I Time-Dependent Lagrangian Leads to Time Dilation?

[stevendaryl]

This is just something unexpected that I noticed recently, and I hadn't heard anyone mention it before.

The relativistic Lagrangian for a particle moving under a scalar potential $\Phi$ is this:

$L = \frac{1}{2} m g_{\mu \nu} \dfrac{dx^\mu}{d\tau} \dfrac{dx^\nu}{d\tau} - \Phi$

This leads to the equations of motion:

$m \dfrac{d^2 x^\mu}{d\tau^2} = - \partial^\mu \Phi$

So that's just the relativist generalization of Newton's $F = m A$, with $F = -\nabla \Phi$. However, a difference is that it's a 4-D equation, rather than a 3-D equation. So let's look at just the 0th component, with $x^0 = t$:

$m \dfrac{d^2 t}{d\tau^2} = - \dfrac{\partial \Phi}{\partial t}$

This is truly unexpected (to me). When there is no potential, $\dfrac{dt}{d\tau}$ is the time dilation factor $\gamma$. The above equation seems to be saying that time dilation depends not only on velocity (or spacetime curvature, if you consider General Relativity, which I'm not doing here) but also on the potential. So even a particle at rest will experience time dilation if it is in a time-varying potential.

Another thing that is surprising is that this time dilation can be positive or negative. So if a particle starts out at rest, with $\dfrac{dt}{d\tau} = 1$, then a negative value for $- \dfrac{\partial \Phi}{\partial t}$ will lead to the particle having $\dfrac{dt}{d\tau} \gt 1$. So time runs faster for the particle, rather than slower.

Is this a real effect? My guess is that it wouldn't be easy to test because there are so few scalar fields (the only one I know of is the Higgs field), and they are not as easily manipulated as the electromagnetic field.

 

[Demystifier]

I have studied this stuff in more detail in
https://arxiv.org/abs/1006.1986
and published as a part of a book chapter
https://arxiv.org/abs/1205.1992

The scalar potential can be viewed as a dynamical mass squared, which can become negative so that particle can exceed the velocity of light.

But note that your parameter $\tau$, called $s$ in my work, is not the usual proper time. Hence, the potential does not modify the time dilation. It only modifies the relation between proper time and this parameter.