# What would a rocket do in curved spacetime? and Aspects of Geometry in Propulsion pdf and LaTeX

AspGeoPropul_exec

I should just put this out there. I need further (negative) feedback on how to improve this or what avenues I should take in investigating this. Having someone else take up some of the load would help too.

One of my ultimate goals/dreams would be to send a rocket into a black hole (or wormhole if there was one or if it was constructed). Yes, just like in the movie Interstellar (2014), but with rockets.

I wanted a manifestly covariant expression for the equations of motion (dynamics) of a rocket (including so-called “thrust”) in curved space-time. What would a (chemical) rocket do in curved space-time?

Also, I have been (trying) to read a lot engineering texts on rocket propulsion on my own because I really, really, really want to work at SpaceX. (P.S. My application is still in processing and I tried calling SpaceX several times but I get an answering machine for the operator and the university recruiters for my alumni school listed all have been gone from SpaceX and there hasn’t been an update on who to talk to. If you are someone working directly at SpaceX, please help!).

Again, I really, really, really want to work at SpaceX. I started off with the rocket propulsion book Jim Cattrell suggested to Musk and see the references attached to the pdf link below. I also have driven in and around Rocket Road in Hawthorne and I saw a guy on a forklift take a 5 meter (in diameter) aluminum hemisphere into the warehouse. We actually still manufacture things in the U.S.! And I’m cognizant of my academic degrees, but I wanted to have the job of that forklift driver or the welder that walked by: I just want to be doing something at SpaceX.

Here’s what I have for the equation of motion of a rocket on a spacetime manifold:

$\begin{gathered} M_0 \mathfrak{a} + \int_B \rho \frac{ \partial u^i}{ \partial t} \text{vol}^n \otimes e_i + \int_B \left( u^i \dot{dm}\otimes e_i + \rho u^j \frac{ \partial u^i}{ \partial x^j} \text{vol}^n \otimes e_i \right) + \int_B \rho u^i \text{vol}^n \otimes \left( \frac{ \partial e_i}{ \partial t } + [\mathbf{u}, e_i ] \right) = \\ = \sum \mathbf{F}_{\text{ext}} = \int_{ \partial (R+B) } *\sigma + \mathbf{F}_{\text{grav}} \end{gathered}$
or using Stoke’s law to get terms involving the boundary $\partial B$ of a n-dim. compact submanifold $\partial B$ of $n+1$-dim. spacetime manifold $M$
$\begin{gathered} M_0 \mathfrak{a} + \int_B \frac{ \partial u^i}{ \partial t} \rho \text{vol}^n \otimes e_i + \int_B \frac{ \partial \rho }{ \partial t} \text{vol}^n \otimes \mathbf{u} + \int_{ \partial B} \rho u^i i_{\mathbf{u}} \text{vol}^n \otimes e_i + \int_B \rho u^i \text{vol}^n \otimes \left( \frac{ \partial e_i}{ \partial t} + [\mathbf{u}, e_i] \right) = \\ = \sum \mathbf{F}_{\text{ext}} = \int_{ \partial (R+B) } *\sigma + \mathbf{F}_{\text{grav}} \end{gathered}$

Explanations on how I got here are in the pdf and LaTeX file. Please tell me what’s wrong with it. Also, if you know of a colleague or professor that’s good with differential geometry and fiber bundles, please let me and him or her know.

AspGeoPropul_exec.tex Aspects of Geometry in Propulsion LaTeX file on Google Drive

AspGeoPropul_exec.pdf Aspects of Geometry in Propulsion pdf file on Google Drive

By the way, for the fluid mechanics part of it, it was neat to read up about relativistic hydrodynamics which appears to be active area of academic research, but I’m not sure if anyone thought about its application in a rocket nozzle in curved spacetime!