Living in Three Dimensions? Think Again...
One of the most common misconceptions regarding the concept of spacetime is that time is the fourth dimension. Perhaps you may have heard someone make this statement before. Well, although this statement is technically true, it is only a partial truth. This misconception arises due to the tendency to consider spacetime as “space-time,” or as the familiar three spatial dimensions in which we perceive ourselves as living joined with one dimension of time. If this view is not entirely correct, then, what is really meant when physicists refer to this concept called spacetime?
What is spacetime?
For the purposes of this discussion, we will be talking purely about flat spacetime (also called Minkowski spacetime). What do I mean by “flat” spacetime? By “flat,” I am referring to the standard conventions of Euclidean geometry, i.e. the interior angles in a triangle sum to 180 degrees, the circumference of a circle is 2πR, etc. This advance identification is important because in so-called “curved” geometries and spacetimes, the results expected from Euclidean geometry are not guaranteed to hold; for instance, the sum of the interior angles in a triangle can be greater or less than 180 degrees in spherical and hyperbolic geometries, respectively. With this clarification in mind, let us return to the task of defining, and consequently constructing, spacetime.
To do this, we will re-examine the misconception that I posited in the introduction: that spacetime consists of three spatial dimensions and one time dimension. The reason that this statement is only a partial truth is that every coordinate in our universe can be described by four spatial coordinates. These coordinates are the traditional three-dimensional spatial coordinates (whether these are Cartesian, cylindrical polar, spherical polar, etc.) along with a fourth, so-called “Lorentz coordinate” given by ct, where c is the speed of light and t is a point in time. The Lorentz coordinate is a spatial coordinate since [c] (meaning “the units of c”) = m/s and [t] = s, so [ct] = (m/s)*s = m, the units of position. Therefore, when physicists mention the concept of spacetime, they refer to the four-dimensional space consisting of the familiar three-dimensional spatial coordinates as well as one coordinate given by an object’s position in time. These four coordinates combined can describe any point (called an “event”) in the observable universe.
Before moving on, I want to address one final idea regarding the construction of Minkowski spacetime. As discussed, the coordinates of an event in spacetime are given by a Lorentz coordinate ct along with the traditional three-dimensional spatial coordinates. By convention, such an event is labeled as x^u, where the superscript u = 0, 1, 2, 3 simply denotes whether the given coordinate is Lorentzian or spatial; in a Cartesian coordinate system, for instance, a superscript of 0 corresponds to the Lorentz coordinate ct while the superscripts of 1, 2, and 3 correspond to the Cartesian coordinates of x, y, and z, respectively. Subsequently, physicists define the quantity (x^u)^2 = -(x^0)^2 + (x^1)^2 + (x^2)^2 + (x^3)^2. The minus sign preceding the square of the Lorentz coordinate presents some interesting geometrical implications, as we shall see.
How can we visualize spacetime?
To visualize spacetime, we will use a tool called a “spacetime diagram.” Specifically, since we have been discussing Minkowski spacetime, we will use a Minkowski spacetime diagram. For simplicity, our spacetime diagram will be two-dimensional; that is, it will consist of a ct-axis and an x-axis. This is completely analogous to a standard graph of the xy-plane in two dimensions, where in this case ct takes the role of y.
As you can see in Fig. 1, our spacetime diagram plots ct as a function of x (the ct’ and x’ axes correspond to “boosted” frames of reference, a topic I will mention shortly; you can ignore these axes for now). The line x = ct is an asymptote with a slope of c, meaning that an object traveling at speed c travels along this asymptote as t increases. For those who are curious, this result is produced by taking the total time derivative of both sides of the equation. Additionally, this asymptote is a consequence of one of the fundamental principles of relativity: the speed of light is c in all inertial reference frames, meaning that regardless of an observer’s speed relative to the speed of light, light will always escape the observer at speed c.
