Fundamental Principles of General Relativity

general principle: laws of physics must be the same for all observers (accelerated or not)general covariance: laws of physics must take the same form in all coordinate systemslocal Lorentz invariance: laws of special relativity apply locally for all inertial observersinertial motion is geodesic motion: world lines of particles unaffected by physical forces are timelike or null geodesics of spacetimespacetime is curved:gravitational effects such as free fall can be described as a form of inertial motion

•The presence of matter changes the geometry of spacetime, this geometry being interpreted as gravity

2-D visualization of space-time distortion. The white lines do notrepresent the curvature of space.Here, spacetime is treated as a 4-D Lorentzian manifold which iscurved by the presence of mass, energy, and momentum (or stress-energy) within it.

The presence of mass causes a curvature of spacetime in thevicinity of the mass and this curvature dictates the spacetimepath that all freely moving objects must follow.

Consequence: light ray passing near the Sun should be deflected by some angle (about 1.75 degrees from Einstein’s calculations). This was confirmed by Eddington as bending of starlight during a total solar eclipse in 1919.

Earth The Sun acted like a gravitational lens.

Sun

Apparent direction to star

Actual direction to star) 1.75

Defining feature of GR

- gravitational ‘force’ is replaced by geometry. In GR,phenomena that in classical mechanics are ascribed to theaction of the force of gravity, e.g free fall, orbital motion,and spacecraft trajectories, are taken to represent inertial motion in a curved spacetime. People standing on Earth perceive the ‘force of gravity as a result of their undergoinga continuous physical acceleration caused by the mechanicalresistance of the surface in which they are standing. -An object is responsible for the field unlike in Newton’sfield force where gravitational force is an action-at a distance

Justification of GR: the Principle of Equivalence

-freely falling observers are the ones in inertial motion. Inertial observers can accelerate with respect to each other.

-in the vicinity of any given point, a gravitational field isequivalent to an accelerated frame of reference in theabsence of gravitational effects.

-gravitational mass and inertial mass are completely identicaland equivalent.

Newton vs. Einstein-both make essentially identical predictions as long as thestrength of the gravitational field is weak. Some divergence:1. The orientation of Mercury's orbit is found to precess

in space over time.This is commonly called the "precession of the perihelion", because it causes the position of the perihelion to move. Only part of this can be accounted for by perturbations in Newton's theory. There is an extra 43 seconds of arc per century in this precession that is predicted by the Theory of General Relativity and observed to occur (a second of arc is 1/3600 of an angular degree). This effect is extremely small, but the measurements are very precise and can detect such small effects very well.

2. Einstein's theory predicts that the direction of light propagation should be changed in a gravitational field, contrary to the Newtonian predictions. Precise observations indicate that Einstein is right, both about the effect and its magnitude. A striking consequence is gravitational lensing.

3. The General Theory of Relativity predicts that light coming from a strong gravitational field should have its wavelength shifted to larger values (what astronomers call a "red shift"), again contrary to Newton's theory. Once again, detailed observations indicate such a red shift, and that its magnitude is

correctly given by Einstein's theory.

4. The electromagnetic field can have waves in it that carry energy and that we call light. Likewise, the gravitational field can have waves that carry energy and are called gravitational waves. These may be thought of as ripples in the curvature of spacetime that travel at the speed of light.

Just as accelerating charges can emit electromagnetic waves, accelerating masses can emit gravitational waves. However gravitational waves are difficult to detect because they are very weak and no conclusive evidence has yet been reported for their direct observation. They have been observed indirectly in the binary pulsar. Because the arrival time of pulses from the pulsar can be measured very precisely, it can be determined that the period of the binary system is gradually decreasing. It is found that the rate of period change (about 75 millionths of a second each year) is what would be expected for energy being lost to gravitational radiation, as predicted by the Theory of General Relativity.

Experimental tests of GR

1. Slight bending of light in the gravitational field of the suna) Eddington’s observations

b) work of Irwin Shapiro and collaborators at NIT c) space probes to Mars d) discovery of gravitational lens in 1979

2. Advance of the perihelion

3. Clocks running slower in a gravitational field; stronger the force of gravity, slower the time flows

Other Implications of the Theory of GR

1. Existence of gravity waves

2. Basis of cosmology

3. Changing universe