General relativity and black holes

  According to the general relativity theory of Einstein, near a large mass, time happens slowly due to the gravitational action.

  Einstein deduced (as we read in his book “The Meaning of Relativity”) the following formula

(7)

  where x = 8 p G / c²

  t ‘= time at a distance r from the gravity center of the mass (a star) producing the gravitational field

  t = objective time  (time in the far reaches of the gravitational field)

s = density of the star

  V = Volume of the star

  r = distance from the center of the star to the point in space we are analyzing.

  Then substituting x by its value is obtained

(8)

  and as  is the mass of the star M divided by the radius r, is obtained

(9)

(Equation that is often currently deducted from the Schwarzschild metric for general relativity)

  and as according to equation (3) 2GM/r = ve2 , where ve is the classical exhaust speed at the distance r from the center of the star, we obtain

(10)   (You can do another derivation of this formula, more didactic, through the principle of equivalence )

  From this, it follows that as a body approaches an astro, time passes more slowly for this body, according to the escape velocity of the star (from a classical point of view), so that when it reaches a distance such that the speed of classical exhaust is equal to the speed of light, the time stops for the object at that location. That is for r = 2GM/c2 that is called  Schwarchild radius . We can see that if in this expression we clear M/r the same relationship we obtained through classical physics and speed limit of light in section special relativity and black holes is obtined. Hence the values ​​of the table of that paragraph are valid, as the equation for the radius of Schwarchild coincidentally is the same calculated using general relativity or by classical mechanics and the speed limit of light.

  Then a spherical surface around the black hole in which time stops, appears. This spherical surface is called the event horizon of the black hole.

  Passing through this horizon, time with imaginary components exist (the calculation of time spent inside the event horizon leads to a square root of a negative number), which leads us to believe that perhaps the time happens inside a black hole may be in a fifth dimension perpendicular to both the three spatial and temporal normal dimension.

  Furthermore, the general theory of relativity tells us that space is curved around a mass in a way that a light beam that passed near that mass deviate twice than it would if it were affected by gravity from a  classical view (like a particle). So Einstein obtained by performing some approaches that the deviation was:

(11)

which it gives us an angle of 1.75 degree seconds for a ray of light passing near the sun. This was verified by observing eclipses.

It is also obtained that the light emitted by a star must have a spectrum slightly redshifted , meaning that the light emitted will be with less frequency than usual because all electrons vibrate slowly due to the partially stopping of time, obtaining the formula :

We can see that if the radius was 2GM/c2 (radius of the event horizon) the frequency would be zero and therefore we would not see the light from the star, another reason for something to be called “black hole”.

It is estimated that in this radius, space curvature will be such that light would be trapped in the hole. In this way as we approach the event horizon the usual three spatial coordinates are curved so that any movement inside the hole would occur towards the center of it.

  Thus everything that surpass the event horizon can never leave.

One thought on “General relativity and black holes

Leave a Reply

Your email address will not be published. Required fields are marked *