Second Differentials

Problem: You are given the velocity distribution of a collection of particles \(f(v)\), which is a Gaussian. Find the distribution of acceleration of those particles \(g(a)\). 

We know by Maxwell's reasoning that \(g(a)\) must be a Gaussian as well. 

Here comes the observation: Although we define second-order derivative from the first-order derivative, we seem not able to derive a function of second-order derivative from a function of the first-order derivative. 

$$v=|\textbf{v}|=\frac{\Delta s}{\Delta t}$$

$$\Delta v^2=\Delta v_x^2 + \Delta v_y^2 + \Delta v_z^2$$

$$\Delta v=\Delta(\frac{\Delta s}{\Delta t})=\frac{\Delta^2 s}{\Delta t}$$

$$(\Delta ^2 s)^2= (\Delta^2 x)^2 + (\Delta^2 y)^2+ (\Delta^2 z)^2$$

Metric of the Tangent Manifold

$$du^2=r_{\mu\nu}dv^\mu dv^\nu$$

$$(d^2 s)^2=r_{\mu\nu} d^2 x^\mu  d^2 x^\nu$$

$$\frac{d^2 v^\mu}{d\tau^2} + \Sigma^\mu_{\rho\sigma}\dot{v}^\rho \dot{v}^\sigma=0$$

Meaning that the tangent space itself is Riemannian, i.e. the spacetime is locally Riemannian. 

Soyjak: Lovelock, page 117