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 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

4. 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

5. 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.”

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

7. 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

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

9. 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)

10. 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!

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

12. 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

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

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

15. Parallel Transport
Vector fields are parallel transported along curves, while mantaining a constant angle with the tangent vector

16. Flat and Curved Spaces
In a flat space, transported vectors are not rotated.
In a curved space they are rotated:

17. Measuring Curvature Parallel Transport
Leading to Riemann Curvature Tensor

18. 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.

19. 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.)

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

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

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

23. 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

24. Connection to Curvature
Einstein’s attempts

25. Connection to Metric

26. 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?