Section 17–1 The geometry of space-time

(Analogy of spacetime / Paths in spacetime / Axes of spacetime)

In this section, Feynman discusses the concept of spacetime from the perspective of an analogy, possible paths, and the axes of spacetime diagram. However, it is closely related to Minkowski diagram, which is a 2-D graph that helps to visualize events, worldlines, and causal structure. For this reason, the section could be titled Minkowski Diagram instead of simply Spacetime Geometry.

 

1. Analogy of spacetime:

“An analogy is useful: When we look at an object, there is an obvious thing we might call the ‘apparent width,’ and another we might call the ‘depth’ (Feynman et al., 1963, section 17–1 The geometry of space-time).”

 

Feynman explains the concept of spacetime using the “apparent width” and “apparent depth” of an object that are not its fundamental properties. In other words, a given depth is a kind of “mixture” of another depth and width that is similar to the two equations: x′ = xcos θ + ysin θ, y′ = ycos θ – xsin θ. However, one may add that there are no absolute space and absolute time that are different from absolute space-time intervals. In general, the concepts of space and time are interrelated such that there is “no space without time” or “no time without space.” In other words, space and time lose their absolute meaning—no observer can access “pure space” or “pure time”. Instead, reality reveals itself through spacetime intervals, which blend space and time into a unified geometric structure.

 

In Minkowski’s (1907) words, “[f]rom now onwards space by itself and time by itself will recede completely to become mere shadows and only a type of union of the two will still stand independently on its own (p. 111).” The shadow projection analogy is a simple way to idealize how different observers perceive spacetime in Minkowski geometry. Just as tilting a light source changes the shape of a shadow, relative motion causes a rotation* in spacetime from the viewpoint of an observer. However, this analogy has some limitations, e.g., it does not explain the hyperbolic geometry involved. The x′-axis and ct′-axis are tilted via hyperbolic functions (cosh, sinh), not the regular sine/cosine. Philosophically, our perceived spacetime is like a projection, similar to the shadows projected by the fire in Plato’s cave.

 

* Poincaré explicitly used the word rotation in his 1905 paper Sur la dynamique de l’électron. He wrote: “We can form combinations devised by Lie, such as … but it is easy to see that this transformation is equivalent to a change of coordinates; the axes are rotating a very small angle around the z-axis.” This shows that Poincaré recognized Lorentz transformations as infinitesimal rotations in four-dimensional space, by using Lie group theory. Later, in his Autobiographical Notes (1946), Einstein recalled: “Minkowski showed that the Lorentz transformation (apart from a different algebraic sign due to the special character of time) is nothing but a rotation of the coordinate system in the four-dimensional space (p. 59).” In effect, Minkowski reinterpreted and expanded Poincaré’s notion of rotation, developed a geometric formulation of special relativity in which Lorentz transformations appear as hyperbolic rotations in four-dimensional spacetime. (Sommerfeld and Varičak made profound and complementary contributions to the development of the concept of hyperbolic rotations in special relativity.)

 

2. Paths in spacetime:

“This new world, this geometrical entity in which the “blobs” exist by occupying position and taking up a certain amount of time, is called space-time (Feynman et al., 1963, section 17–1 The geometry of space-time).”

 

According to Feynman, the geometric entity in which “blobs” exist—occupying a range of positions over a period of time—which is called spacetime. (In other words, spacetime is the unification of the three dimensions of space and the one dimension of time into a single, four-dimensional continuum that serves as the stage for all physical events.) A specific point in space-time, defined by coordinates (x, y, z, t) is referred to as an event. To visualize this concept, we often use a Minkowski diagram: a two-dimensional graph that simplifies space to a single spatial dimension (x) and combines it with time (t). This diagram plots a sequence of events (also called world-points) that represent the history of an object. The continuous path that an object traces through space-time on such a diagram is known as its world line (Minkowski, 1907).

 

“If the particle is standing still, then it has a certain x, and as time goes on, it has the same x, the same x, the same x; so its “path” is a line that runs parallel to the t-axis (Feynman et al., 1963, section 17–1 The geometry of space-time).”

 

Feynman describes a stationary object in a space-time diagram as having a constant position x; as time progresses, it has the same x whereby its “path” is a line that is parallel to the t-axis. In contrast, a path parallel to the x-axis would imply that the object is present at all positions simultaneously at a single instant in time—an impossibility for any physical object with mass. To reflect the finite speed of light, the time axis is often scaled by c, so the vertical axis is labeled ct. In this convention, a light ray's world line bisects the angle between the x and ct axes, emphasizing that light moves at the same speed in all inertial frames. This is related to the amazing fact that the Lorentz transformations preserve the spacetime interval c²t²-x² that is invariant under all inertial frames.

