Random General Relativity conceptual questions

[coopre]

So I want to know if these are true or false, if false why are they

A geodesic is a path between two points in spacetime that maximizes the invariant
distance ds2.

A massive particle's rest mass increases with velocity.

If I re a laser beam in the general direction of a black hole, there is no way I
will get hit by any of the light from my laser.

A 4-vector can be both perpendicular and parallel to itself.

For a massive particle with angular momentum L orbiting a point mass, the
corrections dues to general relativity always make the radius of a stable circular
orbit (assuming it exists) smaller than it would be in Newtonian gravity.

 

[Dale]

A geodesic is a path between two points in spacetime that maximizes the invariant distance ds2.

Technically it extremizes it. Whether the extremum is a maximum or a minimum depends on your sign convention (+--- vs -+++) and whether the geodesic is timelike or spacelike.

A massive particle's rest mass increases with velocity.

No, the rest mass is an invariant quantity and is the same in all frames regardless of the velocity in that frame.

If I re a laser beam in the general direction of a black hole, there is no way I will get hit by any of the light from my laser.

Depends where you and the laser and the black hole are positioned relative to each other.

A 4-vector can be both perpendicular and parallel to itself.

How do you figure this?

 

[christopherV]

For a massive particle with angular momentum L orbiting a point mass, the
corrections dues to general relativity always make the radius of a stable circular
orbit (assuming it exists) smaller than it would be in Newtonian gravity.

Relativity can't always predict the effects of angular momentum correctly. The underlying Riemannian geometry just does not handle spin-orbit coupling. Check out Einstein-Cartan Theory for an (hypothetical) extension of Relativity and Riemannian Geometry.

 

[yuiop]

If I re a laser beam in the general direction of a black hole, there is no way I
will get hit by any of the light from my laser.

It is possible for the laser beam to bend round the back of the black hole and come back at you. In extreme circumstances the photons can circumnavigate the black hole several times before coming back at you.

For a massive particle with angular momentum L orbiting a point mass, the
corrections dues to general relativity always make the radius of a stable circular
orbit (assuming it exists) smaller than it would be in Newtonian gravity.

If you can confirm that you are talking about a test particle orbiting a massive body such as a large black hole (and not an elementary particle such as an electron orbiting a nucleus) then I may be able to point you to the equations to resolve this question.