- #1

tomdodd4598

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- TL;DR Summary
- How does one consistently solve the equations of motion for a massless particle in a non-constant potential?

The Lagrangian for a massless particle in a potential, using the ##(-,+,+,+)## metric signature, is

$$L = \frac{\dot{x}_\mu \dot{x}^\mu}{2e} - V,$$

where ##\dot{x}^\mu := \frac{dx^\mu}{d\lambda}## is the velocity, ##\lambda## is some worldline parameter, ##e## is the auxiliary einbein and ##V## is the potential.

The EL equations give us the EOMs

$$\dot{x}_\mu \dot{x}^\mu = 0,$$

$$\ddot{x}^\mu + \Gamma^\mu_{\sigma\rho} \dot{x}^\sigma \dot{x}^\rho - \frac{\dot{e}\dot{x}^\mu}{e} + e\partial^\mu V = 0,$$

where ##\Gamma^\mu_{\sigma\rho}## are the Christoffel symbols of the metric ##\eta_{\mu\nu}## for some choice of coordinates. After this, I'm not sure how to proceed, for the following reasons.

In the ##V=\text{constant}## case, the system is underdetermined, and we are free to choose some ##e##, such as setting ##e=1##. We then get the consistent EOMs

$$\dot{x}_\mu \dot{x}^\mu = 0,$$

$$\ddot{x}^\mu + \Gamma^\mu_{\sigma\rho} \dot{x}^\sigma \dot{x}^\rho = 0.$$

In the general case, however, we seem to end up with the two EOMs being inconsistent.

For example, choose Cartesian coordinates and an ##z##-direction potential such as ##V = z##, and choose initial conditions satisfying the null velocity condition such as ##\dot{x}^\mu \left(0\right) = \left(1,0,0,1\right)##. The EOM for the coordinates becomes

$$\ddot{x}^\mu = \left(0,0,0,-1\right),$$

yielding ##\dot{x}^\mu \left(\lambda\right) = \left(1,0,0,1-\lambda\right)##, which fails to satisfy the null velocity condition for ##\lambda \neq 0##.

We can solve for ##e## by multiplying the second EOM by ##\dot{x}_\mu## and plugging in the first EOM:

$$\dot{x}_\mu \left( \ddot{x}^\mu + \Gamma^\mu_{\sigma\rho} \dot{x}^\sigma \dot{x}^\rho \right) + e \dot{x}_\mu \partial^\mu V = 0,$$

$$e = -\frac{\dot{x}_\mu \left( \ddot{x}^\mu + \Gamma^\mu_{\sigma\rho} \dot{x}^\sigma \dot{x}^\rho \right)}{\dot{x}^\mu \partial_\mu V}.$$

Unfortunately, ##e## is then undefined for ##\dot{x}^\mu \partial_\mu V = 0##, and the system of ODEs is horrendous regardless (Mathematica flat out refuses to solve them), and so I find it hard to believe that this is the correct approach.

$$L = \frac{\dot{x}_\mu \dot{x}^\mu}{2e} - V,$$

where ##\dot{x}^\mu := \frac{dx^\mu}{d\lambda}## is the velocity, ##\lambda## is some worldline parameter, ##e## is the auxiliary einbein and ##V## is the potential.

The EL equations give us the EOMs

$$\dot{x}_\mu \dot{x}^\mu = 0,$$

$$\ddot{x}^\mu + \Gamma^\mu_{\sigma\rho} \dot{x}^\sigma \dot{x}^\rho - \frac{\dot{e}\dot{x}^\mu}{e} + e\partial^\mu V = 0,$$

where ##\Gamma^\mu_{\sigma\rho}## are the Christoffel symbols of the metric ##\eta_{\mu\nu}## for some choice of coordinates. After this, I'm not sure how to proceed, for the following reasons.

In the ##V=\text{constant}## case, the system is underdetermined, and we are free to choose some ##e##, such as setting ##e=1##. We then get the consistent EOMs

$$\dot{x}_\mu \dot{x}^\mu = 0,$$

$$\ddot{x}^\mu + \Gamma^\mu_{\sigma\rho} \dot{x}^\sigma \dot{x}^\rho = 0.$$

In the general case, however, we seem to end up with the two EOMs being inconsistent.

For example, choose Cartesian coordinates and an ##z##-direction potential such as ##V = z##, and choose initial conditions satisfying the null velocity condition such as ##\dot{x}^\mu \left(0\right) = \left(1,0,0,1\right)##. The EOM for the coordinates becomes

$$\ddot{x}^\mu = \left(0,0,0,-1\right),$$

yielding ##\dot{x}^\mu \left(\lambda\right) = \left(1,0,0,1-\lambda\right)##, which fails to satisfy the null velocity condition for ##\lambda \neq 0##.

*This section below was part of the original post, but Orodruin spotted an issue with the line of reasoning.*

We can solve for ##e## by multiplying the second EOM by ##\dot{x}_\mu## and plugging in the first EOM:

$$\dot{x}_\mu \left( \ddot{x}^\mu + \Gamma^\mu_{\sigma\rho} \dot{x}^\sigma \dot{x}^\rho \right) + e \dot{x}_\mu \partial^\mu V = 0,$$

$$e = -\frac{\dot{x}_\mu \left( \ddot{x}^\mu + \Gamma^\mu_{\sigma\rho} \dot{x}^\sigma \dot{x}^\rho \right)}{\dot{x}^\mu \partial_\mu V}.$$

Unfortunately, ##e## is then undefined for ##\dot{x}^\mu \partial_\mu V = 0##, and the system of ODEs is horrendous regardless (Mathematica flat out refuses to solve them), and so I find it hard to believe that this is the correct approach.

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