General RelativitySpecial Relativity is the study of behaviour in uniformly moving frames of reference. Now let us consider the situation for an accelerating frame of reference. We will now revisit Einstine's famous thought experiment. (Einstein, 1916) Therefore, by the principle of equivelance, because a gravitational field produces an acceleration, it can be regarded as an accelerating frame of reference. So the relationships within a gravitational field can be used to describe the behaviour of any accelerating frame of reference. If a particle, initially at rest at a distance r from a mass is allowed to fall toward a that mass there is a transfer of energy from potential to kinetic. The initial potential energy of the particle is given by: .....(18) Where G = gravitational constant The final kinetic energy of the particle at the end of its fall will be: ....(19) Where v = final velocity of the particle. If there is no other transfers of energy the initial potential energy must be equal to the final kinetic energy. Therefore: or .....(20) Spacial EffectsIf we consider equation (12) we see: ...(12 from Special Relativity) Substituting for v^{2} gives: .....(21) Therefore space is contracted within a gravitational field. Temporal EffectsSimilarly, if we consider equation (13) we see ...(13 from Special Relativity) Substituting for v^{2} gives: .....(22) So we see that time is dilated within a gravitational field. Frequency EffectsIf we consider equation (14) we see: ...(14 from Special Relativity) Substituting for v^{2} gives: .....(23) So we see that ligh within a gravitational field will also experience a Red Shift. Mass EffectsIf we consider equation (15) we see: ...(15 from Special Relativity) Substituting for v^{2} gives: .....(24) Thus, when the mass of an object is measered within a gravitational field it will seem greater than when measured in space free of gravitational fields. Light SpeedFrom wave theory we note: .....(25) where c = speed of light
...(26) This can be expanded into a series, similar to equation (16), whose first approximation is: ...(27) Therefore the speed of light is slower inside a gravitational field. Bending of LightThis should result in a measurable bending effect as light passes into and out of a gravitational field. Consider a wave of light. If we look at the wave front of a ray of light we note that the speed of light will vary across the length of the wave as indicated in equation (27). Thus the wave front will be deflected by an angle da in a time dt. The angle of deflection in the distance C x dt is given by:
As da -> 0 And : therefore: and the unit length The deflection per unit length is found by dividing da by C. ...(28) Now from equation (27) we get:
therefore: ...(29) Now the total deflection is given by the integral along the total path the light travels of the deflection angles da. Now
where D = distance of closest approach to the mass M therefore Therefore ....(30) Now: ....(31) Therefore:
The total angle of deflection of the light ray is given by: ...(32) ...(33) where r = distance from the centre of the body. So we should be able to detect a bending of a ray of light caused by the gravitational field of a massive object. The only object in our solar system which produces a strong enough gravitational field to measure such a deflection is the sun. In this case: D = 6.96 x 10^{8} m (the radius of the sun). Radians Or of arc Einstein's Field EquationEinstein quickly realised that these relationships reflected a distortion of space and time and that this distortion could be represented as a "curviture" in the space-time domain. To truely explore the relationships involved required an examination of the relationships between adjacent points in space-time. From geometry it is known that the distance between two points in three dimensions can be calculated knowing that: .....(34) By extrapolation, it can be shown that for an arbitary number of dimensions this equation becomes: ......(35) In polar co-ordinates:
Extending this to 4 dimensions gives: ......(36) At this stage the sign of the change in time is not yet determined. If the motion of a test particle is purely radial the angular terms in this equation can be ignored. Thus: ........(37) Now from equation (21): Or And from equation (22): Or So equation (37) becomes: ....(38) Or in its more general form: ....(39) The only thing remaining is to determine the sign of the time variable. We note that as r increases the unit of time t decreases with respect to the time measured by universal observer at a remote location and as r decreases the unit of time t increases with respect to the time measured by universal observer at a remote location. Therefore dt must always have the opposite sign to dr. Thus equation (39) becomes: ....(40) Black HolesClose examination of equation (40) shows that the metric for ds^{2} has problems at two conditions:
In fact, if we consider equation (26) carefully we note that when that c (the speed of light) = 0. Thus, light can not escape from the surface of such a body. For this reason they are refered to as "Black Holes". This critical radius is refered to as the "Event Horizon", for the simple reason that an observer outside the event horizon could not see (or measure) anything which takes place beyond this point. At r = 0, inside the event horizon, is the point called the "singularity". This is a mathematical term which indicates that a function tends toward infinity. In this case, the space-time functions tend toward infinity. Space becomes infinately contracted, while time becomes infinately dilated. If all the mass of a body is compressed inside the event horizon radius, light will not escape from the surface of this body. For the sun this radius is approximately 2.95 x 10^{3} m. Presumably, all the matter within the body will then continue to fall into the singularity, but there is no way of really knowing this from the outside of such an object. Matrix Form of the Field EquationsReturning to equation (40), this type of equation can be better represented in matrix form. After some time, Einstein did manage to produce a mathematical equation to describe the behaviour of time and space in the presence of objects in matrix form. The equation is: ...(41) where the terms, (Riemann's power tensor), and are all matricies. The format of the equation is that matter "tells" space-time how to distort (curve) while the curviture of space-time "tells" matter how to move. |
Created: 23 - Jan - 1997. |