetic calorimeter, and connected them with an event vertex where an 125 GeV resonance decayed into them.

The opposite can be done, but is not so spectacular. The particle data book is full of scattering electrons on positrons, mesons on protons etc, and the consecutive appearance of resonances with larger mass than the sum of the incoming masses. Look at the plots of page 5 in the link

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