Curvature and Energy Tensors
We still have a way to go before we can actually introduce Einstein's field equations of general relativity in any meaningful way. Therefore, before proceeding with the next instalment, it might be worth reminding ourselves that 4-dimensional spacetime curvature is not an intuitive concept and that many aspects of general relativity will contradict our normal intuition.
Unfortunately, the mathematical concepts that underpin Einstein’s idea can also appear to be equally baffling and not always readily understood without first having served a lengthy apprenticeship in mathematics. However, we shall continue to try to provide an ‘interpretative framework’ of the mathematics in which general relativity has been formulated. While we have outlined some of the mathematical tools in differential geometry for describing spacetime in terms of a metric, we have not really addressed curved spacetime or the role of mass-energy on this curvature.
We will start using an initial analogy to aid the present discussion by imagining an insect crawling over the surface of a large sphere, which from our perspective we can see is not flat, although this fact may be far from obvious to the insect. As such, the geometry of our spherical surface will have some properties that differ from a flat surface:
- Shortest distance between two points is not a straight line
- Sum of the angles in a triangle is more then 180 o
- Parallel lines converge and distance is not bounded
In this context, differential geometry provides the mathematical notation to extend the description of flat spacetime, as outlined in terms of the Pythagorean and Minkowski metrics of flat space and flat spacetime to many different types of curved spacetime, which may then help us answer questions such as:
What is the length of a line in a given curved geometry?
Most of us will have some understanding of the basic concepts of
calculus, which allows us to address such problems and, in some ways,
differential geometry is an extension of this concept. For example,
we might start with a small area of flat spacetime, for which we know
the rules, and then integrate the results over a larger area of curved