 

Note: One might think of the x-axis as a “line of now,” and the t-axis as a “line of here.” (Mermin, 2009).

 

3. Axes of space-time:

“Note, for example, the difference in sign between the two, and the fact that one is written in terms of cos θ and sin θ, while the other is written with algebraic quantities. (Of course, it is not impossible that the algebraic quantities could be written as cosine and sine, but actually they cannot.) (Feynman et al., 1963, section 17–1 The geometry of space-time).”

 

A given event can be represented using different axes of x′ and t′, but it is not exactly the same mathematical transformation as shown by the two equations: x′ = xcosθ + ysinθ, y′ = ycosθ – xsinθ, where the Pythagorean distance is preserved: x²+y²=constant. Feynman notes that one might expect Lorentz transformations to be expressed with sine and cosine in the same way, but they cannot. The key difference is that spacetime preserves a different distance, namely the invariant interval (ct)²−x²=constant, which has a minus sign instead of a plus. Because of this, Lorentz transformations are not circular rotations but hyperbolic rotations, described by hyperbolic functions: ct′=ctcosh⁡ϕ−xsinh⁡ϕ, x′=−ctsinh⁡ϕ+xcosh⁡ϕ, where ϕ is called the rapidity (or hyperbolic angle). In short, rotations in ordinary space are described by cos and sin, whereas transformations in spacetime prefer cosh and sinh.

 

“In fact, although we shall not emphasize this point, it turns out that a man who is moving has to use a set of axes which are inclined equally to the light ray, using a special kind of projection parallel to the x′- and t′-axes (Feynman et al., 1963, section 17–1 The geometry of space-time).”

 

Feynman could have explained the tilt of the x′ and t′ axes using reasoning similar to that as shown below:

 

 

Review questions:

1. How would you provide an analogy of space-time?

2. How would you describe different paths of an object in space-time diagrams?

3. How would you explain the axes of x′ and t′ in space-time diagrams?

 

Key Takeaways (In Feynman’s Style):

The power of Minkowski diagrams lies in their link to Lorentz transformations—which preserve the spacetime interval. This is more than algebra—it’s geometry in motion! Think of spacetime as a four-dimensional stage where events shift under transformations that resemble hyperbolic rotations. Unlike ordinary rotations in space, this is a hyperbolic transformation in a spacetime geometry—stranger than our everyday experience, but beautifully precise. Still, Minkowski diagrams remain an idealization, because they:

• neglect gravitational curvature,
• leave out visual/optical appearance,
• exclude non-inertial frames and accelerations,
• assume perfect measurements, and
• ignore quantum effects.

 

“We shall not deal with the geometry, since it does not help much; it is easier to work with the equations (Feynman et al., 1963, section 17–1 The geometry of space-time).”

Interestingly, Feynman remarked that the geometry of spacetime does not help much in special relativity. Einstein also initially resisted Minkowski’s geometric geometric interpretation: “Since the mathematicians have invaded the relativity theory, I do not understand it myself any more.” For Einstein, special relativity in 1905 was a physical theory grounded in concrete equations rather than abstract geometry. He even described Minkowski’s formulation as superfluous learnedness (Pais, 1982, p.152). Yet, by 1910–1912, Einstein began to appreciate that the diagrammatic picture (worldlines, light cones, tilted axes) made the symmetry and invariants of spacetime visually and conceptually clear. This change of perspective carried a profound lesson—equations alone can state relations, but geometry can uncover hidden structure. Indeed, Einstein’s eventual embrace of Minkowski’s geometric viewpoint paved the way for general relativity, where the curvature of spacetime itself became the essence of gravitation.

 

In short: Einstein was initially skeptical of spacetime diagrams and favored equations, he later embraced Minkowski's geometric framework, recognizing its unique power to visualize spacetime’s structure.

 

References:

Einstein, A. (1969). Autobiographical notes. In Albert Einstein: Philosopher-Scientist. Paul A. Schilpp, ed., 3rd ed. Illinois: Open Court.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Mermin, N. D. (2009). It’s about time: understanding Einstein’s relativity. Princeton: Princeton University Press.

Minkowski, H. (1907). Space and Time. In Petkov, V., Ed. Minkowski’s Papers on Relativity. Moscu: Minkowski Institute Press.

Pais, A. (1982/2007). " Subtle is the Lord...": the science and the life of Albert Einstein. Oxford: Oxford University Press.

Sommerfeld, A. (1910). Zur Relativitätstheorie. I. Vierdimensionale Vektoralgebra. Annalen der Physik, 337(9), 749-776.

Varićak, V. (1912). Uber die nichteuklidische Interpretation der Relativtheorie. Jahrb. dtsch. math. Verein, 21 (1912) 103-127.