# Is there a general relativistic relativistic rocket equation?

I am not to familiar with rocket equations but I know that if you combine special relativity with classical mechanics you get "c" as a speed limit. Basically the closer you get to "c" the more you increase in relativistic momenta without increasing so much in velocity.

Now in general relativity there are two other effects that I can think of. We consider the case where the rocket is moving in a spherically symmetric gravitational field.

1. The speed of light is anisotropic and varies with altitude. In the Scharzschild solution in Schwarzschild coordinates in the radial direction in coordinate time you have $$v_{light}=c(1-\frac{2GM}{rc^2})$$ and in the direction transverse to the radial direction you have $$v_{light}=c\sqrt{1-\frac{2GM}{rc^2}}$$. The fact that the speed limit varies in this way should affect the rocket equation somehow.
2. The gravitational time dilation/gravitational red shift I think that in coordinate time the gravitationl acceleration for an object standing on the face of a planet such as the Earth is the same as classically, $$\frac{dv}{dt}=-\frac{GM}{r^2}$$. However in proper time you have $$\frac{dv}{dt}=-\frac{GM}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{rc^2}}}$$. I think this is correct. Equivalently the energy of the fuel in the rocket will be "redshifted" so the rockets will basically not contains as much energy as classically. This might somehow affect the rocket equation. This is sometimes a confusing subject.

Question: Is there a general relativistic, relativistic rocket equation and what does it look like?

I guess effects such as those mentioned might be to miniscule to bother with in the solar system but at least you should be able to prove that you can disregard them.

Edit: While not exactly a "rocket-equation" a common way to include effects of acceleration due to relativity is to use the post-Newtonian expansion. I wrote down the 3PN-accelerations for the most simple case of one test body in a spherical field on the physics site. At the 1PN level the acceleration becomes:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}+\frac{v^2}{c^2}\right)\hat{r} +\frac{4GM}{r^2}\left(\hat{r}\cdot \hat{v}\right)\frac{v^2}{c^2}\hat{v}$$

This is an expression that JPL always uses for orbit computation. When you do not have only one massive body of spherical symmetry there are more terms. For a rocket standing still, $$v=0$$, on the face of the earth you get:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}\right)\hat{r}$$

and for a rocket travelling radially upwards you get:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}-3\frac{v^2}{c^2}\right)\hat{r}$$

Question: How do these two extra terms make it harder/easier for a rocket to escape the gravitational field of the Earth?

• Yikes, great question :-) See the last paragraph in @DavidHammen's answer. Yikes again!
– uhoh
May 14 '19 at 0:40
• In which way is the relativistic rocket equation as given by wikipedia unsatisfactory? Your point (2) is explicitly mentioned in the derivation and I am not sure (1) matters. May 14 '19 at 16:08
• @Polygnome Can you point me to it? The first sentence in the article says "Relativistic rocket refers to any spacecraft that travels at a velocity close enough to light speed for relativistic effects to become significant." but the second effect should maybe be there even if you go very slow as compared to the speed of light. You might need more fuel in your rocket if you try to escape from a planet governed by GR and not by Classical Newton. If there is no gravitational field present or it can be neglected then I guess SR should be enough. May 14 '19 at 16:31
• If you rocket is traveling radially upward from a planet your velocity as a fraction of the speed of light can decrease even if you speed up, as measured in coordinate time. In "proper time" of the rocket things might be different. I do not know how much 1 matters or not, but if it do not matter, it would be great if someone could prove it. May 14 '19 at 16:42
• @Polygnome: The article presents a SRT rocket, OP asks for a GRT extension. May 18 '19 at 11:39