C Speed, Gravitational Time Dilation & 4-Velocity

[sawer]

Hi

1-)If an object's total velocity through space-time(four-velocity)is c, for example even we stand still we move with velocity c (through time) and if mass slows down time, can we say mass also increase our velocity in space?

2-) Is Four-velocity magnitude constant in General Relativity? Does gravitational time dilation increase velocity in space dimension while slows speed of time flow?

 

[mfb]

You are mixing statements that apply to different reference frames here.

Mass influences spacetime around it. In some aspects this can be interpreted as slowing down time as seen from observers far away, but also slowing down the speed of light because everything happens slower.

can we say mass also increase our velocity in space?

No.

Is Four-velocity magnitude constant in General Relativity?

Locally space always looks like it does in special relativity. If you consider a process locally nothing changes. If you ask "what does X think about Y" then things start to depend on your favorite reference frame.

 

[Dale]

Hi

1-)If an object's total velocity through space-time(four-velocity)is c, for example even we stand still we move with velocity c (through time) and if mass slows down time, can we say mass also increase our velocity in space?

2-) Is Four-velocity magnitude constant in General Relativity? Does gravitational time dilation increase velocity in space dimension while slows speed of time flow?

The constant magnitude of the four velocity is not a particularly meaningful thing. I know that a lot of pop-sci sources present it as some sort of big “wow” thing, but it really isn’t.

Geometrically, the four velocity is just the unit tangent vector to your worldline. Unit vectors, by definition, have unit length. And in relativity the unit of speed is c. So the magnitude of the four velocity is c just because it is a unit vector.

For your specific question, no, mass does not change the unit vector. And no, it is more complicated than that. The determination of what is a unit vector depends on the metric, so a gravitational field can influence the components of the four-velocity but it can be more complicated