Tidal Forces In A Black Hole



Consider a spherical distribution of noninteracting particles falling towards the Earth.










Each particle moves along a straight line to the center of the Earth. Particles nearer the center fall faster than the other particles since gravity is more intense for them.










Gravity produces a tidal force in the distribution of particles that forces the spherical distribution into an ellipsoidal shape as time progresses. This effect also occurs in General Relativity and it is especially obvious close to a Black Hole region.


To see this we consider the equation that governs motion of particles when gravitational tidal forces are present. This equation is the geodesic deviation equation.








We can evaluate this equation for the case where the deviation vector  is projected out into the 3-space orthogonal to the particle velocity vectors Error! Objects cannot be created from editing field codes..  Assuming that this is done we consider the three dimensional effect of falling near a Black Hole. Let the three-dimensional orthogonal separation vector be  where the index a runs from 1 to 3.  The geodesic deviation equation takes the form:







We now switch to Schwarzschild coordinates and jump into the frame of reference of the falling particles. It can be shown that the above equation reduces to the following three equations, where we have replaced the coordinate numbers 1, 2, and 3 with spherical polar coordinate symbols for the first, second and third Schwarzschild coordinates.







This equation describes the tidal force in the radial direction. The deviation change is posite in this case. It represents a radial tension or stretching effect.







This equation describes a compression type pressure aligned along the transverse  






This equation also describes a compression type pressure this time aligned along the transverse  direction. Notice that all three effects become extreme, as we get closer to  where the effect turns infinite. Any person falling into the Black Hole will get elongated and simultaneously squashed in the middle.



This is sometimes referred to as the 'toothpaste effect'. Eventually the falling body will be torn to pieces right down to the constituent atoms.







Tuesday, January 07, 2003