Invariant Viewpoint

In the Star Trek Paradox mentioned previously, it became obvious that Kirk’s concept of time and space were different from the Klingon’s concept. More specifically, Figure 1.1 depicts Kirk’s view of reality while Figure 1.2 depicts the Klingon view. Furthermore, the two views of reality appear to be philosophically incompatible. If Kirk’s view is true, the Klingon’s view surely cannot be, and vice versa. Yet, the Theory of Relativity asserts that both of these viewpoints are equally valid descriptions of physical reality. How can this be possible?

Clearly it cannot be possible if everyone is talking about the same, single circle of light. Therefore, we conclude that Kirk and the Klingon must be talking about two different circles of light. In other words, space is relative. Kirk’s concept of space (which contains the circle of light he is centered on) must be different from the Klingon’s concept of space (which contains the circle of light he is centered on).

In order to see how this is possible, let us stop focusing our attention on the relativistic differences due to the two different viewpoints and try to identify those absolute, invariant aspects that everyone agrees upon. For example, both Kirk and the Klingon agree that the initial incident took place. Kirk sees the incident as shown in Fig. 1.1(a) and the Klingon sees it as shown in Fig. 1.2(a), but both agree that the incident was real and that a photon torpedo exploded at the instant the two spaceships were closest together.

In physics, we call an incident that takes place at a specific place at a specific time an event. Different observers may describe a given event differently, but all will agree that the event really and truly takes place. Therefore, events are one aspect of physical reality that are absolute. In other words, events are invariant, completely independent of who observes them. In fact, their existence is valid even if no one ever observes them.

So, if we picture the above strafing incident in terms of events rather than centers of circles, we may gain a clearer understanding of the situation. One way to do this is to plot the various events as a function of their location in space and time. In other words, we construct a spacetime diagram that maps not only the initial incident and the final situation, but also all of the events in between. Figures 1.3(a) and 1.3(b) show how this can be done from Kirk’s point of view and from the Klingon’s point of view, respectively.

Fig. 1.3 Spacetime diagrams of the strafing incident.

As you can see, for every significant event that occurs in reality, from the beginning through the end of the process, there is one and only one point on each of the diagrams. In other words, both diagrams depict the same set of events.

Since four independent coordinates are required to specify an event, the reality in which they are imbedded must be some kind of four-dimensional “event space”. This 4-D space is called spacetime because it contains events located in both space and time. And like the events that comprise it, spacetime is invariant. In other words, there are certain relationships between the events in spacetime that are completely independent of who may or may not observe them.

For example, Fig. 1.3 shows that the events comprising the expanding circle of light assume the shape of a cone, and that those comprising each spaceship assume the shape of a line (or tube). These light cones and worldlines are invariant aspects of spacetime, even though their appearance and orientation on a spacetime diagram may not be.

Notice that each diagram is drawn with the observer’s time-axis vertical. Therefore, the worldline of the Enterprise is vertical in the first diagram and slanted in second, while that of the Warbird is slanted in the first and vertical in second. In other words, the Enterprise’s worldline is in the same direction as the Enterprise’s concept of time, while the Warbird’s worldline is in the direction of the Warbird’s concept of time. This is another way of saying that time is relative – one observer’s concept of time is different from another observer’s concept of time.

We can also see from the diagrams how the two views of space are different. The Enterprise’s concept of space is a horizontal surface in Fig. 1.3(a) while the Warbird’s concept of space is a horizontal surface in Fig. 1.3(b). Since vertical lines in one diagram are not vertical in the other, it should be fairly obvious that horizontal lines in one diagram are not horizontal in the other.

In order to illustrate this fact, let us reconstruct the spacetime diagrams of Fig. 1.3 and include on both diagrams the two different circles of light. In order to simplify our terminology, let us adopt the convention of associating non-primed terms with the Enterprise and primed terms with the Warbird.

Fig. 1.4. Two different maps of the same spacetime.

As you can see, the light circle centered on the Enterprise is horizontal in the first spacetime diagram and slanted in the second, while the light circle centered on the Warbird is slanted on the first diagram and horizontal on the second.

The direction of slant and the angle of slant in each case must be as shown in the diagrams in order to keep the respective spaceships centered. Later we will quantify this statement, but in the meantime notice the following qualitative features. On a spacetime diagram, time lines and their corresponding space planes always tilt toward each other. In other words, if a timeline is rotated clockwise, the corresponding space plane is rotated counterclockwise, and vice versa. Furthermore, we will see later that the angle of rotation is the same for both time and space if the spacetime diagram is normalized by plotting both time and space to the same scale.

From Fig. 1.4 it now should be obvious how both Kirk and the Klingon can remain centered as the light expands outward in time. They are not centered on the same expanding circle of light. They are each centered on different circles.

Although we reached this same conclusion earlier, now we can understand how the two circles are related to one another. They are simply two different cross sections of the same light cone.

Since both of these arbitrary cross sections are perfect circles in spacetime, we conclude that the diameter of the light cone must be the same invariant constant in every direction of spacetime. Therefore, spacetime must have an invariant property corresponding to “the length of a line” or “the distance between events.” We call this invariant property the interval between events.

Closely related to the concept of distance in normal Euclidean three-dimensional space are the concepts of displacement and vectors. Such concepts also pertain to four-dimensional spacetime. We call these four-dimensional vectors four-vectors or 4-vectors and conclude that they are another invariant aspect of spacetime.

When two or more vectors are combined into a single entity in normal three-dimensional space, the result is a tensor. Such tensors have components whose values depend upon the particular coordinate system used to describe them. But the tensors, themselves, are physical quantities completely independent of the coordinate system. The same is true for tensors in four-dimensional spacetime, their components are relative quantities but the tensors themselves are invariant.

Other invariant aspects of four-dimensional spacetime exist, but these examples should suffice to illustrate the fact that not everything in Relativity is relative. There are many invariant concepts in Relativity that help us visualize and understand many of the mysterious aspects of reality that are completely hidden when we view the universe only from a time-varying –three-dimensional perspective.

Questions:

Q1. When we say events are invariant, we mean (A) the detailed aspects of events appear the same to all observers, (B) events occur at the same place and same time according to all observers, (C) all observers agree that the same events occur in reality, (D) More than one of these. (E) None of these.
C 516

Q2. Which of the following are not invariant aspects of spacetime? (A) events, (B) light circles, (C) worldlines, (D) intervals, (E) absolute future.
B 516

Q3. Which are true regarding spacetime diagrams? (A) Tilted time lines always represent moving observers. (B) Tilted space planes always represent moving observers. (C) Light cones always represent rest observers? (D) Two of these. (E) Three of these.
D=A+B 516

Q4. According to special relativity, we live in (A) a time-varying three-dimensional reality, (B) an invariant four-dimensional reality, (C) a multiple universe in which reality is dependent upon the perception of the observer, (D) More than one of these. (E) None of these.
B 516

Events are Invariant –.Events are things that happen at a particular place and time, they are like points in reality space
Spacetime is Invariant –.Spacetime is the four-dimensional space consisting of all the events that occur in reality.
Light Cones are Invariant –.A light cone consists of all the light rays emanating from or converging on a given event.
Worldlines are Invariant –.A worldline is a line in spacetime, especially a line associated with location of a particle existing in time.
Intervals are Invariant –.An interval is the distance between two events in spacetime.
Four-Vectors are Invariant –.A four-vector is the four-dimensional extension of the concept of a vector.
Tensors are Invariant –.A tensor is a physical quantity simultaneously associated with two or more directions.

ModPhy1/Unit1/SpecialRelativity/