General Relativity Talk
Emiliano Garcia
Outline
• Brief Introduction
• Basics of General Relativity
• Gravity Probe B
Brief Introduction
• Newton’s Universal Law Of Gravity
– Two bodies attract each other with a force that is
directly proportional to the product of their masses
and inversely proportional to the square of the
distance between them.
Brief Introduction
• Special Relativity
• Einstein published his first paper on
relativity in 1905.
– First Postulate: The laws of physics are the
same to all inertial observers
– Second Postulate: The speed of light is the
same to all inertial observers
Brief Introduction
• Conflict
– No signal can travel faster than light, but in
Newton's theory, gravity is a force transmitted
instantaneously over vast distances.
– Example: Masses at the ends of the universe
– What’s wrong?
Basics of General Relativity
• Mach’s Principle
– Newtonian inertial frames not rotating with
respect to the “fixed” stars
– Inertial properties are determined by the
presence of other bodies in the universe
Basics of General Relativity
• Principle of Equivalence
– “We shall therefore assume the complete
physical equivalence of a gravitational field
and the corresponding acceleration of the
reference frame. This assumption extends the
principle of relativity to the case of uniformly
accelerated motion of the reference frame.”
Basics of General Relativity
• Principle of Covariance
– In special relativity, all inertial observers
equivalent
– Here, all observers observe the same laws of
physics.
– Laws of physics can be expressed in terms of
tensors, since tensors are defined
independent of any coor. system
Basics of General Relativity
• Correspondence Principle
– In weak gravitation fields with velocities<<c ,
general theory should correspond with the
predictions of Newtonian mechanics
– As gravitational fields go to zero, general
theory should corresponds with the
predictions of special relativity
Basics of General Relativity
• Principle of minimal gravitational
coupling
– No terms explicitly containing the curvature
should be added when going from special to
general
Basics of General Relativity
• Leads to the idea that space-time is curved
• Field Equation; where Rik is the Ricci curvature tensor, R
is the scalar curvature, gik is the metric tensor, Λ is the
cosmological constant, Tik is the stress-energy tensor
describing the non-gravitational matter, energy and
forces at any given point in space-time
Basics of General Relativity
Gravity Probe B
• Proposed by Stanford
University Physicist
Leonard Schiff
• The experiment will
measure how space and
time are warped by the
presence of the Earth,
and how the Earth's
rotation drags space-time
around with it.
Gravity Probe B
• A gyroscope (actually four) and telescope will be
placed in a satellite and sent into a polar orbit 400
miles (640 km) above the Earth .
• The telescope and the gyroscope’s spin axis will be
perfectly aligned, and both will point at a distant
star (IM Pegasi).
– Classically, the underlying principle of a gyroscope is
that rotating systems, free from disturbing forces,
should stay pointing in the same direction in space.
– However, as mentioned before, space-time is warped
-- and may even be set in motion by moving matter. A
gyroscope orbiting the Earth will find two space-time
processes acting on it-- frame-dragging and the
geodetic effect -- gradually changing its direction of
spin.
Gravity Probe B
• Frame Dragging
– In 1918, Josef Lense and Hans Thirring
predicted that masses in space-time would
not only curve the structure of space-time
around them, but would also twist the local
space-time around them if they are rotating
– This twisting would alter measurements of
distance, time and direction in local spacetime
– Example: Racquetball in honey….
Gravity Probe B
• The Geodetic Effect
– When a mass “sits” in space-time, it causes
the space-time “fabric” to deform, changing
the space and altering the passage of time
around that object.
– In the case of the Sun’s “dip”, as the planets
travel across space-time, they respond to the
“dip” and follow the curve in space-time and
travel around the Sun.
Gravity Probe B
Gravity Probe B
• Note: Why IM Pegasi?
– Stars are not fixed in the sky.
– IM Pegasi was chosen because it is one of
the brightest stars in the sky, in the microwave
range.
– Very-Long-Baseline Interferometry (VLBI)
measurements since 1997 were used to map
IM Pegasi’s motion in the sky.
Gravity Probe B
• Note: Perfect Gyroscopes
– In order for GP-B to measure anything, its
gyroscopes must be nearly perfect
– Must not wobble or drift more than 1e-11
degrees in an hour while its spinning, since
the predicted twist in space-time is on the
order ~ 1e-6 degrees each year.
Perfect Gyroscopes
• Must be near perfect
spherically and
homogeneity
• Gyro Size: 1.5 inches
of fused quartz
• Sphericity: <40
atomic layers from
perfect
Perfect Gyroscopes
• Quartz Purity: Within
2 parts per million
• Gyro Spin Rate:
~10,000 rpm
• Gyro Drift Rate:
<10e-12
degrees/hour
Gravity Probe B
• Note: How do you measure the twist?
• Superconductivity
– Gyroscopes coated with a sliver thin layer
(1270nm) of niobium.
– As the gyroscope spins, the niobium layer
generates a magnetic field, whose axis is
exactly aligned with the spin axis.
Superconductivity
• A thin superconducting metal loop
encircles each gyroscope.
• The loop is connected to an external
SQUID (Superconducting Quantum
Interference Device)
• If and when the gyro tilts, the magnetic
field changes with it, which affects the
current in the loop
Superconductivity
Gravity Probe B
• According to Einstein’s theory , the gyroscope should
turn in two directions simultaneously. As it travels
through curved space time , it will turn slightly along
one axis . As “frame -dragging” occurs, it will also turn
slightly along a perpendicular axis.
• Schiff calculated that at a 400-mile altitude, space time
curvature would turn the gyroscope 6.6 arcseconds
per year in one direction, and the “frame-dragging ”
will turn at .042 arcseconds per year in a
perpendicular direction.
Gravity Probe B
Conclusions
• Conflicts between the Newton’s Universal
Law of Gravity and Einstein’s Special
Theory of Relativity lead to the
formalization of General Relativity
• GP-B is almost 34 weeks in orbit, and is
scheduled to orbit for a little over a year.
Sources
• Gravity Probe B – einstein.stanford.edu
• Goldstein Poole & Safko, Classical Mechanics
3rd Edition. Addison Wesley, 2002
• Steven Weinberg, Gravitation and Cosmology.
John Wiley & Sons, Inc.,1972
• Tim Maudlin, Quantum Non-Locality & Relativity
2nd Edition. Blackwell Publishing, 2002
• Wikipedia entry on “General Relativity” –
wikipedia.org