# Gradients of gravitational forces - focusing of geodesics

 AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy Gravity, black holes and physics

2.6. Deviation and focusing of geodesics
Now, that we already have available to Einstein's equations of the gravitational field, we still return for a moment to the question of the influence of the gravitational field on the motion of bodies, which we began in §2.4. The homogeneous gravitational field can be "filtered out" in the whole space by transition to a global inertial system. Even in inhomogeneous gravitational field according to the equivalence principle can introduce local inertial system, in which all the physical phenomena take place (local) under the laws of special relativity. Therefore, since the gravitational field
does not exist locally, the question arises, how does the real (inhomogeneous) gravitational field actually affect physical processes? Or otherwise: how the curvature of spacetime manifests itself in the laws of physics? How do we distinguish the effects of gravity from the influence of the frame of reference? It is clear that locally in no way. Therefore, a non-local approach must be used to study the effect of gravity. E.g. instead of one test particle (which can always be considered locally free and which therefore tells us nothing about the curvature of spacetime), we will observe the difference in behavior of the two test particles.

Gravitational force gradients, deviations of geodesics
Let us have two test particles, one of which is located at the point x
i (here we will consider it as a reference) and the other at the near point xi + ei , where ei is a four-vector space-time "distance" of these two test particles. If we let these two adjacent test particles "fall" freely together in the gravitational field (Fig.2.17), they will move along the world lines xi (l) and xi (l) + ei (l) (the four-vector ei (l) connects the points of both trajectories with the same l), which are geodesics : If we subtract these equations from each other and omit the higher order members in ei (we mean the limit transition ei ® 0), we get  Fig.2.7. Two close test particles loosely "falling" next to each other in a gravitational field (curved spacetime) will move along geodesics that will not be completely parallel, but will deviate from each other (deviation of geodesics).

We are interested in the difference in the motion of both test particles (deviation of both geodesics), ie the covariant derivative of their deviation ei. Using equation (2.15) for the covariant (absolute) derivation of the vector along the curve, we get the final result

 D2 e i                       dxk   dxm               --------       -       Riklm -----  ----- el   =   0   , dl2                          dl   dl (2.57)

which is the so-called equation of deviation of geodesics. According to her, two bodies falling freely next to each other in an inhomogeneous gravitational field will approach or move away from each other (their geodesics will deviate from each other), while the mutual acceleration of both particles D2e i /dl2 is proportional to the curvature tensor Ri klm. The components of the curvature tensor thus describe the gradients of gravitational forces (tidal forces). In planar spacetime, all components of the curvature tensor are equal to zero - there is no deviation of geodesics.

To obtain a more complete picture of the influence of gravity on the motion of matter, let us now move from a pair of nearby geodesics to the whole congruence of such geodesics. We can imagine this as a point test particle is placed at each point in space and their worldlines form this congruence of geodesics in space-time - a certain geodesy passes through each point (event) of space-time. If we determine a unit tangent vector Vi for each geodetic at each of its points, we get the whole field of tangent vectors Vi(xk) for the congruence of geodetics. The divergence ("running away") or convergence ("gathering on each other") C of geodesics in a given congruence is determined by the covariant four divergences of the vector field V i :

 C = def    V i ; i   . (2.58)

The convergence of the congruence of time-type geodetics follows the important Raychaudhuri differential equation , 

 d C / ds  =  R ik V i V k + 2 s 2 + (1/3) C 2  , (2.59a)

where s describes the mutual "slip" of the geodetic (we do not consider rotation here; if the congruence of the geodetic rotated, the term -2 w2 would still be on the right side). An analogous equation also applies to the convergence of light (zero) geodesics :

 d C / d l  =  R ik V ik V k + 2 s 2 + 1/2 C 2  , (2.59b)

where V i are isotropic tangent vectors.

Energy conditions
Let us now notice some general properties of equations (2.59a, b). The last two terms on the right-hand sides of both equations are nonnegative, so the final sign of the convergence changes is decided by the term
RikVi Vk. In order for this term even to be nonnegative, according to Einstein's equations (2.50b), the gravitational field must be excited by an energy-momentum tensor satisfying the inequality

 (Tik - 1/2 T gik ) Vi Vk   ł   0 (2.60a)

for all vectors of V i time type, resp. inequality

 T ik V i V k    ł    0 (2.60b)

for isotropic vectors V i (for which gik Vi Vk = 0). For the energy-momentum tensor of the shape of an ideal fluid (1.108), this is satisfied when the density and pressure satisfy the relations

 r  ł  0 ,  r + pa ł 0 ,  resp. r  ł  0 ,  r + a=1S3pa  ł  0   . (2.61a, b)

This so-called weak energy condition (2.60) is fulfilled for all yet observed forms of matter, because the energy density is non-negative and a large negative pressure p *) would be needed to disturb the energy condition.
*) However, a weak energy condition would not be met, for example, for a hypothetical C-field in the "steady-state" model of the universe (mentioned in Chapter 5), which has a negative energy density Too, or for a "false vacuum" field in inflationary expansion of universe (see §5.5). And also not for the hypothetical so-called exotic mass used for antigravity "reinforcement" of traversible wormholes (.......).
In practice, an even stronger condition of so-called energy dominance is probably met: for each vector Vi of time type is Tik Vi Vk ł 0, and the vector Tik Vk is of time or isotropic type, ie the local energy density is non-negative and in addition, the local energy flow takes place only inside or on the mantle of the light cone. Thus, T°°> = |Tik| applies, ie energy "dominates" over the other components of the energy-momentum tensor. For the most common type of energy-momentum tensor (1.108) this is satisfied when r ł 0 , - r Ł pa Ł r (a = 1,2,3), ie when the pressure does not exceed the energy density, the speed of "sound" does not exceed the speed of light.
Note: A total of 4 energy conditions are sometimes discussed in the literature (....., ....,). However, for our analysis of the geometry of spacetime around black holes, as well as global cosmological geometry, a weak energy condition and a condition of energy dominance will suffice.

Focusing of geodetics
We see, therefore, that the member
RikVi Vk in equations (2.59a, b) is also nonnegative when reasonable energy conditions (2.60) are met, so that the inequality holds for the rate of change of convergence of geodetics

 d C / d s   ł   1/3 C2   ł   0  , (2.62)

according to which convergence grows monotonically along the congruence of geodesics. We can thus declare the statement :

 Theorem 2.5 (focusing of geodesics)
 If the energy conditions (inequalities) for the energy-momentum tensor are satisfied, such that according to Einstein's equations R ik V i V k ł 0 applies for each vector V i time resp. isotropic type, gravity has an attractive character and on geodesics of the time resp. isotropic type has a focusing effect .

The statement about the focus of geodesics is of great importance for the geometric- topological logical structure of spacetime - it is used, for example, in proving the 2nd law of black hole dynamics (§4.6) or in proving theorems about the existence of singularities of spacetime (§3.8).   2.5. Einstein's equations of the gravitational field 2.7. Gravitational waves

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Vojtech Ullmann