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 free-fall, non-accelerating frame. Ask, what happens in the free-fall frame? In the free-fall 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 free-fall 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 freely-falling 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 free-fall frames along its path. But the light rays converge in the gravitating frame. Einstein's conclusion: gravity curves spacetime.

## How mass-energy 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 mass-energy-momentum curves spacetime. Einstein's equations have an iconic status, second only to his E = mc2 equation. They look like this:
 Gmn = 8pG Tmn
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 = energy-momentum 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 super-fast 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 mass-energy 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