# Relative velocity and proper time derivative of geodesic deviation?

From wiki

To quantify geodesic deviation, one begins by setting up a family of closely spaced geodesics indexed by a continuous variable s and parametrized by an affine parameter $$\tau$$. That is, for each fixed $$s$$, the curve swept out by $$\gamma _s(\tau)$$ as τ varies is a geodesic. When considering the geodesic of a massive object, it is often convenient to choose $$\tau$$ to be the object's proper time. If $$x^μ(s, τ)$$ are the coordinates of the geodesic $$\gamma_s(\tau)$$, then the tangent vector of this geodesic is:

$$T^\mu = \frac{\partial x^\mu(s,\tau)}{\partial \tau}$$

If $$\tau$$ is the proper time, then $$T_\mu$$ is the four-velocity of the object traveling along the geodesic. One can also define a deviation vector, which is the displacement of two objects traveling along two infinitesimally separated geodesics:

$$X^\mu = \frac{\partial x^\mu(s,\tau)}{\partial s}$$

Now, naively

$$v^\mu = T^\beta \nabla_\beta X^\mu$$

Should $$v^\mu$$ be the relative velocity? If so, how does that fit in with this definition of relative velocity? (when their geodesics intersect)

$$v:=-\frac{1}{g(u^′,u)} u^′ - u \tag{1}$$

I feel I've managed to confuse myself.

• Found a relevant lecture: youtube.com/… Oct 3, 2022 at 13:18

Indeed, you seem to have confused yourself :-)

I edited several of your formulas. Check them out.

The equation

$$v^\mu = T^\beta \nabla_\beta X^\mu$$

measures the rate of separation between 2 neighboring geodesics, so it measures their relative 4-velocity. The quoted question and the paper mentioned therein take the matter to a deeper level, however... you have to realize that the "relative ordinary velocity" of an object passing close to you, regardless if it is moving on a geodesic or not, is simply the spacelike component of its 4-velocity (corrected by a $$\gamma$$ factor). This assumes that you are using a LIC (locally inertial coordinates system). You are gently guided to this formula in MTW Gravitation Ex. 2.5 page 65 and it coincides with your eq. 1.

So eq. 1 is simply the gamma corrected projection of the equation $$v^\mu = T^\beta \nabla_\beta X^\mu$$

You can better see their relation, by observing that - since $$X$$ and $$T$$ vectors commute (they form a 2D coordinate system) - you have that $$v^\mu = X^\beta \nabla_\beta T^\mu$$ which represent the difference in 4-velocity on neighboring geodesics, similar to eq. 1. You can read about this in MTW Box 11.4, for example.

• Unlike in equation (1) where u and u' are specified, the quantity $v^{\mu}$ here represents relative velocity b/w which two neighboring tangent vectors? It seems that $v^{\mu}$ only provides the rate of change and not exactly the difference
– KP99
Oct 3, 2022 at 11:36
• Indeed eq 1 shows the difference between 2 generic world lines, while $v^\mu$ shows the limit of (the difference between infinitesimally close world lines divided by $\Delta s$), so it is a ratio, but the idea is the same. Anyway, when I have more time I will improve the answer Oct 3, 2022 at 12:23