The Theory of General Relativity
General relativity is a theory about gravity, which at its centre is the idea that gravity is the geometry of curved 4-dimensional spacetime. As such, Newton’s ideas about gravity as a force might be said to have been replaced by Einstein’s idea of the curvature of spacetime in which mass particles follow the shortest path in spacetime as defined by a geodesic.
So what is the nature of spacetime curvature?
While it is not the goal to answer this question within the context of an introduction, the following concepts might provide an initial guide to the discussions to follow:
- Gravity reflects the geometry of spacetime:
Therefore,phenomena previously associated with gravitational forces
are now described in terms of the curved geometry of 4-dimensional
- Mass-Energy are the cause of spacetime
curvature: Mass is the primary source of spacetime
curvature. However, on the basis of Einstein’s most famous equation
[E=mc2], any form of energy is also a source of spacetime
- Mass moves along the shortest path in curved spacetime: In general relativity, the Earth orbits the Sun, not because of a gravitational force exerted by the Sun, but because it is following the shortest path in the curved spacetime produced by the Sun.
As indicated, the question above encapsulates the scope of general relativity that this entire section seeks to consider, even if it possibly falls short of a definitive answer. However, in contradiction to the assertion that gravity is just the curvature of spacetime, the following discussions will still make reference to the Newtonian concept of a gravitational force. This juxtapositioning of two different concepts is primarily to help in the transition between what amounts to two different scientific descriptions of cause and effect. However, the following quote by Kip Thorne, an American theoretical physicist, possibly puts these apparently differences into some perspective:
|"Einstein and Newton, with their very different viewpoints on the nature of space and time, give different names to the agent that causes test particles to accelerate towards or away from one another in a frame that is not quite free-float. Einstein called it spacetime curvature, Newton calls it tidal acceleration. But there is just one agent acting. Therefore, spacetime curvature and tidal acceleration must be precisely the same thing, expressed in different languages."|
While the implications of special relativity were a radical departure from classical physics, the mathematics that support its assumptions can still be generally understood by most people. However, it is probably true to say that this is not the case with general relativity.
Does this mean that a person without an understanding of advance mathematics cannot understand the implications of general relativity?
While it is true that mathematics is an essential tool of modern science, it should be possible to translate the knowledge it imparts into terms that can be more generally understood. For example,  is representative of the mathematical form of Einstein’s field equation of general relativity, where the left-hand side defines the curvature of spacetime with the energy-momentum of matter on the right:
In practice,  is masking the complexity of 10 non-linear, partial differential equations for any given solution in the form of a spacetime metric, but comprehending this shorthand demands an understanding of the tensor notation embedded in differential geometry. Therefore, fully understanding the mathematical implications of  is non-trivial and, as such, the overview of the mathematics associated with general relativity will be deferred until after some of the more basic concepts have been outlined. Of course, some may argue that mathematics associated with general relativity is essential from the outset, if any real understanding of the theory is to be acquired. However, the counter argument is that if something cannot be generally explained without the abstraction of advanced mathematics, then maybe it is not so well understood after all. With this basic argument tabled, the intention is to begin with an initial introduction of some of the basic principles, which underpin general relativity. The next step will be to examine some of the key implications of a specific solution of Einstein’s field equations of general relativity, know as the `Schwarzschild Metric`, which will also examine some of the most extreme relativistic effects associated with objects referred to as ‘black holes’. The intention is that the first 2 sections will only require a basic understand of algebra in order to manipulate some of the established equations. Finally, in the last section, an attempt will be made to introduce ‘some’ of the complexity of the mathematics used to describe the curvature of spacetime by mass-energy, as formulated in Einstein field equations. Here is a summary of the discussions to follow:
- Classical Foundations:
- Newtonian Equations
- Basic Concepts
- Basic Model
- The Schwarzschild
- The Metric
- Effective Potential
- Gullstrand-Painleve Coordinates
- Black Holes
- Mathematical Overview:
- Basic Concepts
- Differential Geometry
- Curvature and Energy
- Solutions to the Field Equations
Note, although general relativity also underpins the idea of an expanding
universe, which is characterised as `Friedmann Cosmology`; this
subject will be addressed under the
Cosmology section of this website.