Sciences

Space Time Theory

What is General Relativity?

 

What's the basic idea?

Both space and spacetime can either be curved or flat.

We've described the Euclidean (or Euclidean-Mesopotamian :-) metric in two space dimensions: dL² = dX² + dY² and we've discussed at some length the complications that arise with the addition of time to space to give the Minkowski metric (shown here in just one space and one time dimension): dS² = c² dT² - dX² What else can we do to our spacetime distance function to make life more interesting (and hopefully solve the problem with Newtonian gravity discussed in the last section)?
What if we play around with the form of the Minkowski metric? It turns out that if the spacetime metric is arranged in the right manner, we can get something called spacetime curvature. And that is what the General Theory of Relativity is all about.
For example, suppose we add some extra space and time dependence to the Minkowski metric to make a new spacetime distance function dS² = g_TT(T,X) c² dT² - g_XX(T,X) dX² Using differential geometry, taking the right combination of first and second derivatives of g_TT(T,X) and g_XX(T,X), we could calculate the what is called the curvature tensor R_uv for this choice of spacetime distance function. The subscripts on R_uv are called tensor indices and refer back to the coordinates used in the above metric. The Minkowski metric corresponds to the choice g_TT = g_XX = 1 and it has R_uv = 0 for all values of the tensor indices. This is why the Minkowski metric is known also as flat spacetime - because the spacetime curvature calculated from this distance function is zero.
In Einstein's time they were already learning about differential geometry, but Einstein motivated this field of mathematics even more when he came up with an equation relating the curvature tensor of the spacetime distance function to the distribution of matter and energy in spacetime, encoded in a tensor T_uv called the stress-energy tensor.
This equation is now called the Einstein equation:

Ruv - (1/2) guv R = (8 Pi G/c4) Tuv

This equation (or actually, set of equations, for there is an equation for every combination of tensor indices u and v) models a lot of phenomena in the Universe that was impossible to describe with mathematics just using Newton's law of gravity. For example, observations of the bending of light by gravity, gravitational radiation emitted by pulsars, new observations of black holes and the observed expansion of the visible Universe can all be modelled rather successfully using this elegant formalism uncovered by Einstein.

What happens to light cones?

Most people have heard the phrase A straight line is the shortest distance between two points. But in differential geometry, they say this same thing in a different language. They say instead Geodesics for the Euclidean metric are straight lines. A geodesic is a curve that represents the extreme value of a distance function in some space or spacetime.
Geodesics are important in the relativistic description of gravity. Einstein's Principle of Equivalence, part of the General Theory of Relativity, tells us that geodesics represent the paths of freely-falling particles in a given spacetime. (Freely-falling in this context means moving only under the influence of gravity, with no other forces involved.)

The shortest path between the two red points in the Poincare upper half plane is on a semicircle, not a straight line.

Space geodesics

When our distance function is the Pythagorean Rule dL² = dX² + dY², also known as the Euclidean metric, straight lines are the curves that give the minimum Euclidean distance between two points.
In two space dimensions there are many metrics one can dream up in addition to the Euclidean metric. For example, take a class of metrics of the form: dL² = (k²/Y²) (dX² + dY²) For Y>0, this distance function or metric is called the Poincare upper half plane. The geodesics for this metric are described by the formula: (X - X₀)² +Y² = k²/h² The geodesics consist of two types of curves: a) semicricles of radius k/h centered at X = X₀ and in the limit h = 0, vertical lines with X = X₀.
The shortest path between any two points on the Poincare upper half plane is along one of those two types of curves, not along the straight line that connects the two points. This is shown in the figure above.

In flat spacetime in two space and one time dimensions, the light cones really do look like cones. If we add the right kind of curvature, we can twist the light cones so that they overlap as shown below.

