From acceleration togeometry

In exploring the equivalence of
gravity and acceleration as well
as the role of tidal forces,
Einstein discovered several
analogies with the geometry of
surfaces. An example is the
transition from an inertial
reference frame (in which free
particles coast along straight
paths at constant speeds) to a
rotating reference frame (in
which extra terms
corresponding to fictitious
forces have to be introduced in
order to explain particle motion):
this is analogous to the
transition from a Cartesian
coordinate system (in which the
coordinate lines are straight
lines) to a curved coordinate
system (where coordinate lines
need not be straight).
A deeper analogy relates tidal
forces with a property of
surfaces called curvature. For
gravitational fields, the absence
or presence of tidal forces
determines whether or not the
influence of gravity can be
eliminated by choosing a freely
falling reference frame. Similarly,
the absence or presence of
curvature determines whether
or not a surface is equivalent to
a plane. In the summer of 1912,
inspired by these analogies,
Einstein searched for a
geometric formulation of
gravity.[12]
The elementary objects of
geometry – points, lines,
triangles – are traditionally
defined in three-dimensional
space or on two-dimensional
surfaces. In 1907, the
mathematician Hermann
Minkowski introduced a
geometric formulation of
Einstein's special theory of
relativity in which the geometry
included not only space, but also
time. The basic entity of this new
geometry is four-dimensional
spacetime. The orbits of moving
bodies are lines in spacetime; the
orbits of bodies moving at
constant speed without changing
direction correspond to straight
lines.[13]
For surfaces, the generalization
from the geometry of a plane –
a flat surface – to that of a
general curved surface had been
described in the early 19th
century by Carl Friedrich Gauss.
This description had in turn been
generalized to higher-dimensional
spaces in a mathematical
formalism introduced by
Bernhard Riemann in the 1850s.
With the help of Riemannian
geometry, Einstein formulated a
geometric description of gravity
in which Minkowski's spacetime is
replaced by distorted, curved
spacetime, just as curved
surfaces are a generalization of
ordinary plane surfaces.[14]
After he had realized the validity
of this geometric analogy, it
took Einstein a further three
years to find the missing
cornerstone of his theory: the
equations describing how matter
influences spacetime's
curvature. Having formulated
what are now known as
Einstein's equations (or, more
precisely, his field equations of
gravity), he presented his new
theory of gravity at several
sessions of the Prussian
Academy of Sciences in late
1915.

Physical consequences

In 1907, Einstein was still eight
years away from completing the
general theory of relativity.
Nonetheless, he was able to
make a number of novel,
testable predictions that were
based on his starting point for
developing his new theory: the
equivalence principle.[7]
The gravitational redshift of a
light wave as it moves upwards
against a gravitational field
(caused by the yellow star below)
The first new effect is the
gravitational frequency shift of
light. Consider two observers
aboard an accelerating rocket-
ship. Aboard such a ship, there is
a natural concept of "up" and
"down": the direction in which the
ship accelerates is "up", and
unattached objects accelerate in
the opposite direction, falling
"downward". Assume that one of
the observers is "higher up" than
the other. When the lower
observer sends a light signal to
the higher observer, the
acceleration causes the light to
be red-shifted, as may be
calculated from special
relativity; the second observer
will measure a lower frequency
for the light than the first.
Conversely, light sent from the
higher observer to the lower is
blue-shifted, that is, shifted
towards higher frequencies.[8]
Einstein argued that such
frequency shifts must also be
observed in a gravitational field.
This is illustrated in the figure at
left, which shows a light wave
that is gradually red-shifted as it
works its way upwards against
the gravitational acceleration.
This effect has been confirmed
experimentally, as described
below.
This gravitational frequency
shift corresponds to a
gravitational time dilation: Since
the "higher" observer measures
the same light wave to have a
lower frequency than the "lower"
observer, time must be passing
faster for the higher observer.
Thus, time runs more slowly for
observers who are lower in a
gravitational field.
It is important to stress that,
for each observer, there are no
observable changes of the flow
of time for events or processes
that are at rest in his or her
reference frame. Five-minute-
eggs as timed by each
observer's clock have the same
consistency; as one year passes
on each clock, each observer
ages by that amount; each
clock, in short, is in perfect
agreement with all processes
happening in its immediate
vicinity. It is only when the clocks
are compared between separate
observers that one can notice
that time runs more slowly for
the lower observer than for the
higher.[9] This effect is minute,
but it too has been confirmed
experimentally in multiple
experiments, as described below.
In a similar way, Einstein
predicted the gravitational
deflection of light: in a
gravitational field, light is
deflected downward.
Quantitatively, his results were
off by a factor of two; the
correct derivation requires a
more complete formulation of
the theory of general relativity,
not just the equivalence principle.

