- #1

- 694

- 114

## Main Question or Discussion Point

I attempted to find a formula relating the velocity of an object released from infinity moving towards a mass ##M##to the distance ##r## from ##M##. According to Newton's law of gravity, ##a=\frac{GM}{r^2}##. Since ##a=v \frac{dv}{dr}## , ## \int a dr = \int v dv = \frac{v^2}{2} + k##. Since initial velocity equaled 0, k=0 . So ##v=\sqrt{2 \int a dr}= \sqrt{\frac{2GM}{r}}## (ignoring the -ve in the integral).

I've checked my maths but I'm still not sure if this formula is correct, because I immediately noticed that the formula appears to violate the second postulate of the special theory of relativity in a particular case. If ##M## is considered to be spherical and have a radius of ##x## where ##x<\frac{2GM}{c^2}##(by substituting v with c) with most of it's mass concentrated at the center, then any object moving towards this mass from infinity will exceed the speed of light when ##r## becomes less than ##\frac{2GM}{c^2}##(i.e. when the object moving towards ##M## enters the region between ##x## and ##\frac{2GM}{c^2}##). This also means that the escape velocity for a distance less than ##\frac{2GM}{c^2}## from ##M## will be unachievable, which doesn't make any sense. Where have I gone wrong?

I've checked my maths but I'm still not sure if this formula is correct, because I immediately noticed that the formula appears to violate the second postulate of the special theory of relativity in a particular case. If ##M## is considered to be spherical and have a radius of ##x## where ##x<\frac{2GM}{c^2}##(by substituting v with c) with most of it's mass concentrated at the center, then any object moving towards this mass from infinity will exceed the speed of light when ##r## becomes less than ##\frac{2GM}{c^2}##(i.e. when the object moving towards ##M## enters the region between ##x## and ##\frac{2GM}{c^2}##). This also means that the escape velocity for a distance less than ##\frac{2GM}{c^2}## from ##M## will be unachievable, which doesn't make any sense. Where have I gone wrong?