# Questions on Mass [duplicate]

In my eyes, mass is a property of "matter" (I suppose) that indicates how an object reacts (accelerates) to an applied force. In special relativity lectures, if I recall correctly:

$F=γ^4m_0a$

So the resistance to acceleration does increase with increasing velocity. But there is this: $m=E/c^2$. Does the mass increase as an object has more internal energy such as warmth? In the case of kinetic energy does this also occur or isn't that internal energy? Should it then be $E_{internal}$ instead of the total energy? Different inertial frames measure an object to have different kinetic energies, how does mass work in that case?

I'm confused about mass and really need some explanation, thanks!

• Possible duplicate of Why is there a controversy on whether mass increases with speed?
– JMac
Jun 13, 2017 at 10:50
• Thanks you, reading that clears up my first question, but I still dont understand how kinetic internal enery (heat) has an effect on this, can you explain ? Jun 13, 2017 at 11:00
• Your formula is incorrect: momentum is $p=\gamma mv$ and then $\frac{dp}{dt}=F$.
– user154997
Jun 13, 2017 at 11:44
• Possible duplicate: physics.stackexchange.com/a/328597/154997
– user154997
Jun 13, 2017 at 12:05

A body's mass m (used to be called $rest\ mass, m_0$) is an intrinsic property of the body, independent of its motion. But it is not simply a sum of masses of its constituent particles. It increases with the body's internal energy (including vibrational energy of particles) according to $\Delta U = c^2 \Delta m$.
A body's total energy, E is given by $E=\gamma m c^2$, in which $\gamma$ has its usual meaning. $\gamma m$ (which does, of course, increase with the body's speed) used to be called the body's relativistic mass, but this usage has largely fallen out of favour, partly because, except for the 'mere' constant, $c^2$, $\gamma m$ is telling us the body's total energy. Why confuse the issue by giving it another name?
If the body is at rest, $\gamma=1$, so the energy of the body at rest is $E=m c^2$. [The relationship, $\Delta U = c^2 \Delta m$, given in the first paragraph follows from $E=m c^2$, applied to an ordinary body.]
The body's KE when moving is therefore $E_k=\gamma m c{^2}-m c^2$.
In SR, a body's momentum is given by $\vec{p}= \gamma m \vec{v}$. Part of the motivation for calling $\gamma m$ "relativistic mass" was that we then preserve in SR the Newtonian momentum formula, with relativistic mass instead of rest mass. But this is now generally considered to be not worth the conceptual disadvantages described above. And, what's more, it doesn't help to preserve other Newtonian formulae, such as that for KE!