Gravity and acceleration

Most effects of gravity vanish in
free fall, but effects that seem
the same as those of gravity
can be produced by an
accelerated frame of reference.
An observer in a closed room
cannot tell which of the following
is true:
Objects are falling to the floor
because the room is resting on
the surface of the Earth and the
objects are being pulled down by
gravity.
Objects are falling to the floor
because the room is aboard a
rocket in space, which is
accelerating at 9.81 m/s2 and is
far from any source of gravity.
The objects are being pulled
towards the floor by the same
"inertial force" that presses the
driver of an accelerating car into
the back of his seat.
Conversely, any effect observed
in an accelerated reference
frame should also be observed in
a gravitational field of
corresponding strength. This
principle allowed Einstein to
predict several novel effects of
gravity in 1907, as explained in
the next section.
An observer in an accelerated
reference frame must introduce
what physicists call fictitious
forces to account for the
acceleration experienced by
himself and objects around him.
One example, the force pressing
the driver of an accelerating car
into his or her seat, has already
been mentioned; another is the
force you can feel pulling your
arms up and out if you attempt
to spin around like a top.
Einstein's master insight was
that the constant, familiar pull
of the Earth's gravitational field
is fundamentally the same as
these fictitious forces.[5] The
apparent magnitude of the
fictitious forces always appears
to be proportional to the mass
of any object on which they act -
for instance, the driver's seat
exerts just enough force to
accelerate the driver at the
same rate as the car. By
analogy, Einstein proposed that
an object in a gravitational field
should feel a gravitational force
proportional to its mass, as
embodied in Newton's law of
gravitation.[6]

General relativity (GR) is a theory
of gravitation that was
developed by Albert Einstein
between 1907 and 1915.
According to general relativity,
the observed gravitational
attraction between masses
results from their warping of
space and time.
By the beginning of the 20th
century, Newton's law of
universal gravitation had been
accepted for more than two
hundred years as a valid
description of the gravitational
force between masses. In
Newton's model, gravity is the
result of an attractive force
between massive objects.
Although even Newton was
bothered by the unknown nature
of that force,[1] the basic
framework was extremely
successful at describing motion.
Experiments and observations
show that Einstein's description
of gravitation accounts for
several effects that are
unexplained by Newton's law,
such as minute anomalies in the
orbits of Mercury and other
planets. General relativity also
predicts novel effects of
gravity, such as gravitational
waves, gravitational lensing and
an effect of gravity on time
known as gravitational time
dilation. Many of these
predictions have been confirmed
by experiment, while others are
the subject of ongoing research.
For example, although there is
indirect evidence for
gravitational waves, direct
evidence of their existence is
still being sought by several
teams of scientists in
experiments such as the LIGO
and GEO 600 projects.
General relativity has developed
into an essential tool in modern
astrophysics. It provides the
foundation for the current
understanding of black holes,
regions of space where
gravitational attraction is so
strong that not even light can
escape. Their strong gravity is
thought to be responsible for
the intense radiation emitted by
certain types of astronomical
objects (such as active galactic
nuclei or microquasars). General
relativity is also part of the
framework of the standard Big
Bang model of cosmology.
Although general relativity is not
the only relativistic theory of
gravity, it is the simplest such
theory that is consistent with
the experimental data.
Nevertheless, a number of open
questions remain, the most
fundamental of which is how
general relativity can be
reconciled with the laws of
quantum physics to produce a
complete and self-consistent
theory of quantum gravity.
From special to general
relativity
In September 1905, Albert
Einstein published his theory of
special relativity, which
reconciles Newton's laws of
motion with electrodynamics
(the interaction between objects
with electric charge). Special
relativity introduced a new
framework for all of physics by
proposing new concepts of
space and time. Some then-
accepted physical theories were
inconsistent with that
framework; a key example was
Newton's theory of gravity,
which describes the mutual
attraction experienced by bodies
due to their mass.
Several physicists, including
Einstein, searched for a theory
that would reconcile Newton's
law of gravity and special
relativity. Only Einstein's theory
proved to be consistent with
experiments and observations.
To understand the theory's basic
ideas, it is instructive to follow
Einstein's thinking between 1907
and 1915, from his simple
thought experiment involving an
observer in free fall to his fully
geometric theory of gravity.[2]
Equivalence principle
Main article: Equivalence principle
A person in a free-falling elevator
experiences weightlessness, and
objects either float motionless
or drift at constant speed. Since
everything in the elevator is
falling together, no gravitational
effect can be observed. In this
way, the experiences of an
observer in free fall are
indistinguishable from those of
an observer in deep space, far
from any significant source of
gravity. Such observers are the
privileged ("inertial") observers
Einstein described in his theory
of special relativity: observers
for whom light travels along
straight lines at constant speed.
[3]
Einstein hypothesized that the
similar experiences of weightless
observers and inertial observers
in special relativity represented
a fundamental property of
gravity, and he made this the
cornerstone of his theory of
general relativity, formalized in
his equivalence principle. Roughly
speaking, the principle states
that a person in a free-falling
elevator cannot tell that he is in
free fall. Every experiment in
such a free-falling environment
has the same results as it would
for an observer at rest or
moving uniformly in deep space,
far from all sources of gravity

Introduction to
general relativity
This article is intended as an
accessible, non-technical
introduction to the subject. For
the main encyclopedia article,
see General relativity.

A TUTORIAL ON GENERAL RELATIVITY

read an introduction to general relativity if you are not much acquainted with general relativity