Edmund Bertschinger MIT Department of Physics and Kavli Institute for Astrophysics and Space Research General Relativity and Applications 2. Dynamics of Particles, Fluids, and Spacetime

2 Dynamics in General Relativity How do particles move in curved spacetime? How do fluids move in curved spacetime? What curves spacetime? How?

3 Lagrangian Dynamics All modern physics theories are based on the Principle of Least Action, which leads directly to Lagrangian Dynamics. Please read Lecture Notes 3: “How Gravitational Forces Arise from Curvature” for a mathematical introduction to the Principal of Least Action.

4 General Relativity “Spacetime tells matter how to move; matter tells spacetime how to curve.” (Wheeler) The laws of physics have the same form in all coordinate systems. Gravity is a fictitious force (like Coriolis). Gravity is a force and it is a manifestation of spacetime curvature. (Force/Geometry duality)

5 Fields in Physics How do particles move under electromagnetic or gravitational forces? How is gravity similar to and different from electromagnetism? Modern perspective: key role of symmetry

6 Symmetries in Physics Symmetry transformation = a change which preserves the equations governing a system. Crucial ingredient of all modern physics theories. Electromagnetism has two symmetries: 1. Lorentz transformations 2. Local gauge transformations (Internal symmetry)

7 Internal Symmetries Lorentz Force Law and Maxwell Equations are unchanged by the local gauge transformation A  (x)  A  (x)+   (x) for any  (x). Gauge transformation: mixing of fields (A 0,A 1,A 2,A 3 ) at each point in spacetime In the Standard Model of particle physics, SU(3) c xSU(2) L xU(1) Y are internal symmetries. Consequences: mixing of quarks, gluons, photon+Z 0

8 Symmetries of General Relativity GR, like electromagnetism, has two symmetries: 1. General coordinate transformations (General Covariance) 2. Local (spacetime-dependent) Lorentz transformations (Internal symmetry) In SR, these two symmetries both reduce to global Lorentz transformations. Local Lorentz symmetry is usually ignored in textbook presentations of GR but is a crucial ingredient of string theory, supergravity, quantum gravity, and an understanding of gravitational forces in GR!

9 General Covariance “Vector equations are valid independently of the coordinate system or basis which one uses.” “The laws of physics have the same form in all coordinate systems.” The action is a scalar under general coordinate transformations.

10 The importance of symmetry General covariance + Local Lorentz symmetry are so powerful that they essentially completely determine the equations of motion in GR. (This is the modern field theoretic perspective, not Einstein’s geometric one!)

11 The richness of theories with symmetry Electromagnetism: Electric + Magnetic Electric forces independent of speed. Magnetic forces proportional to speed. Electric charge is conserved. General relativity: Electric (Newtonian) + Magnetic (gravitomagnetism) + Tensor (gravitational waves). Energy-momentum is locally conserved.

12 Newtonian Gravity (“scalar”) Uniform mass sheet Trajectory of a massive particle

13 Surprises of scalar gravity in GR 1. The trajectory of a massless particle (e.g. photon) is also deflected by gravity (gravitational lensing). In a static nonuniform field the deflection is twice the naïve prediction for a particle moving at speed v=c. 2. Time slows down in a gravitational field: just as the accelerating twin ages less in special relativity, so too does one who lives in a strong gravitational field. 3. The frequency of light waves (as measured locally by observers at rest) decreases when light climbs out of a gravitational field (gravitational redshift).

14 Gravitomagnetism (“vector”) Rotating uniform-mass sphere H Inwardly moving body is deflected out of this plane

15 Gravitomagnetic Spin and Orbit (Lense-Thirring) Precession H s Magnetic Torque causes spin to precess – basis of Nuclear Magnetic Resonance (NMR, MRI). Gravity Probe-B is measuring this for gravity! Orbital angular momentum vector precesses (Lense & Thirring 1918) These effects are weaker than Newtonian gravity by (v/c) 2 Spinning mass Orbiting satellite L

16 Gravity-Probe B Gravitomagnetic precession

17 Was Einstein right? Find out in 2006… Gravity-Probe B

18 Gravitational Radiation (“tensor”) Newtonian gravity and gravitomagnetism are action at a distance, in clear violation of the principle of relativity. How does general relativity fix this?  By adding WAVES that travel at the speed of light How are they produced and how are astrophysicists preparing to detect them?  Produced by accelerating masses: for example, two black holes merging  Detected by their TINY effect on test masses, using LASERS bouncing back and forth between moving mirrors

19 Neutron Binary System – Hulse & Taylor (Nobel Prize) PSR Timing of pulsars   17 / sec Neutron Binary System separated by 10 6 miles m 1 = 1.4m  ; m 2 = 1.36m  ;  = Prediction from general relativity spiral in by 3 mm/orbit rate of change orbital period ~ 8 hr Gravitational waves — the Evidence

20 Effect of a GW on matter

21 Stress-Energy Tensor Source of gravity: energy-momentum-stress (pressure) in a local Lorentz frame  = mass-energy density f i = momentum density [f i = (+p)v i for a perfect fluid] p = pressure  ij = shear stress [ ij = 0 for a perfect fluid] Newtonian gravity:  is source, however under a Lorentz transformation    2  (E=m and volume Lorentz contracts)

22 Energy-Momentum Conservation (fluid equations):  T  =0 Flat spacetime: Perfect fluid: Continuity Euler Curved spacetime: same idea, with gravitational forces

23 Einstein Field Equations G  =8GT  in the Weak Field Limit (This is most, but not all, of the content of the Einstein Field Equations) Compare with Maxwell equations Transverse Transverse- Traceless

24 Physical Content of the Einstein Equations Source of Newtonian-like gravity: +3p Minus signs because like charges attract Gravitational Ampère Law lacks Maxwell Displacement Current  g and H are action-at-a-distance Waves of spatial strain s ij travel at speed of light. Two indices  spin-2 field.

25 Summary Dynamics of general relativity is based on two symmetries: General covariance (coordinate-independence) Local Lorentz invariance GR extends Newtonian gravity: Gravitomagnetism (similar to magnetism except no q/m and no displacement current) Gravitational radiation: propagating waves of tidal shear Experiments are currently testing these new phenomena: Gravity Probe-B (gravitomagnetism) LIGO/Virgo (gravitational radiation)