# Mass/Energy in special relativity

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1. Jun 23, 2014

### Entanglement

I only understand mass and energy according to Newtonian mechanics,
Where mass is the resistance of a body to accelerate, or how much matter there is, and anything has mass is a matter.
Energy is the ability to do work and it's not a matter.
It was believed that mass and energy are two different separated things until Einstein who explained that they are the same thing, I don't understand how are they same thing, and what are their new conceptions according to Einstein and to special relativity.
I'm still a freshman in special relativity, so I barely know much about it, I'd like a simple intuitive explanation if possible, and thanks !

Last edited: Jun 23, 2014
2. Jun 23, 2014

### Staff: Mentor

Actually, in modern usage mass and energy are not the same thing. They are decidedly distinct.

You may be aware that the Lorentz transform unifies space and time. Not that they are the same thing, but they are different parts of a unified spacetime.

In the exact same way energy and momentum are united in the four-momentum. Energy is one component of the four-momentum vector and momentum is the other three components. Mass is the "length" of the four-momentum.

3. Jun 23, 2014

### Matterwave

Einstein basically showed, in his first paper on the subject, that a bound state (say an atom) which emits radiation (a form of energy) of energy E, will have less inertial mass in its rest frame by a factor of E/c^2. This means that even though the particle this atom lost (photon) is mass-less, and the only thing that this particle carried away is energy, still, the atom has lost mass. This is what is meant by mass-energy equivalence. One should be careful, though, because energy is not measured in absolute quantities like mass. We know what 0 mass means, and we know what 2kg means. But for energy, only differences in energy matter. So we can't naively, for example, associate kinetic energy of an unbound state with mass. I mention this because a lot of people will learn energy-mass equivalence only from a very superficial level, and then declare "photons have energy, therefore they have mass!". This is not true.

There is more subtlety in the subject than that.

4. Jun 23, 2014

### pervect

Staff Emeritus
In Newtonian mechanics, "quantity of matter", "force required to accelerate", and mass are all the same thing.

In special relativity, they're three different things. This is probably the first lesson to learn.

The equations are not that complicated to present, and hopefully will lead to understanding of the testable, scientific issues. Note that we try to avoid extended discussion of the non-scientific, non-measurable philosophical issues here on PF, mainly because such issues tend to wind up in endless arguments with no clear resolution. This happens precisely because the issues can't be resolved by experiment.

Here are the details.

A point particle (or an isolated system that isn't a point) has a characteristic property, called its "mass", (more precisely, it's invariant mass) that is a property of the particle which is independent of the observer.

Momentum is a concept shared by Newtonian mechanics and relativity, though the formulae are different the ideas remain the same.

The momentum of such a particle is given by the expression p = $\gamma m v$ where $\gamma = 1 / \sqrt{1-v^2/c^2}$ where m is the invariant mass, and v is the velocity of the particle. Note the difference from Newtonian physics, which doesn't include the factor of $\gamma$.

Force is the rate of change of momentum with time. This is true in both special relativity and Newtonian mechanics. One may or may not be used to describing force in this manner in Newtonian mechanics, but it becomes worthwhile to learn this because this definition works for special relativity too.

Using the chain rule from calculus we can write:

$F = dp/dt = (d/dt) (\gamma m v) = (d\gamma/dt) m v + \gamma (dm/dt) v + \gamma m (dv/dt)$

Thus F = ma no longer works in special relativity, so you need to disambiguate "mass" from "force required to accelerate", they're different concepts now. As you can see the force expression is complicated - forces are usually replaced with 4-fources for this reason, but at this point I feel a further explanation of this point would be more distracting then helpful.

Energy for a moving particle with velocity v in special relativity is given by the formula E = \gamma m c^2 This is different from the Newtonian formula in that the energy is not zero when the velocity is zero, it is instead given by mc^2. When you regard energy as the integral of work $E = \int dW$, the change in the energy at zero velocity is just a change in the constant of integration.

However, if you have a particle of mass m, and an anti-particle of mass m, (neither of which is moving so there is no significant kinetic energy) and you annihilate them in a particle-anti-particle reaction, the released energy will be 2*m*c^2. Thus this simple additive constant does has some physical interpretation in keeping with the "ability to do work" paradigm, as includes the energy (ability to do work) available in the system via the mechanism of particle-antiparticle annihilation. This is more significant for particle physics than everyday physics, but the definition of the energy being equal to mc^2 at zero velocity in special relativity remains as the default choice even in non-particle physics applications.

This choice of energy = mc^2 at zero velocity is also necessary for the well-known equation
E^2 = (p c)^2 + (m c^2)^2 to work, E being the energy, p being the magnitude of the momentum, and m being the invariant mass.