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.)
Space geodesicsWhen 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/h2The 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.
Spacetime geodesicsThings 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.