General Theory of Relativity

PhysicsMontwood High School

R. Casao

The special theory of relativity deals with uniformly moving reference frames; the frames of reference are not accelerated.The general theory of relativity, a new theory of gravitation presented by Einstein, incorporates accelerating reference frames.Einstein believed that the effects of gravitation and acceleration could not be distinguished from one another.

Principle of EquivalenceEinstein imagined himself in a space vehicle far away from gravitational influences.In the ship, at rest or in uniform motion relative to distant stars, he and everything in the ship would float freely.

When the rocket motors were turned on and the ship accelerated, a phenomenon similar to gravity would be observed.The wall adjacent to the motor would push up against Einstein and become the floor, while the opposite wall would become the ceiling.Einstein would be able to stand on the floor and jump up and down.If the acceleration of the ship were equal to g (9.8 m/s2), Einstein could be convinced that the ship was not accelerating but was at rest on the surface of the Earth.

Consider dropping 2 balls inside the spaceship, one wood ball and one lead ball.When the balls are released, they continue to move upward side by side with the velocity of the ship at the moment of release (principle of inertia).

Because the ship is accelerating, the floor moves upward faster than the balls, with the result that the floor soon catches up with the balls.Both balls meet the floor at the same time, regardless of their mass.Occupants

might believe gravity is responsible for the balls hitting the floor at the same time.

The two explanations for the falling balls are equally valid and Einstein incorporated this equivalence, the impossibility of distinguishing between gravitation and acceleration, in the foundation of his general theory of relativity.The principle of equivalence states that observations made in an accelerated reference frame are indistinguishable from observations made in a Newtonian gravitational field.

Bending of Light by GravityA ball thrown sideways in a stationary spaceship in a gravity-free region will follow a straight-line path relative to an observer inside the ship and to a stationary observer outside the ship.

If the ship is accelerating, an observer outside the ship still sees a straight-line path, but an observer in the accelerating ship sees a curved path (a parabola).This same principle holds true for a beam of light.

Suppose a light ray enters the ship horizontally through a side window, passes through a glass sheet, leaving a visible trace, and reaches the opposite wall.An outside observer sees the light ray enter the window and move horizontally along a straight line with constant velocity toward the opposite wall.

The ship is accelerating upward and during the time it takes for the light to reach the glass sheet, the ship moves up some distance and during the time needed for the light to reach the far wall, the ship moves up a greater distance.To observers in the accelerating frame of reference of the ship, the light has followed a downward curving path to the floor.

An observer inside the ship feels “gravity” because of the ship’s acceleration.The observer is not surprised by the deflection of the thrown ball, but may be surprised by the deflection of light.According to the principle of equivalence, if light is deflected by acceleration, it must be deflected by gravity.According t Newton’s physics, gravitation is an interaction between masses; a moving ball curves because of the interaction between its mass and the mass of the Earth.So why does light bend if it is massless? Einstein recognized that light was massless, but not “energyless”.Gravity pulls on the energy of light because energy is equivalent to mass (E = mc2).

Einstein stated that light bends when it travels in a spacetime geometry that is bent due to the presence of mass.The mass of Earth is too small to appreciably bend the spacetime around it, so any such bending of light in our immediate environment is not ordinarily detected.Close to bodies of mass much greater than Earth’s, the bending of light is large enough to detect.Light bends in Earth’s gravitational field, but is hard to detect.

For example: in a constant gravitational field of 9.8 m/s2, a beam of horizontally directed light will “fall” a vertical distance of 4.9 m in 1 s, but it will travel a horizontal distance of 3 x 108 m in that time.

Gravity and Time: Gravitational Red ShiftAccording to the general theory of relativity, gravitation cause time to slow down.If you move in the direction in which the gravitational force acts – from the top of a skyscraper to the ground floor – time will run slower at the point you reach than at the point you left behind..Imagine our accelerating reference frame to be a large rotating disk with three identical clocks, one placed on the center of the disk, one on the rim of the disk, and one at rest on the ground nearby the disk.

From special relativity, the clock attached to the center, since it is not moving with respect to the ground, should run at the same rate as the clock on the ground – but not at the same rate as the clock attached to the rim of the disk.The clock at the rim is in motion with respect to the ground and should be observed to be running more slowly than the ground clock and the clock in the center of the disk.Although the clocks on the disk are attached to the same frame of reference, they do not run synchronously; the outer clock runs slower than the inner clock.