Fall 2009 ASTR 2030 Homepage
The Principle of Equivalence of gravity and acceleration
How do you apply the Principle of Equivalence?
Suppose you are in a gravitating system.
The Principle of Equivalence
asserts that a gravitating frame is equivalent to an accelerating frame.
Go into a freefall, nonaccelerating frame.
Ask, what happens in the freefall frame?
In the freefall frame,
unaccelerated objects move in straight lines at constant velocity.
Whatever happens there tells you what happens
back in the gravitating/accelerating frame.
A cannonball, seen through a window.



The same cannonball,
but now you and the window are falling freely.
In the freefall frame,
the cannonball follows a straight line trajectory.





Two consequences of the Principle of Equivalence:
gravitational redshift, and the gravitational bending of light
A and B (who are moving to the right slowly)
are in a gravitational field that points downward.
Equivalently, A and B
are accelerating upward.
A sends a light ray to B.
This shows the situation as seen in a freelyfalling frame.
The light follows a straight line.
During the time that the light goes from A to B,
B has accelerated upwards a bit.
Since B is moving upwards,
away from A's original position,
B sees the light redshifted compared to A.



The same situation,
but now seen in the gravitating/accelerating frame.
B sees the light redshifted (to lower energy) compared to A.
The light follows a curved line.





Third consequence of the Principle of Equivalence:
spacetime is curved
Two light rays approach the Earth on parallel lines.
Each light ray follows a straight line relative to freefall frames along its path.
But the light rays converge in the gravitating frame.
Einstein's conclusion:
gravity curves spacetime.



How massenergy curves spacetime: Einstein's (1915) equations
The Principle of Equivalence determines only how objects move in the curved
spacetime produced by a gravitational field,
not how mass curves spacetime.
It took Einstein several years, until 1915,
eventually to come up with his “Einstein equations”,
which describe mathematically how massenergymomentum curves spacetime.
Einstein's equations have an iconic status, second only to his
E = mc^{2}
equation.
They look like this:
These equations actually represent not one but ten distinct equations.
You do not have to understand the mathematical meaning of the equations,
which is notoriously difficult.
However, their physical meaning is this:
Geometry
(curvature)
of spacetime

=

energymomentum
content
of spacetime


In general relativity,
spacetime can in principal be arbitrarily curved.
General relativity opens the possibility
of pulling spacetime around, like playdoh, into fantastical shapes.
For example,
one place could be linked by a wormhole shortcut to another far distant place,
allowing superfast intergalactic travel.
Or, the future could be linked to the past,
allowing time travel.
Unfortunately,
Einstein's equations cut short such dreams.
You are free to specify whatever curved spacetime you like,
but then Einstein's equations tell you how you must arrange massenergy
in order to produce your desired spacetime.
There are two problems with this.
First,
curving spacetime substantially
requires a lot of mass in a small space
—
like a black hole.
Second,
making something like a wormhole
generically requires gravitationaly repulsive negative mass
(which Thorne calls “exotic matter”).
Fall 2009 ASTR 2030 Homepage
Updated 2009 Sep 24