Spacetime geodesics

Things get more complicated when we graduate from space to spacetime. Remember that the Minkowski metric has a spacetime distance function dS² that can be negative, positive or zero, whereas the distance functions in space dL² can only be positive.
The means we have to separate our geodesics on the basis of whether the distance function dS² is positive, negative or zero. Goedesics with dS² < 0 are called spacelike geodesics. Goedesics with dS² = 0 are called null geodesics. Goedesics with dS² > 0 are called timelike geodesics. The behavior of timelike and null geodesics are the most important for understanding time travel.
Timelike geodesics behave the opposite from geodesics in space. They actually represent the longest spacetime distance between two spacetime events.
In Minkowski spacetime, all of the geodesics all straight lines, whether timelike, spacelike or null. The light cones are made of the null geodesics, and they rigidly separate the past from the future. In flat spacetime in two space and one time dimensions, the light cones really do look like cones, as shown in the top figure to the right.
But in a generic curved spacetime, the null geodesics won't as a rule be straight lines, sometimes they can be more interesting. The light cones made from null geodesics from a spacetime metric in a curved spacetime can even be made to have the past and future light cones overlap. An example of a spacetime satisfying the Einstein equations in three spacetime dimensions (two space and one time) where the past and future light cones overlap is shown in the figure on the bottom right. We'll see more of this spacetime later. See how it's twisted? Angular momentum can twist light cones and even make time travel possible in theory if not in practice.

How does that relate to Newton's model?

General Relativity didn't make Newtonian gravity false. Newton's model is still a good model for most measurements and calculations of the motion of planets and rockets and so forth, and General Relativity must agree with Newton's model in that limit. Here is proof that these two very different models agree on the motion of the Earth around the Sun but disagree in the region where a new relativistic phenomenon - the black hole - is present in Einsetin's model but not in Newton's.

Newtonian and relativistic potentials are almost the same at large distance scales like the radius of the Earth's orbit R = 1.5 1013 cm.

The Earth and Sun in the Newtonian model

In Newton's model of nature based on the math he invented, the differential calculus, the primary equation by which motion of objects is calculated is Force = mass x acceleration This gives a set of second-order differential equations and when we solve that system, we get the motion of the object in question when subjected to the force in question.
If the object in question is the Earth and the force in question is the gravitational force of the Sun on the Earth, then after writing the above equation in spherical coordinates, we get the following equation for the change in time of the radial coordinate r for the position of the Earth: m (dr/dt)² = 2 (E - V_N(r))
V_N(r) = - M m G/r + m (L/m)²/(2 r²) G is Newton's gravitational constant, M stands for the mass of the Sun and m is the (approximately) the mass of the Earth. Because the Newtonian force law for gravity depends only on radial distance, not on time or angle, there are two constants of motion called angular momentum L and energy E.
The function V(r) is plotted above for the Earth/Sun system. Notice that whenever V(r) = E, the radial coordinate stops changing because dr/dt = 0. These are called turning points. The turning points classify the type of orbit, as shown in the above figure.
Notice that it is the L²/r² term in the potential that causes the turning points for smaller r. Angular momentum acts almost like a force of repulsion to counter gravitational attraction. That will be important later as an agent of causality violation in the relativistic model.

Newtonian and relativistic potentials begin to disagree very close to the center of attraction.

The Earth and Sun in the relativistic model

In General Relativity, the first ingredient in model-making is the spacetime metric. This metric must solve the Einstein equation that relate the spacetime curvature to the matter and energy present.
If the matter and energy present is idealized by a pointlike object of mass M, then the spacetime metric that solves the Einstein equation is called the Schwarzschild metric.
We won't display that metric here for the sake of brevity. The timelike geodesics are easy to calculate for the Schwarzschild metric. We wind up with an equation that looks a lot like the Newtonian formula above: m (dr/dt)² = 2 (E - V_R(r))
V_R(r) = - M m G/r + m (L/m)²/(2 r²) - M G m (L/m)²/(2 c² r³) At large distances from the Sun, the last term in the potential V_R(r) can be neglected, and the classification of orbits is the same as for the Newtonian case.
But if the Sun really were a pointlike mass, the last term in the potential V_R(r) would give us trouble. Notice in the bottom figure the red line, which represents the relativistic potential, veers down to become infinitely negative, instead of infinitely positive as with the Newtonian potential. This behavior is the signal of a black hole. We can't go into that in depth right here, but you can read about it in books specializing in black holes.
The Sun is not pointlike, the mass of the sun is spread within a radius of about 700,000 kilometers. The black hole behavior would not set in unless the mass of the Sun were confined to a within radius of about three kilometers.