The Meaning of Einstein’s Equation*

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

2. 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?

3. Preliminaries • Special Relativity • No absolute velocities, • Only relative • Described by 4-vectors • Depends on inertial coordinate systems • Field of clocks at rest with respect to each other

4. Preliminaries • General Relativity • Not even relative velocities • Except for two particles at same point • Compare vectors by moving to same point • Need effects of parallel transport • On curved spacetime – path dependent • Einstein’s Equation • Relative acceleration of nearby test particles in free fall

5. Einstein’s Equation – “Plain English” • Consider small round ball of test particles rel. at rest • Volume V(t), t – proper time for center particle • In free fall it becomes an ellipsoid • relative velocity starts out zero => 2nd order in time

6. Summary of Einstein’s Equation • Flows – diagonal elements of Tmn • Px = Flow of momentum in x direction = pressure • r = Flow of t-momentum in t-direction = energy density “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.”

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

8. 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

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

10. 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)

11. 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!

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

13. 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

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

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

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

17. 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

18. Measuring Curvature Parallel Transport Leading to Riemann Curvature Tensor

19. 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.

20. 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.)

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

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

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

24. 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

25. Connection to Curvature • Einstein’s attempts

26. Connection to Metric

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