# How does General Relativity explain escape velocities?

In general relativity, objects follow the shortest possible path through curved space-time called a geodesic and that there exists no such force of gravity which pulls objects, it is just because their timelines in the warped space-time fabric are towards the object causing the curvature. If this is so, then how are objects able to escape a body's gravitational pull by reaching a certain velocity what we call the "Escape Velocity", do they change their geodesic trajectories through space-time?

• You might find this question relevant and comparable to yours. – ohneVal May 2 at 9:55

Yes. Remember that geoodesics are space-time paths: the velocity is embedded in how they are slanted relative to the time direction.

So if you look at some point in spacetime, there are geodesics leaving it in all future time-like directions that corresponds to paths a particle emitted in that direction and velocity would take. Some of them may escape the vicinity of a heavy object and go off to infinity. The slowest such geodesic defines the escape velocity.

(Geodesics are often described as the shortest path between a starting and ending point, but you can also define them by a starting point and initial direction and then continue them as long as it is feasible using the geodesic equation.)

I read in your profile that you hate mathematics but love it in physics. It's rather hard for me to understand it, but I don't want to insist on this point. Just some ideas deriving from what I've seen in my long years of practice with theoretical physics.

Just to give you an example. When you talk of geodesics, that's a genuine mathematical concept. It's wonderful that it found so a beautiful application in GR. Maybe you don't know that initially Einstein was reluctant to approach his physical problem (to reconcile gravitation with relativity) using such abstract instruments. So I can hope that continuing your studies you'll change your mind. It's simply impossible to deepen one's knowledge of the topics you like without a serious knowledge of mathematics.

Incidentally you wrote

objects follow the shortest possible path

but the contrary is true. In spacetime a geodesic representing a body's path isn't the shortest path between two assigned points, but the longest. Anders Sandberg very aptly remarked that there is another approach to the concept of a geodesic. The one based on its (maximal or minimal) length among all paths connecting two given points is one, usually referred to as variational approach.

The other, more useful for your question, characterizes geodesics as special curves among all those starting from a given point with a given direction. It's difficult to give you a more precise idea (without using the relevant mathematics). I'll try the following.

In a flat space or spacetime there are straight lines, with the properties you know. In a curved space (or spacetime) straight lines don't exist (geometry isn't euclidean). We can try what in a sense is the best possible approximation to a straight line, i.e. a curve that saves some of the properties of the straight line. That it were possible was discovered few years before Einstein confronted himself with GR problem. Unfortunately I'm afraid it's impossible to go deeper. I limit myself to give you two magic words: "parallel transport".

• Thanks for taking the efforts to explain it intuitively without using mathematics and ah! I wrote my profile 2 years ago when I was a high school teen :D.....my thoughts still havent changed though but now I am starting to fall in love with the mathematics Einstein used in his EFEs, currently I am studying Science in a junior college and simultaneously trying to understand the Differential Geometry and Tensor Calculus behind the mathematics of GR – Ajinkya Naik May 3 at 17:43
• The doubt I encountered was due to an Instagram post by a physics page that said "Objects fall to the ground because their future is on the ground as per Einstein's GR" .I interpreted it thinking that objects move towards a massive body through geodesics which I thought of as a rail track and that each object is confined to only one railtrack or geodesic and is obliged to move through this spacetime wherever it goes or curves, so this sort of created a confusion regarding escape velocities, which gave the impression of objects derailing or leaving their geodesic. – Ajinkya Naik May 3 at 17:48