Relativity and Black Holes
Spacetime and the Interval

In 3-dimensional geometry, positions are represented by points (x,y,z). In physics, we are interested in events which have both time and position: (t,x,y,z). The collection of all possible events is called spacetime.

With an event (t,x,y,z) in spacetime we associate the units of cm with coordinates x, y, z. In addition, we express time t in terms of cm by multiplying it by the speed of light. (In fact, many texts use coordinates (ct, x, y, z) for events.) These common units (cm for us) are called geometric units. We express velocities in dimensionless units by dividing them by c. So for velocity v (in cm/sec, say) we associate the dimensionless velocity . Notice that under this convention, the speed of light is 1.
This leads us to restate time dilation and length contraction as:
and

The concept of spacetime was introduced by Hermann Minkowski.

Minkowski, in fact, was one of Albert Einstein's mathematics professors at the Zurich Politechnikum in 1900. Einstein's lack of respect for authority and his casual attitude toward coursework caused Minkowski to label Einstein a "lazy dog"... certainly an incorrect assessment!
In Gottingen, Minkowski studied Einstein's 1905 paper on special relativity and was impressed. This study lead Minkowski to his formulation of an absolute four-dimensional spacetime.

In an address to the 80th Assembly of German Natural Scientists and Physicians at Cologne in September 1908, Minkowski opened with this statement:

When Einstein learned of Minkowski's work, he was not impressed. To Einstein, Minkowski was merely rewriting the laws of special relativity in a new mathematical language. Einstein felt that the abstract mathematics obscured the underlying physics. However, we will soon see that Einstein had to change his mind about the usefulness of spacetime.

As a sad footnote, Hermann Minkowski died of appendicitis in 1909 at the age of 45. This was just a matter of months after his address in Cologne. Minkowski is honored today by the fact that his spacetime is often called Minkowski spacetime or the Minkowski vector space.

Let's explore the structure of this Minkowski spacetime.

Suppose events A and B occur in inertial frame S at (t1, x1, y1, z1) and (t2, x2, y2, z2), respectively. Then define the interval (or proper time) between A and B as the possibly imaginary quantity
where , , and so forth. Notice that the interval squared is simply the difference of time separation squared and spatial separation squared.
We have seen that measurements of time and space are relative to the inertial frame in which an observer makes the measurements. However, it can be shown that the interval between two events is the same, regardless of the inertial frame of an observer. That is, the INTERVAL IS AN ABSOLUTE in Minkowski spacetime! The interval is to spacetime geometry what the distance is to Euclidean geometry. So, in a real sense, the special theory of relativity is misnamed. It is actually a theory of an absolute. Though the familiar quantities of time and distance are relative, the interval is an absolute quantity (or better yet, an invariant quantity).

An interval in which time separation dominates and is called timelike. An interval in which space separation dominates and is called spacelike. An interval for which is called lightlike. A single photon can be present at two events which are separated by a lightlike interval.

If we have a pair of axes, one of which represents space (the x-axis) and one of which represents time (the t-axis), then we can discuss spacetime diagrams. We can use the origin O to partition this two dimensional spacetime into several sets using the interval between event O and the other events in the plane. If the interval between O and an event is timelike, then the event will lie either in region F or region P. Region F consists of the "future events for O." Region P consists of the "past events for O." The boundary of these regions (here in red) is the collection of events separated from O by a lightlike interval. The events in region E are separated from O by a spacelike interval. The events in region F are those that event O can influence. The events in region P are the events which can influence event O. The events in region E are too far from event O to either be influenced by O or to influence O. That is, there isn't enough time for anything (not even photons) to be present at both event O and an event in region E. For this reason, there are certain causality restrictions between the different events in this spacetime. If someone were to shine a flashlight at O then the wavefront would travel out along the boundary between region F and region E. Notice that the slopes of these lines are ±1. This is because, in geometric units, the speed of light is 1.

We can introduce a second spatial axis and use spacetime diagrams to create light cones for given events. Here, the upper cone is the future light cone of the event at point O. The lower cone is the past light cone of event O. The surface of the lower lightcone is the collection of events which an observer at O can see, and the surface of the upper lightcone are events at which an observer could see event O.

If S and S' are inertial frames with S' moving relative to S with velocity v as shown here, then the coordinates of a given event which has coordinates (t,x,y,z) in the S frame and coordinates (t', x', y', z') in the S' frame will satisfy the equations:
This collection of equations is called the Lorentz Transformation. We can verify that the interval between two events is fixed under the Lorentz transformation.

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