General Relativity - Introduction

[Swapnam]

The following is a elucidation of GR as gleaned from Susskind's book and some parallel readings. The focus is on the theory's overall conceptual logic rather than heavy mathematical techniques.

The objective of General Relativity (GR) is to extend the study of spacetime in Special Relativity (SR) by including acceleration within the purview of analysis. The flat Minkowski geometry is replaced by a curved structure in the presence of gravitating bodies. A pithy lesson is "Spacetime tells mass how to move, Mass tells spacetime how to curve". Or even simply Gravity = Acceleration = Geometry.

This implies a revision of old ideas such as a global Lorentz reference frame, where all clocks run at the same rate. Now we have to contend with transformations across curvilinear coordinate systems, while ensuring the laws of physics retain the same form. (Compare with Galilean principle of relativity, one of the axioms of SR : The laws of physics take the same form for all observers in a state of uniform relative motion.)

[Swapnam]

The starting point is the Equivalence Principle, which states that no physical experiment can differentiate a reference frame in a uniform gravitational field from one in a state of uniform acceleration (Compare with SR, where there is no notion of absolute velocity).

To see this, recall the equivalence of the two masses of a body appearing in Newton's law of gravitation : ma = Gmn/r^2, where the m on the left side is the inertial mass, representing resistance to acceleration (by Newton's second law), whereas the m on the right side is the gravitational mass (An analogue of a particle's charge for Coulomb's electrostatic force. Notice that the inertial mass and charge of a particle have no relation).
It is a peculiar feature of gravity that these two are experimentally found to be exactly equal (accuracy upto one part in 10^12), which implies that all objects with various masses and constitutions freely falling in a gravitational field produced by mass n experience the same acceleration a = Gn/r^2. (Recall the apocryphal account of Galileo dropping different balls from Tower of Pisa and reporting they reach the ground at the same time.)
But this implies that an object falling in a uniform gravitational field of strength g has the same experience as falling in an elevator accelerating upwards at g in empty space.

Immediately we are led to a famous prediction of GR : Since a light ray fired horizontally in an elevator accelerating up will appear as bent down (which the observer in the elevator attributes to the presence of an external force i.e. gravity, while the observer outside blames on viewing a straight line in an accelerated frame i.e. "wrong" coordinate choice), the very same observation holds for light travelling in a gravitational field - Gravity curves light away from its straight line trajectory. The verification of this effect (gravitational lensing) was one of the triumphs of Einstein's theory.

[Swapnam]

However, note that this equivalence is a local approximation, since if two particles starting from an initial separation d are freely falling towards a source of a gravitational field e.g. the Earth, they'll both be drawn towards its center, and by a careful measurement of the decrease in d over time, one will be able to detect the presence of a real gravitational field as opposed to merely an accelerating frame of reference where no such central convergence occurs.

Obviously, the uniformity of the gravitational field and the "point particle" are themselves an idealization, as points at various distances from a mass will experience different field values, and likewise for different parts of any extended rigid body leading to its distortion, which will cause an observational difference from a uniformly accelerating reference frame. In summary, tidal forces that are a physical manifestation of a real gravitational field can be used as a test for it.

[Swapnam]

Next we come to the issue of curvilinear coordinate systems. Note that we can represent even flat spacetime with curved coordinates - what differentiates flat from curved space is the possibility that we can always find globally applicable Lorentz coordinates in the former if we wish to, which is not possible in a curved space.

As an example, consider the surface of a flat paper rolled up without deformities. It appears curved when we see it from our 3d space in which it is embedded i.e. it has "extrinsic curvature", but an observer living on the surface on the paper would notice no differences in his observations before and after the rolling. Compare this to the surface of a doughnut with a hole, which is "intrinsically curved".

So we need to differentiate between real vs. apparent curvature in spacetime, which in fact is precisely the difference between a real gravitational field vs. an accelerating reference frame in flat space, i.e. between fictitious forces (e.g. centrifugal force in a rotating frame) due to coordinate choice and physically real tidal forces due to gravitation.

[Swapnam]

The characterization of the geometry of a surface is done through an object called the "metric tensor", which has a collection of components, each of which is a function corresponding to a unique combination of the coordinates.

For Euclidean space, this is trivially the identity matrix (recall the distance element : ds^2 = dx^2 + dy^2 i.e. the only non-0 components are those where the x and y coordinates match with themselves, and these are 1), for Minkowski geometry we get a -1 in the (t, t) position and identity matrix elsewhere (recall the definition of proper time/distance interval in SR : ds^2 = -dt^2 + dx^2 + dy^2), and for a generally curved spacetime it can be a complicated matrix.

Thus an analogue of the problem in GR had already been tackled by Gauss, Riemann etc. in the development of Differential Geometry and can be restated as : Can we find a coordinate transformation such that the metric tensor becomes the identity matrix everywhere in the space? If so, the metric, and hence the space, was only apparently curved and actually flat, and if not, then the curvature is real, and so is the gravitational field.

To establish the above needs a framework for the transformation of objects across coordinate systems, which is the subject of Tensor Analysis.