What is Earth's escape velocity?

Since space is curved within the Earth's gravitational field, every body that moves there will follow the curvature of space no matter what speed it has, so what will its trajectory be, how will it be straight, only if the launch is made absolutely vertically towards sea level? Only then can it escape? In that case, since gravity is not a force, no escape velocity is required?

Escape velocity from Earth is about 11 km/s in any direction that doesn't go straight in to the ground. That is the same in GR as in Newtonian gravity.

binis said:

Since space is curved within the Earth's gravitational field

It is not space that is curved, it is spacetime that is curved. You are neglecting the time part of spacetime and drawing the wrong conclusions.

Two objects that are traveling in the same direction in space, but at different speeds, are going in different directions in spacetime.

binis said:

every body that moves there will follow the curvature of space no matter what speed it has

No. Different speeds are different directions in spacetime and the curvature of spacetime can be different in different directions. This is essentially why the Newtonian gravitational deflection for light is off by a factor of 2 from the relativistic deflection.

binis said:

only if the launch is made absolutely vertically towards sea level? Only then can it escape?

No, that is not necessary. The spatial direction is only relevant to avoid colliding with the earth.

Ibix said:

Escape velocity from Earth is about 11 km/s in any direction that doesn't go straight in to the ground. That is the same in GR as in Newtonian gravity.

How is it calculated by GR? Please provide a link.

Dale said:

No, that is not necessary. The spatial direction is only relevant to avoid colliding with the earth.

What about the last query? " In that case, since gravity is not a force, no escape velocity is required?"

binis said:

How is it calculated by GR? Please provide a link.

Chapter 7 of Carroll's online lecture notes on GR. From memory, equations 7.43, 7.44, 7.47 and 7.48 are the relevant ones; you'll need to set ##\epsilon=1## for a timelike particle, ##E=1## for a "stop at infinity" case, and calculate the magnitude of the three velocity measured by an observer hovering at ##r=R_E##.

In fact you end up only needing 7.43 and the Schwarzschild metric in Schwarzschild coordinates, which is how it's easy to see it's independent of ##L##.

binis said:

What about the last query? " In that case, since gravity is not a force, no escape velocity is required?"

It makes no sense. Assuming there's a way to define "arriving at infinity" you can always ask what velocity achieves it. In the Schwarzschild case (in my last post) it's even done in the background by a conservation of energy argument.

binis said:

What about the last query? " In that case, since gravity is not a force, no escape velocity is required?"

It is a non sequitur, which is a form of logical fallacy. Gravity not being a force in no way implies that there is no escape velocity required.

binis said:

Since space is curved within the Earth's gravitational field, every body that moves there will follow the curvature of space no matter what speed it has, so what will its trajectory be, how will it be straight, only if the launch is made absolutely vertically towards sea level? Only then can it escape? In that case, since gravity is not a force, no escape velocity is required?

You have to consider space-time, not just space.

On the diagram below the vertical axis is radial-space (black), while the circumferential axis is time (blue). The red & green lines are worldlines of two objects thrown vertically upwards at different speeds. Those worldlines of free falling object are geodesics (locally straight) in space-time, as modeled by the little car driving on the curved surface without steering. The green object has a higher initial upwards speed, and thus a larger initial angle to the time axis.

In this example both objects come down. But it's easy to see that if you increase the angle (intial upwards speed) even more, the car will never return the blue baseline, just keep spiraling up along that funnel, and escape to infinity.

From: https://www.relativitet.se/spacetime1.html