Why Is the Norm of the Tangent Vector Constant in Geodesic Equations?

[Nabigh R]

I am trying to derive the geodesic equation by extremising the integral
$$ \ell = \int d\tau $$
Now after applying Euler-Lagrange equation, I finally get the following:
$$ \ddot{x}^\tau + \Gamma^\tau_{\mu \nu} \dot{x}^\mu \dot{x}^\nu = \frac{1}{2} \dot{x}^\tau \frac{d}{ds} \ln \left| \dot{x}_\nu \dot{x}^\nu \right| $$
where $\dot{x}^\tau \equiv \frac{d x^\tau}{ds}$ and $s$ is a parameter. Now I get the geodesic equation if the right-hand side vanishes, and the only way that happens is if $\dot{x}_\nu \dot{x}^\nu$ is constant. Now the question is why is it constant?

 

[stevendaryl]

I am trying to derive the geodesic equation by extremising the integral
$$ \ell = \int d\tau $$
Now after applying Euler-Lagrange equation, I finally get the following:
$$ \ddot{x}^\tau + \Gamma^\tau_{\mu \nu} \dot{x}^\mu \dot{x}^\nu = \frac{1}{2} \dot{x}^\tau \frac{d}{ds} \ln \left| \dot{x}_\nu \dot{x}^\nu \right| $$
where $\dot{x}^\tau \equiv \frac{d x^\tau}{ds}$ and $s$ is a parameter. Now I get the geodesic equation if the right-hand side vanishes, and the only way that happens is if $\dot{x}_\nu \dot{x}^\nu$ is constant. Now the question is why is it constant?

Well, $s$ is an arbitrary parameter. We are free to choose any parametrization we like. One particular parametrization, which is possible for slower-than-light geodesics, is to let

$ds = \sqrt{|dx^\mu dx_\mu|}$

For this choice, $|\frac{dx^\mu}{ds} \frac{dx_\mu}{ds}| = 1$

 

[Bill_K]

$$ \ddot{x}^\tau + \Gamma^\tau_{\mu \nu} \dot{x}^\mu \dot{x}^\nu = \frac{1}{2} \dot{x}^\tau \frac{d}{ds} \ln \left| \dot{x}_\nu \dot{x}^\nu \right| $$
where $\dot{x}^\tau \equiv \frac{d x^\tau}{ds}$ and $s$ is a parameter. Now I get the geodesic equation if the right-hand side vanishes, and the only way that happens is if $\dot{x}_\nu \dot{x}^\nu$ is constant. Now the question is why is it constant?

The geodesic equation takes its usual form only when s is chosen to be an affine parameter. For a timelike geodesic this means s must be a linear function of proper time, s = aτ + b where a and b are constants. If s is not an affine parameter, the geodesic equation has the extra term you mentioned.

 

[Nabigh R]

Thanks Bill, Wikipedia too says a similar thing, but what exactly is an affine parameter?

 

[sardar]

Thank you for sharing your progress in deriving the geodesic equation. It is a fundamental equation in the study of curved spaces and plays a critical role in understanding the motion of objects in these spaces.

To answer your question about why $\dot{x}_\nu \dot{x}^\nu$ is constant, we need to look at the definition of this quantity. It is the squared norm of the tangent vector to the curve, which represents the velocity of the object. In other words, it measures the speed of the object along the curve. Since we are assuming that the parameter $s$ is proportional to time, this means that the speed of the object is constant along the curve. This is a natural assumption in classical mechanics, where objects tend to move at a constant speed unless acted upon by external forces.

Moreover, in the context of general relativity, we know that the curvature of spacetime is related to the energy and momentum of objects in that spacetime. If we consider an object with a constant speed, it means that its energy and momentum are also constant along the curve. This implies that the curvature of spacetime along the curve is also constant, which leads to the object following a geodesic, the path of least resistance in curved space.

In summary, the constancy of $\dot{x}_\nu \dot{x}^\nu$ is a direct consequence of the object's constant speed along the curve and the relationship between energy, momentum, and curvature in general relativity. I hope this helps clarify the significance of this quantity in the derivation of the geodesic equation. Keep up the good work in your studies!