Decelerating at 1g into a black hole

[DiamondGeezer]

In Kip Thorne's book "Timewarps" he describes a hypothetical spaceship decelerating almost to the event horizon of a supermassive black hole, and says that the manoevre takes 13 years. I assume that the motion is along a geodesic.

In terms of the Schwarzschild metric, how would this motion be described and how would the elapsed time be calculated?

 

[Fredrik]

It's definitely not along a geodesic. That means free fall.

 

[tim_lou]

as for the proper elapsed time, just calculate the integral of the norm for the tangent vector along the path taken.

 

[DiamondGeezer]

How exactly?

The metric is $$d \tau ^2 = \left(1- \frac{2M}{r}\right)dt^2- \frac{dr^2}{\left(1- \frac{2m}{r}\right)} -r^2 d \theta^2$$

Assuming $$d \theta^2 = 0$$, the remainder can be divided through by $$d \tau ^2$$ to give

$$1 = \left(1- \frac{2M}{r}\right)\left(\frac{dt}{d\tau}\right)^2 - \frac{1}{\left(1- \frac{2m}{r}\right)}\left(\frac{dr}{d \tau}\right)^2$$

The fraction $$\left(\frac{dr}{d \tau}\right)$$ looks something like a velocity and not an acceleration. Is a second derivative is added to the equation somewhere?

What are the constants of the motion? Is total energy conserved?