Why does Special Theory of Relativity leave out Potential Energy?

[Saptarshi Sarkar]

TL;DR Summary

Why does STR leave out Potential Energy while Classical and Quantum Mechanics don't?

While studying Special Theory of Relativity I came across the formula for the energy of a particle. The total energy of a relativistic particle in STR contains the Rest Mass energy and the Kinetic energy. But, in Classical and Quantum Mechanics, we consider the total energy of the particle to be the sum of the Kinetic Energy and the Potential Energy.

Why does STR not include the potential energy in the total energy of the particle?

 

[mfb]

Potentials in classical mechanics and non-relativistic quantum mechanics are assumed to be instantaneous - you have a potential energy that is simply the function of the position of two particles that attract or repel each other. With relativity this is no longer true, which makes the whole concept of potentials more problematic. You can't just consider an electric potential any more for example - you need to consider electromagnetism. If gravity becomes relevant you need general relativity. And so on. You still have potentials, but the simple picture of a particle having a potential energy rarely works when relativistic effects are relevant.

 

[vanhees71]

As explained by @mfb in general the idea of instantaneous interactions at a distance are not a useful concept in relativistic physics. That's why the natural formulation of relativity are fields not point particles.

The most simple example, where this picture can be used is only a charged particle moving in an external electromagnetic field as long as the radiation reaction (i.e., the interaction of the charged particle with its own electromagnetic field, including radiation when it is accelerated) can be neglected. Then you can formulate everything with a lorentz-invariant action
$$S=\int \mathrm{d} \lambda [-m c^2 \sqrt{\dot{x}^{\mu} \dot{x}_{\mu}} -q \dot{x}^{\mu} A_{\mu}(x)],$$
where $x^{\mu}(\lambda)$ is the world line of the particle, parametrized with an arbitrary scalar parameter $\lambda$ and $A_{\mu}$ the four-potential of the electromagnetic field.

In the case, where you have only an electrostatic field, you can choose the gauge, wehre $A_0(x)=\phi(\vec{x})$, $\vec{A}=0$, where $\phi(\vec{x})$ is the electrostatic potential.
Then you have the case which is closest to the non-relativistic motion in an external potential.

 

[Mister T]

Summary:: Why does STR leave out Potential Energy while Classical and Quantum Mechanics don't?

The total energy of a relativistic particle in STR contains the Rest Mass energy and the Kinetic energy.

Let's take a simple example of two interacting particles. They might be Earth and a baseball. When we increase the potential energy of the system by changing the height of the baseball, we increase the rest energy (and equivalently the mass) of the system. The rest energy of the system equals the rest energy of Earth plus the rest energy of the baseball plus the potential energy of the system. In other words, the potential energy makes a contribution to the rest energy (mass) of the system.

So the short answer is that the potential energy is part of the rest energy, so that's why you don't see a separate term for the potential energy in the expression for the total energy of a system of interacting particles. And of course, if your system consists of only one particle, there is no potential energy to consider.

(Note: when I use the term mass I'm talking about what some people call the rest mass. It's the only kind of mass needed in special relativity.)

 

[Nugatory]

But, in Classical and Quantum Mechanics, we consider the total energy of the particle to be the sum of the Kinetic Energy and the Potential Energy.

Even in classical mechanics, associating potential energy with a single particle (as opposed to the entire system including whatever is applying force to the particle) is a dubious idea. We can get away with it when one part of the system is enormously larger than the other, for example when considering a small object on the Earth's surface, but it's not quite right and will lead to confusion in more general problems.

To see the difficulties in counting the potential energy towards the total energy of a particle, consider a system of two masses connected by a massless spring. We pull the masses apart, increasing the potential energy of the system - but there is no sensible way of describing this as an increase in the total energy of one or the other masses considered in isolation.

 

[formodular]

One can add potentials in SR: to add scalar potentials one considers the action $S = - mc \int ds$ we and on adding a scalar potential $-\int V(x^{\mu}(\tau)) d \tau$, with $\tau$ proper time, this leads to $S = - mc \int ds - \int V \frac{d s}{c} = - c \int (m + \frac{V}{c^2}) ds$. The result is that scalar potentials increase the mass (i.e. energy) of a relativistic particle, and of course adding a vector potential instead of a scalar potential in the form $-\frac{e}{c} \int A_{\mu} dx^{\mu}$ gives electromagnetism.