# Free Fall Acceleration to c

#### Goalie33

This question has bothered me for few weeks:

Lets say I was at some height above a super massive planet that has no atmosphere (i.e. no air resistance. If I were to begin falling what would stop me from being accelerated by the gravitational field of this planet to the speed of light? Assume that the mass of the planet and my height above the surface are enough to get me past c (if only the classical calculation is done.)

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#### sylas

This question has bothered me for few weeks:

Lets say I was at some height above a super massive planet that has no atmosphere (i.e. no air resistance. If I were to begin falling what would stop me from being accelerated by the gravitational field of this planet to the speed of light? Assume that the mass of the planet and my height above the surface are enough to get me past c (if only the classical calculation is done.)
The classical calculation is incorrect for that scale. You need physics that can deal with velocities close to c, and that requires relativity.

One way to think about it is this. You never get past c because you never get to infinite kinetic energy. No matter how far you fall, there's a finite amount of potential energy involved, according to any stationary observer. (The velocity is going to depend on who is measuring it and with what co-ordinates.)

Cheers -- sylas

#### kanato

F is not equal to ma once you reach relativistic speeds. As you get close to the speed of light your acceleration will decrease, even if the gravitational field does not change. IIRC the result is $$F = \gamma^3ma$$ [1], where $$\gamma$$ is the usual Lorentz factor, which gets large as you approach the speed of light, causing the acceleration to get small, so you never actually reach the speed of light.

[1] This is for the force, acceleration and velocity all in the same direction. If you want to work with vectors, then the relation becomes more complex, and depends on the angle between the force and the velocity.

#### Goalie33

Thanks to both of you,

I understood that it would involve Special Relativity. (I knew classical mechanics would break down, I know now that the comment towards the end was misleading.) I just didn't know how to incorporate SR.

I think Ive got it now, thanks again!

#### granpa

F is not equal to ma once you reach relativistic speeds.
what if you use proper acceleration instead?

#### Nickelodeon

F is not equal to ma once you reach relativistic speeds. As you get close to the speed of light your acceleration will decrease, even if the gravitational field does not change. IIRC the result is $$F = \gamma^3ma$$ [1], where $$\gamma$$ is the usual Lorentz factor, which gets large as you approach the speed of light, causing the acceleration to get small, so you never actually reach the speed of light.

[1] This is for the force, acceleration and velocity all in the same direction. If you want to work with vectors, then the relation becomes more complex, and depends on the angle between the force and the velocity.
There are two realities here, one from an observers point of view as described above and one from the person in freefall who is quite unaware of any force acting on him. In his freefalling environment he is quite at liberty to achieve a speed of separation faster than c (relative to his original frame/starting point). For instance, if he was falling towards a black hole, say from near infinity, I would imagine that he would exceed c once past the event horizon.

#### JesseM

There are two realities here, one from an observers point of view as described above and one from the person in freefall who is quite unaware of any force acting on him. In his freefalling environment he is quite at liberty to achieve a speed of separation faster than c (relative to his original frame/starting point). For instance, if he was falling towards a black hole, say from near infinity, I would imagine that he would exceed c once past the event horizon.
The speed of light limit is only meant to apply in inertial coordinate systems, even in flat spacetime with no gravity it's quite possible for objects to have a coordinate velocity greater than c if you pick the right non-inertial coordinate system. In GR, no global coordinate system in curved spacetime can ever qualify as inertial, but because of the http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html [Broken] an observer in freefall who is only paying attention to a small region of spacetime around him can have a "locally inertial" frame in that region where the laws of physics should look the same as in SR (assuming the region is chosen small enough that tidal forces are undetectable), and any objects passing by him in that region will be moving at less than c (or exactly c in the case of photons or other massless particles) in his locally inertial frame.

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#### Mentz114

Gold Member
A general relativistic calculation shows that a body falling from rest at infinity towards a Schwarzschild source, reaches c wrt to the source exactly on the event horizon (!).

See ( for instance ) arXiv:gr-qc/0411060.

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