Einstein's Theory of General Relativity

The assignment content discusses the theory of general relativity by Einstein, which unites concepts of space and time into a four-dimensional flat space-time. It highlights the concept of gravitation as curvature of space-time, the gravitational redshift, and the bending of light around massive objects. The summary also touches on the equivalence principle, the replacement of Newtonian theory with Einstein's field equations, and the verification of predictions through experiments.

According to newtonian theory, gravitational effects propagate from place to
place instantaneously. With the advent of Einstein's special theory of
relativity in 1905, a theory uniting the concepts of space and time into that
of four dimensional flat space-time (named Minkowski space-time after the
mathematician Hermann Minkowski), a problem became discernible with
newtonian theory. According to special relativity, which is the current
guideline to the form of all physical theory, the speed of light, c = 3 ×
1010 cm s-1, is the top speed allowed to physical particles or forces: There
can be no instantaneous propagation. After a decade of search for new
concepts to make gravitational theory compatible with the spirit of special
relativity, Einstein came up with the theory of general relativity (1915), the
prototype of all modern gravitational theories. Its crucial ingredient,
involving a colossal intellectual jump, is the concept of gravitation, not as a
force, but as a manifestation of the curvature of space-time, an idea first
mentioned in rudimentary form by the mathematician Ceorg Bernhard
Riemann in 1854. In Einstein's hands gravitation theory was thus
transformed from a theory of forces into the first dynamical theory of
geometry, the geometry of four dimensional curved space-time.
Why talk of curvature? One of Einstein's first predictions was the
gravitational redshift: As any wave, such as light, propagates away from a
gravitating mass, all frequencies in it are reduced by an amount proportional
to the change in gravitational potential experienced by the wave. This
redshift has been measured in the laboratory, in solar observations, and by
means of high precision clocks flown in airplanes. However, imagine for a
moment that general relativity had not yet been invented, but the redshift has
already been measured. According to a simple argument owing to Alfred
Schild, wave propagation under stationary circumstances can display a
redshift only if the usual geometric relations implicit in Minkowski space-
time are violated: The space-time must be curved. The observations of the
redshift thus show that space-time must be curved in the vicinity of masses,
regardless of the precise form of the gravitational theory.
Einstein provided 10 equations relating the metric (a tensor with 10
independent components describing the geometry of space-time) to the
material energy momentum tensor (also composed of 10 components, one of
which corresponds to our previous ). These Einstein field equations, in
which both of the previously mentioned constants G and c figure as
parameters, replace Poisson's equation. Einstein also replaced the newtonian
law of motion by the statement that free test particles move along geodesics,
the shortest curves in the space-time geometry. The influential gravitation
theorist John Archibald Wheeler has encapsulated general relativity in the
aphorism "curvature tells matter how to move, and matter tells space-time
how to curve." The Eotvos-Dicke-Braginsky experiments demonstrate with
high precision that free test particles all travel along the same trajectories in

space-time, whereas the gravitational redshift shows (with more modest
precision) these universal trajectories to be identical with geodesics.
Despite the great contrast between General Relativity and Newtonian
theory, predictions of the former approach the latter for systems in which
velocities are small compared to c and gravitational potentials are weak
enough that they cannot cause larger velocities. This is why we can discuss
with newtonian theory the structure of the earth and planets, stars and stellar
clusters, and the gross features of motions in the solar system without fear of
error.
Einstein noted two other predictions of General Relativity. First, light beams
passing near a gravitating body must suffer a slight deflection proportional
to that body's mass. First verified by observations of stellar images during
the 1919 total solar eclipse, this effect also causes deflection of quasar radio
images by the sun, is the likely cause of the phenomenon of "double
quasars" with identical redshift and of the recently discovered giant arcs in
clusters of galaxies (both probably effects of gravitational lensing), and is
part and parcel of the black hole phenomenon. In a closely related effect first
noted by Irwin Shapiro, radiation passing near a gravitating body is delayed
in its flight in proportion to the body's mass, a time delay verified by means
of radar waves deflected by the sun on their way from Earth to Mercury and
back.
The second effect is the precession of the periastron of a binary system.
According to newtonian gravitation, the orbit of each member of a binary is
a coplanar ellipse with orientation fixed in space. General relativity predicts
a slow rotation of the ellipse's major axis in the plane of the orbit
(precession of the periastron). Originally verified in the motion of Mercury,
the precession has of late also been detected in the orbits of binary pulsars.
All three effects mentioned depend on features of General Relativity beyond
the weak equivalence principle. Indeed, Einstein built into general relativity
the much more encompassing "strong equivalence principle": the local
forms of all nongravitational physical laws and the numerical values of all
dimensionless physical constants arc the same in the presence of a
gravitational field as in its absence. In practice this implies that within any
region in a gravitational field, sufficiently small that space-time curvature
may be ignored, all physical laws, when expressed in terms of the space-
time metric, have the same forms as required by special relativity in terms of
the metric of Minkowski space-time. Thus in a small region in the
neighborhood of a black hole (the source of a strong gravitational field) we
would describe electromagnetism and optics with the same Maxwell
equations used in earthly laboratories where the gravitational field is weak,

and we would employ the laboratory values of the electrical permittivity and
magnetic susceptibility of the vacuum.