# Interpretation of conserved covariant tangent vector norm in the presence of an EM field

The motion of a point particle in curved spacetime can be obtained by extremising

$$S = \int L d\lambda= \int \left( \frac{m}{2}g_{\mu\nu}(x) \dot{x}^\mu \dot{x}^\nu \right) d\lambda,\tag{1}$$ where $$\dot{x}^\mu =dx^\mu/d\lambda$$ and $$\lambda$$ is an (affine?) parameter for the trajectory. Using the e.o.m one can show that $$L$$ itself is a constant of the motion (I think corresponding to the translational symmetry in $$\lambda$$):

$$\frac{1}{2} g_{\alpha \beta,\mu}\dot{x}^\alpha \dot{x}^\beta -\frac{d}{d\lambda}(g_{\mu\alpha}\dot{x}^\alpha) = 0.\tag{2}$$ contracting with $$\dot{x}^\mu$$ and rearranging one has: $$\implies \frac{d}{d\lambda}(\dot{x}^\mu \dot{x}_\mu )=0.\tag{3}$$ Thus, the tangent vector norm is constant along the (geodesic) curve. We then say $$\dot{x}^\mu \dot{x}_\mu = \mp 1,0\tag{4}$$ depending on whether it is timelike, spacelike or null.

It is my understanding that in the presence of an EM field $$A_\mu$$ the Lagrangian becomes:

$$L = \int \left( \frac{m}{2}g_{\mu\nu}(x) \dot{x}^\mu \dot{x}^\nu + q A_\mu\dot{x}^\mu \right) d\lambda,\tag{5}$$

The trajectory should now deviate from the geodesic curve due to a force. Finding the equations of motion

$$\frac{m}{2} g_{\alpha \beta,\mu}\dot{x}^\alpha \dot{x}^\beta +q A_{\alpha,\mu} \dot{x}^\alpha -\frac{d}{d\lambda}(mg_{\mu\alpha}\dot{x}^\alpha + qA_\mu) = 0,\tag{6}$$

contracting again with $$\dot{x}^\mu$$ the terms with $$A$$ cancel and I again find that the norm of the tangent vector is conserved as before $$\dot{x}^\mu \dot{x}_\mu = c.\tag{7}$$

How do I interpret $$c$$ in this case? Can I still use it to distinguish massless from massive worldlines?

I think you are here "rediscovering" the fact that worldlines in GR always have the same size of 4-velocity, i.e. $$-c^2$$ for timelike worldlines, as long as we take 4-velocity to be the tangent vector using proper time as affine parameter, and zero for null lines. So yes, the constant you called $$c$$ (an unfortunate choice of letter here by the way!) does distinguish between massless and massive worldlines.
• This must only hold for certain types of 'forces' though? If instead of the extra EM term $q A_\mu \dot{x}^\mu$ I had added a 'potential-like' term $V(x)$ then the tangent vector would not have been constant. On the other hand, if we define (usually called proper time/ distance) $d \tau^2 = g_{\mu\nu}dx^\mu dx^\nu$, then it would seem to me that by definition any path, as long as it was never null, would have tangent vector $dx^\mu/d\tau$ with constant unit norm everywhere. Aug 3 '19 at 17:48
• Perhaps we should start with a reparametrisation invariant $L$ really (The ones I wrote down are not). Because otherwise there's no guarantee that $\tau$ can always be identified with the parameter $\lambda$ in my question... Aug 3 '19 at 18:02