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| Fig. 1. A standard Minkowski spacetime diagram. |
Returning to the notion of boosted frames of reference I mentioned previously: we must first develop an understanding of how boosted frames are represented in Minkowski spacetime diagrams before we can explore a truly remarkable application of such a diagram. Say we have two observers, O and O’ (read “O prime”), where observer O is stationary in his frame of reference and O’ is moving past him with velocity v. Without delving too deeply into the specifics, the faster that O’ moves relative to O, the shorter his length appears to O. Likewise, in O’’s frame of reference, he is stationary while O moves past him with velocity -v, resulting in O’s length appearing shorter to O’. This is an example of the special relativistic phenomenon of length contraction, whereby objects moving with velocity v close to the speed of light appear length contracted to stationary observers. The consequence of this relativistic phenomenon for our Minkowski spacetime diagram is that the reference axes for the observer moving with velocity v are “boosted” toward the x = ct asymptote: as the observer’s speed approaches the speed of light, the reference axes converge upon this asymptote. Thus, in our spacetime diagram (Fig. 1), the primed axes x’ and ct’ are the reference axes for an observer O’ moving with velocity v relative to a stationary observer O whose reference axes are x and ct.
The Relativistic Rocket: An Application of the Spacetime Diagram
We are now ready to demonstrate a remarkable use of the Minkowski spacetime diagram: the visualization of length contraction and the relativity of simultaneity (another relativistic phenomenon) through the case of the relativistic rocket. Imagine that we once again have two observers, O and O’, where O’ is traveling on a rocket with velocity v past O, who is stationary in his own frame of reference. According to O’, the rocket has length L0. Additionally, the rocket that O’ is traveling on has a flashlight attached to either end, and each flashlight emits a light signal at the instant that the origins of the reference frames of O and O’ overlap, i.e. (0, 0) = (0, 0)’. Let these emissions be denoted as events A and B. The Minkowski spacetime diagram can help us answer two distinct questions about this situation: 1) How does the length of the rocket as measured by O compared with the length measured by O’, and 2) are events A and B simultaneous for both observers if they each receive both signals simultaneously?
Suppose observer O measures the length of the rocket to be L. In order to compare the value of L with L0, let us consult our spacetime diagram for the situation, Fig. 2. As can be seen in the diagram, events A and B each occur at a constant position x’ according to O’.
The distance x between lines of constant x’ is the length of the rocket as measured by O, and this measured length L is in fact shorter than L0, the length as measured by O’. This is the result we expect, since objects moving at velocity v past a stationary observer appear length contracted to that observer. A note of caution: while it might be tempting to use the Pythagorean theorem to solve for L, the fact that the square of the Lorentz coordinate is preceded by a minus sign (mentioned previously) means that L^2 - (ct)^2 = (L0)^2; using the Pythagorean theorem suggests, erroneously, that L is actually longer than L0, which we know is not the case.
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| Fig. 2. Spacetime diagram for a relativistic rocket. |
Now for the remarkable question: do events A and B occur simultaneously for both O and O’? In O’’s frame of reference, the rocket lies on a line of constant ct’, meaning that events A and B also lie on a line of constant ct’. This means that, according to O’, events A and B occur simultaneously since ct’ is the same for each event, and thus no time interval occurs between events. In O’s frame of reference, however, something surprising happens: as you can see in Fig. 2, event B occurs at an earlier time ct than event A. In fact, O reasons that since the light signals reached him simultaneously, event B must have occurred first in order for the light from B to reach him at the same time as the light from A. Thus, there is a time interval between events A and B in O’s frame of reference, and the events are therefore NOT simultaneous according to O. Therefore, we have seemingly discovered another fundamental principle of special relativity using our Minkowski spacetime diagram: simultaneity is relative. How exciting!
Lengths are contracted at relativistic speeds, and simultaneity is relative. These are two incredibly remarkable results of simple thought experiments using the powerful concept of four-dimensional spacetime. Other famous results, such as the radius of the event horizon of a black hole, also known as the Schwarzschild radius, emerged from calculations performed in curved spacetimes. The crucial concept to learn from all of this is that we truly do live in a four-dimensional universe, and it is precisely the four-dimensional nature of spacetime that makes it such a powerful and versatile tool.
Bonus Comic
Revisiting the Relativistic Rocket
For those who would like a deeper look into the example of the relativistic rocket that I discussed in this article, please click here to watch a YouTube video that addresses the topic in more detail.
Sources
Brown, H.R., Pooley, O. (2006) Chapter 4: Minkowski Space-Time: A Glorious Non-Entity. Philosophy and Foundations of Physics, Vol. 1, pp. 67-89. https://doi.org/10.1016/S1871-1774(06)01004-7
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