Einstein's Equivalence Principle
Gravitational forces are locally equivalent to inertial forces1 which are
equivalent to non-inertial frames which are equivalent to curvilinear
coordinates.
Inertial forces
Consider the motion of a free test body in an inertial frame K with
Cartesian coordinates: its equation of motion is [([(x)\vec])\ddot]=0 which
is an equation for a straight line.
Now, consider the same body from a frame K′ that accelerates with
respect to K with acceleration [(a)\vec]. In this frame the equation
of motion for a free body is [([(x)\vec])\ddot]=−[(a)\vec] (not an equation
for a straight line). This equation can be written with the help of a
fictitious inertial force [(F)\vec]I=−m[(a)\vec] as the Newton's law:
m[([(x)\vec])\ddot]=[(F)\vec]I, where m is the mass of the body.
Inertial forces have the following properties:
- Inertial forces are proportional to the masses of bodies, or,
in other words, under inertial forces all bodies move with the same
acceleration.
- Inertial forces completely disappear after a coordinate
transformation to an inertial (flat) frame.
- Inertial forces appear in the equations of motions not due to some
real physical fields affecting the body but as additional (geometrical)
terms because the frame is not-inertial (curved).
Einstein's equivalence principle
Galileo's "Pisa experiment" showed that all bodies move in a gravitational
field with the same acceleration, which is the first property of
inertial forces. Einstein has postulated that all three properties of
inertial forces are fulfilled for gravitational forces. In other words,
gravitational forces are unlike other physical forces but much like
fictitious inertial forces. This is called the Einstein's equivalence
principle. It can also be formulated as:
- In free fall the effects of gravity disappear in all possible local
experiments and general relativity reduces locally to special relativity.
- An accelerated frame is locally equivalent to a frame in a
gravitation field.
- Gravitation field is locally equivalent to a non-inertial frame.
- Gravitation forces are equivalent to inertial forces.
Footnotes:
1
Inertial forces are fictitious forces which are thought to affect
the bodies when looking at their motion from an non-inertial frame.
Examples: centrifugal, Coriolis and elevator forces.
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