The Meaning of Einsteins Equation Partially based on

The Meaning of Einstein’s Equation* *Partially based on an article by Baez and Bunn, 2006

Overview n Einstein’s Equation: Gravity = Curvature of Space n What Does Einstein’s Equation Mean? n Needs Full Tensor Analysis n Consequences n Tidal Forces and Gravitational Waves n Gravitational Collapse n Big Bang Cosmology … and more! n Stress and Curvature Tensors n What Have We Learned?

Preliminaries n Special Relativity n No absolute velocities n Described by 4 -vectors n Depends on inertial coordinate systems n General Relativity n Not even relative velocities n Except for two particles at same point Need effects of parallel transport n Curvature of spacetime n Relate to energy density n

Parallel Transport Vector fields are parallel transported along curves, while mantaining a constant angle with the tangent vector www. to. infn. it/~fre

Flat and Curved Spaces In a flat space, transported vectors are not rotated. In a curved space they are rotated: www. to. infn. it/~fre

Einstein’s Equation n Consider small round ball of test particles n In free fall it becomes an ellipsoid n relative velocity starts out zero => 2 nd order in time

Summary of Einstein’s Equation n Flows – diagonal elements of Tmn n Px = Flow of momentum in x direction = pressure “Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the x direction at that point, plus the pressure in the y direction, plus the pressure in the z direction. ”

Tidal Forces and Gravitational Waves n Test particle ball initially at rest in a vacuum n No energy density or pressure n But curvature still distorts ball n Vertical Stretching n Horizontal Stretching n Gravitational Waves n Space-time can be curved in vacuum n Heavy objects wiggle => ripples of curvature

Gravitational Collapse n Typically, pressure terms small n Reinsert units: c = 1 and 8 p. G = 1 n P dominates => neutron stars and black holes

Newton’s Inverse Square Law n Consider planet with mass M and radius R, uniform density n Assume weak gravitational effects R>>M, neglect P n Consider n n n Sphere S of radius r >R centered on planet Fill with test particles, initially at rest Apply to infinitesimal sphere (green) within S S

Inverse Square Law (cont’d) n The whole sphere of particles shrinks n Green spheres shrink by same fraction r

Measuring Curvature Parallel Transport

Relative Velocity Consider two nearby particles in free fall starting at “rest”. Relative velocity.

Relative Acceleration n Geodesic Deviation n Second Derivative of Volume

What is Rtt? Usual Form Where Implies Thus, in every LIF for every point Or,

Tensor Formulation – Flat Space n Stress Tensor – for a continuous distribution of matter – perfect fluid (density, pressure) n Symmetric n 4 -momentum density

Stress Tensor Properties n Divergence free n Continuity n Newtonian limit n Equation of Motion n Newtonian Limit, Euler’s Equation

Tensor Formulation – Curved Space n Fluid particles pushed off geodesics by pressure gradient Start with continuity and equation of motion to claim divergence free n Leads to more general formulation n Need Covariant Derivatives

Connection to Curvature n Einstein’s attempts

Connection to Metric

What Have You Learned? n Special Relativity n Space and Time n General Relativity n Metrics and Line Elements n Geodesics n Causality – Light Cones n Cosmology n Einstein’s Equation n Gravity = Curvature n What Next?