Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 139where these flows are measured at the center of the ball at time t = 0,using local inertial coordinates. These flows are the diagonal componentsof the SEM–tensor T. Its components T αβ tell us how much momentumin the α−direction is flowing in the β−direction through a given point ofspace–time. The flow of t−momentum in the t−direction is just the energydensity, T 00 = ρ. The flow of x−momentum in the x−direction is the‘pressure in the x−direction’, T 11 = P 1 ≡ P x , and similarly for y and z.In any event, we may summarize the Einstein equation (3.1) as¨V∣ = − 1 V t=0 2 (ρ + P x + P y + P z ) ≡ − 1 2 (T 00 + T 11 + T 22 + T 33 ). (3.2)This new equation tells us that positive energy density and positive pressurecurve space–time in a way that makes a freely falling ball of point particlestend to shrink. Since E = mc 2 and we are working in normal units, ordinarymass density counts as a form of energy density. Thus a massive object willmake a swarm of freely falling particles at rest around it start to shrink.In short, (3.2) tells us that gravity attracts (see e.g., [Misner et al. (1973);Baez (2001)]).To see why equation (3.2) is equivalent to the Einstein equation (3.1),we need to understand the Riemann curvature tensor and the geodesic deviationequation. Namely, when space–time is curved, the result of paralleltransport depends on the path taken. To quantify this notion, pick twovectors u and v at a point p in space–time. In the limit where ɛ −→ 0, wecan approximately speak of a ‘parallelogram’ with sides ɛu and ɛv. Takeanother vector w at p and parallel transport it first along ɛv and then alongɛu to the opposite corner of this parallelogram. The result is some vectorw 1 . Alternatively, parallel transport w first along ɛu and then along ɛv.The result is a slightly different vector, w 2 . The limitw 2 − w 1limɛ−→0 ɛ 2 = R(u, v)w (3.3)is well–defined, and it measures the curvature of space–time at the point p.In local coordinates, we can write it asR(u, v)w = R α βγδu β v γ w δ .The quantity Rβγδ α is called the Riemann curvature tensor. We can use thistensor to calculate the relative acceleration of nearby particles in free fall ifthey are initially at rest relative to one another. Consider two freely fallingparticles at nearby points p and q. Let v be the velocity of the particle at