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General Relativity and
Applications
2. Dynamics of Particles, Fluids,
and Spacetime
Edmund Bertschinger
MIT Department of Physics and
Kavli Institute for Astrophysics and
Space Research
Dynamics in General Relativity
How do particles move in curved
spacetime?
How do fluids move in curved spacetime?
What curves spacetime? How?
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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.
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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)
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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
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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)
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Internal Symmetries
Lorentz Force Law and Maxwell Equations are
unchanged by the local gauge transformation
Am(x)  Am(x)+m P(x) for any P(x).
Gauge transformation: mixing of fields
(A0,A1,A2,A3) at each point in spacetime
In the Standard Model of particle physics,
SU(3)cxSU(2)LxU(1)Y are internal symmetries.
Consequences: mixing of quarks, gluons,
photon+Z0
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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!
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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.
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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!)
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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.
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Newtonian Gravity (“scalar”)
Trajectory of a
massive particle
Uniform mass sheet
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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).
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Gravitomagnetism (“vector”)
Inwardly moving body is
deflected out of this plane
H
Rotating uniform-mass sphere
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Gravitomagnetic Spin and Orbit
(Lense-Thirring) Precession
H
s
L
Orbiting
satellite
Spinning mass
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
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Gravitomagnetic
precession
Gravity-Probe B
http://einstein.stanford.edu/
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Gravity-Probe B
http://einstein.stanford.edu/
Was Einstein
right?
Find out in 2006…
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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
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Gravitational waves — the Evidence
Neutron Binary System – Hulse & Taylor (Nobel Prize)
PSR 1913 + 16 -- Timing of pulsars
17 / sec


~ 8 hr
Neutron Binary System
• separated by 106 miles
• m1 = 1.4m; m2 = 1.36m; e = 0.617
Prediction from general relativity
• spiral in by 3 mm/orbit
• rate of change orbital period
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Effect of a GW on matter
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Stress-Energy Tensor
Source of gravity:
energy-momentum-stress (pressure)
in a local
Lorentz
frame
r = mass-energy density
fi = momentum density [fi = (r+p)vi for a perfect fluid]
p = pressure
Sij = shear stress [Sij = 0 for a perfect fluid]
Newtonian gravity: r is source, however under a Lorentz
transformation r  g2r (E=gm and volume Lorentz contracts)
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Energy-Momentum Conservation
(fluid equations): nTmn=0
Flat spacetime:
Perfect fluid:
Continuity
Euler
Curved spacetime: same idea, with
gravitational forces
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Einstein Field Equations Gmn=8pGTmn
in the Weak Field Limit
Transverse
TransverseTraceless
(This is most, but not all, of the content of the Einstein Field Equations)
Compare with Maxwell equations
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Physical Content of the Einstein Equations
Source of Newtonian-like gravity: r+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 sij travel at speed
of light. Two indices  spin-2 field.
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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)
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