The Meaning of Einstein’s Equation* *Partially based on an article by Baez and Bunn, AJP 73, 2005, 644

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

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

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

Summary of Einstein’s Equation Flows – diagonal elements of Tmn 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.”

Consequences Gravitational Waves Gravitational Collapse The Big Bang Newton’s Inverse Square Law

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

Gravitational Collapse Typically, pressure terms small Reinsert units: c = 1 and 8pG = 1 P dominates => neutron stars Above 2 solar masses => black holes

The Big Bang Homogeneous and Isotropic Expanding Assume observer at center of ball of test particles. Ball expands with universe, R(t) Introduce second ball – r(t)

Equation for R Equivalence Principle – “at any given location particles in free fall do not accelerate with respect to each other” So, replace r with R. Nothing special about t=0. Assume pressureless matter Universe mainly galaxies – density proportional to R-3 Get Newtonian Gravity!

Cosmological Constant Last model inaccurate Pressure of radiation important Expansion of universe is accelerating! Need to add L L>0 leads to exponential expansion

Newton’s Inverse Square Law Consider planet with mass M and radius R, uniform density Assume weak gravitational effects R>>M, neglect P Consider 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) The whole sphere of particles shrinks Green spheres shrink by same fraction r

Mathematical Details Parallel Transport Measuring Curvature Riemann Curvature Tensor Geodesic Deviation Stress Tensor Connection to Curvature

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

Measuring Curvature Parallel Transport Leading to Riemann Curvature Tensor

Compute Relative Acceleration Consider two nearby particles in free fall starting at “rest”. Particles are at points p and q. Relative velocity. Moving particles are later at p’ and q’. Compute relative acceleration using parallel transport.

Relative Acceleration Geodesic Deviation Equation Second Derivative of Volume Thus, Ricci => how volume of ball of freely falling particles starts to change. (Weyl Tensor describes tidal forces and gravitational waves.)

What is Rtt? Einstein Equation where or Thus, in every LIF for every point Or,

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

Stress Tensor Properties Divergence free Continuity Equation Newtonian limit (small v, p) Equation of Motion Newtonian Limit, Euler’s Equation for perfect fluid

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

Connection to Curvature Einstein’s attempts

Connection to Metric

What Have You Learned? Special Relativity General Relativity Space and Time General Relativity Metrics and Line Elements Geodesics Classic Tests Gravitational Waves Cosmological Models Einstein’s Equation Gravity = Curvature What Next?