Curved Spacetime... A brief introduction

Curved Spacetime... A brief introduction May 5, 2009

Inertial Frames and Gravity In establishing GR, Einstein was influenced by Ernst Mach. Mach s ideas about the absolute space and time: Space is simply the separation between bodies, and Time is merely the succession of events i.e. neither space nor time have an independent existence in its own right and relative motions is all that matters the property of Inertia has nothing to do with absolute space but arises from some kind of interaction between each individual body and all the other members of the universe. i.e. if there are no other masses, an isolated body would have no inertia This is in contrast to Newton s view that the body will still have inertia because of the effect of absolute space. Mach s principle : the measure of inertia is somehow related to the background produced by the universe. Remove the background and the inertia disappears, so inertia is not just a property of matter!

Inertial Frames and Gravity An inertial frame is defined as one in which a free particle moves with constant velocity. But gravity is a long-range force and cannot be screened. Consequently, the only way to visualize an inertial frame is to imagine it far away from any mater, which is of no use for our typical experiments and for astronomy. According to Einstein, if gravitation is long-range and unscreened, it has somehow a permanent character, and it must be intrinsic to the region in which is located. Einstein identified this intrinsic property of the space-time by its geometry.

Einstein s Principle of Equivalence Although, the introduction of inertial frames in the strict Newtonian formulation is impossible, we can still cover the space-time with patches of local inertial frames. The introduction of local inertial frames depends on the equivalence of inertial m I and gravitational mass, m g. Newtonian version: In a small laboratory falling freely in a gravitational field, mechanical phenomena are the same as those observed in an inertial frame in the absence of a gravitational field. Einstein s version: In a small laboratory falling freely in a gravitational field, the laws of physics are the same as those observed in an inertial frame in the absence of a gravitational field. There are no observations or experiments the observer could perform in a local inertial frame that could indicate whether the effects were those of gravity or those of acceleration. Within a small closed cabin the effects of gravity and acceleration are indistinguisable

Bending of a Light Beam Consider an observer at rest in a cabin and a light ray passing in his neighborhood. If the cabin is not accelerated will find that the ray is moving on a straight line If the cabin is accelerated the observer will see the light ray moving in a curved trajectory Two observers, one accelerated (A) and the other in a gravitational field (B) will observed the same phenomenon.

Gravitational Redshift In an accelerated cabin a signal sent from a point A to a point B (of distance h) parallel to the direction of the acceleration g the light will suffer a frequency shift ν/ν. The light will travel for t = h/c and the point B will gain an additional velocity u = g t. This leads to : ν ν = u c = gh c 2 (1) By using the equivalence principle we can imagine that the same observations can be made in a gravitational field. Note that the term gh is the difference in the gravitational potential between A and B, say Φ. I.e. when the light falls through a gravitational potential difference Φ then it gains energy and becomes bluer by the amount Φ/c 2 (and vv). In the same way clocks will run slower (as measured for a distant observer/clock) when they are at t(r) = ( 1 Φ c 2 ) t( ) = ( 1 GM ) rc 2 t( ) (2)

The Concept of Curved Space-Time In special relativity we have been introduced to the notion of space-time and that the space-time is described by the Minkowski metric ds 2 = c 2 dτ 2 = c 2 dt 2 ( dx 2 + dy 2 + dz 2) (3) where dτ is a measure of the proper time. But due to eqn (2) ( ds 2 = c 2 dτ 2 = 1 2GM ) rc 2 c 2 dt 2 (4) Since the intervals of proper time are affected by gravitational field, we may say that the presence of the gravitational field influences the geometry of space-time and we may write ( ds 2 = 1 2GM ) rc 2 c 2 dt 2 ( dx 2 + dy 2 + dz 2) (5) which is no longer Minkowskian.

The Concept of Curved Space-Time Since the gravitational field is influencing time we may also assume that it will influence the measure of the length. Space and time coordinates must be treated on equal footing without any intrinsic preference of one over the other. Thus we expect the line element to be of the form: ds 2 = Ac 2 dt 2 ( Bdx 2 + Cdy 2 + Ddz 2) (6) where the coefficients A, B, C and D depend on the form and strength of gravitational field and are functions of the space-time variables t, x, y, z and the tend to unity at large distances from the source of the gravitational field. But since the gravitational field is equivalent to a certain non-inertial frame, the system of coordinates that will represent it will be curvilinear and the line element is, in general, a quadratic form in the differentials of the coordinates x 0, x 1, x 2, x 3 of the general type: ds 2 = µ,ν g µν dx µ dx ν = g µν dx µ dx ν (7) where g µν is function of the coordinates and is called the metric of the spacetime manifold.

For example, if (x, y, z, ct ) are the coordinates of a frame rotating uniformly with angular velocity ω along the z-axis., the transformation to this frame is given by x = x cos ωt y sin ωt, y = x sin ωt + y cos ωt, z = z and the line element becomes ds 2 = [ c 2 ω 2 (x 2 + y 2 ) ] dt 2 dx 2 dy 2 dz 2 + 2ωy dx dt 2ωx dy dt (8)

The Concept of Curved Space-Time The geometry of a spacetime in which a metric such (7) can be defined is called Riemannian geometry. The accelerated non-inertial frames are equivalent to gravitational fields thus the gravitational effects are to be described by the metric g µν. In this framework, gravitation is to be understood as a deviation of the metric of the space-time manifold from the flat Minkowski metric. Therefore the metric g µν is not fixed arbitrarily on the whole space-time, but it depends on the local distribution of matter. Figure: Curved space-time

The Concept of Curved Space-Time The metric g µν is symmetric in the indexes µ and ν i.e. g µν = g νµ. In an inertial frame, when we use Cartesian coordinates, the quantities g µν are locally g 00 = 1, g 11 = g 22 = g 33 = 1, g ik = 0 for i k (9) With an appropriate choice of coordinates, we can always bring the metric g µν in the form (9) locally. A gravitational field cannot be eliminated by any coordinate transformation. In the presence of a gravitational field, space-time is such that the quantities g µν cannot by any coordinate transformation, be brought to the (9) globally.

The Principle of General Covariance Einstein postulated that all frames of reference are equally good for the description of nature and the laws of physics should have the same form in all. The requirement that, under a general coordinate transformation, the laws of physics must remain covariant i.e. the have and invariant form, is called the principle of general covariance. This is the mathematical representation of the principle of equivalence If we use tensorial quantities for expressing the laws of physics, the principle of general covariance will yield the simplest tensor equations that generalize the special relativistic versions. (Check Maxwell equations in tensor form according to special and general relativity)