Parameter Question In General Relativity

Question

I am a math student taking a course in General Relativity. I haven't taken many physics/applied maths courses before, so I am not sure if I can describe this question well, but I am slightly confused by a kind of usage of parameters. We use $c\tau=s$, where $\tau$ is proper time. In most cases, we have derivatives with respect to $\tau$ or $s$ or $t$, which I can understand. However, a result states that $g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}$ is always a constant in GR ($g$ is the metric), and setting this constant to $1$ identifies $\lambda$ to $s=c\tau$, and setting the constant to $0$ will distinguish $\lambda$ from $s$. I am really confused by this $\lambda$ here: what is it exactly, and why do we use it?

Similar problems also appear in other places. For example when we were deriving the circular orbit equation $(\frac{d\phi}{dt})^2=\frac{1}{2}\frac{Rc^2}{r^3}$ in Schwarzchild metric, We used E-L equation $\frac{d}{d\lambda}(\frac{\partial L}{\partial\dot{r}})=\frac{\partial L}{\partial r}$, where L is a big chunk of formula (kinetic energy with unit mass, I believe). Although this $\lambda$ cancels in the end, I am generally really confused by this $\lambda$. My lecturer just said it is a "time" parameter not called "t" to avoid confusion with the time coordinate, which is just more confusing to me... Thank you in advance!

Answer 1

In general relativity $x^\mu(\lambda)$ is a curve in spacetime parametrized by $\lambda$. From mathematical point of view, the parametrization of the curve is part of its definition and cannot be changed without changing the curve. Not so from physical point of view. There, the parametrization of the curve is to a certain point given by units or description you use. The physics is the same, it is just the description that might differ. So when physicist does not care about the description but only about underlying physics, he just uses some generic parameter he calls $\lambda$. But at certain point he might want to pick one particular description/units and then he will substitute his choice for the generic parameter.

The property $g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}$ is not constant for just about any curve, but for affinely parametrized geodesic. So $\lambda$ is affine parameter of the geodesic. The natural choice of parametrization for timelike geodesic is proper time (which you multiply by $c$ to measure it in units of length) and you will get that in this case $g_{\mu\nu}\frac{dx^\mu}{ds}\frac{dx^\nu}{ds}=1.$ For spacelike geodesics there is no proper time defined, so you cannot pick proper time as affine parameter. But you can define proper length for them and this would be natural parametrization for spacelike curves, in which case you would get $g_{\mu\nu}\frac{dx^\mu}{dl}\frac{dx^\nu}{dl}=-1$. For null curve the scalar product of tangent vector with itself is 0 for whatever the (nicely behaved) parametrization and therefore $g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}=0$ no matter which affine parametrization you pick.

Of course you can choose the other attitude, when you demand $g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}=1$ and then you seek what kind of choice of parametrization this requirement leads to, as you did in your question.

In case of circular orbits it is basically the same. The orbit is again some curve with some parametrization. From mathematical point of view, curves with different parametrization are different. From physics point of view, orbit is just image set of the curve and parametrization of the curve is immaterial. So you work in some generic parametrization and pick one particular choice only once it is convenient. In this case the choice was made to parametrize the orbit by $t$ coordinate.