Section 17–2 Space-time intervals

(Invariant intervals / Simplified intervals / Signs of interval squared)

In this section, Feynman discusses space-time intervals from the perspectives of invariance, simplification using the speed of light = 1, and the signs of interval squared.

1. Invariant intervals:

“…we have here, also, something which stays the same, namely, the combination c²t²−x²−y²−z² is the same before and after the transformation: c²t′²−x′²−y′²−z′² = c²t²−x²−y²−z² (Feynman et al., 1963, section 17–2 Space-time intervals).”

Feynman defines mathematically a space-time interval as c²t²−x²−y²−z². This quantity is invariant and real like the distance in three-dimensional space; it is also called the interval between the two space-time points whereby one of which is at the origin. We can use Lorentz transformation equations to demonstrate that the combination c²t²−x²−y²−z² is the same before and after the transformation: c²t′²−x′²−y′²−z′² = c²t²−x²−y²−z². As an alternative, we can represent the space-time interval as (Ds)² = c²(Dt)²−(Dx)² in which Dt = t(event 2) – t(event 1) and Dx = x(event 2) – x(event 1). Furthermore, some textbook authors may define the space-time interval using different signs: +x²+y²+z²−c²t² (e.g., Thornton, & Marion, 2004).

Feynman explains that the space-time interval is similar to the square of the distance x²+y²+z² that remains unchanged if we rotate the axis, such as x, y, and z. Thus, it is possible to have some functions of coordinates and time which are independent of the coordinate system based on the Euclidean geometry. Specifically, the geometry of space-time is hyperbolic geometry such that the space-time interval is invariant. Simply put, the space-time interval is the same from the perspectives of all inertial observers that travel at different speeds. The space-time interval is invariant because the speed of light is constant in all inertial frames.

2. Simplified interval:

“If time and space are measured in the same units, as suggested, then the equations are obviously much simplified (Feynman et al., 1963, section 17–2 Space-time intervals).”

Feynman suggests getting rid of the c in the space-time interval such that we can have a wonderful space with x’s and y’s that can be interchanged. It helps to see the clarity and simplicity of the space-time interval instead of measure space and time in two different units. If we were to measure all distances and times in the same units, say seconds, then the unit of distance is equivalent to 3×10⁸ meters, and the interval would be simpler. Later, Feynman adds that “[i]nstead of having to write the c², we put E = m, and then, of course, if there were any trouble we would put in the right amounts of c so that the units would straighten out in the last equation, but not in the intermediate ones (Feynman et al, 1963, section 17–4 More about four-vectors).” Similarly, we have chosen the appropriate units such that F = kma is simplified to F = ma.

Feynman simplifies the space-time interval using a system of units in which c = 1 to obtain t′²−x′²−y′²−z′² = t²−x²−y²−z². He explains that it is much easier to remember the equations without the c’s in them, and it is always easy to put the c’s back, by simply checking the dimensions. For example, we cannot subtract a velocity squared as in √1−u², which has units, from the pure number 1; thus, we must divide u² by c² in order to achieve unitless in the expression. However, Feynman has also used ct instead of t for the vertical axis of space-time diagrams. Some physicists explain that it is convenient to use ct instead of t for the vertical axis in space-time diagrams.

3. Signs of interval squared:

“… if two objects are at the same place in a given coordinate system, but differ only in time, then the square of the time is positive and the distances are zero and the interval squared is positive… (Feynman et al., 1963, section 17–2 Space-time intervals).”

Feynman mentions that the square of an interval (t²−x²−y²−z²) may be either positive or negative, unlike distance, which is positive. When an interval is imaginary, it means that two events have a space-like interval between them because the interval is more like space than like time. On the other hand, if two events occur at the same place, but differ only in time, then the square of the time is positive and the distances are zero and the interval squared is positive; this is called a time-like interval. In short, the squared interval s² > 0 means that the “time part of interval is greater than the space part” (t² > x²+y²+z²) and it is known as a time-like interval (Taylor & Wheeler, 1992). If the squared interval s² < 0, it means that the “space part of interval is greater than the time part” (x²+y²+z² > t²) and it is known as a space-like interval.

Feynman elaborates that there are two lines at 45^o in the space-time diagrams (in four-dimensional space-time, there will be light “cones”) and points on these two lines are at zero interval from the origin. In other words, the locations where light reaches are always separated from its origin by a zero interval as expressed by t²−x²−y²−z² = 0. Importantly, the speed of light is the same in all inertial frames means that the interval is zero in all inertial frames, and thus, to state that the speed of light is invariant is equivalent to saying the space-time interval is zero. In addition, we may add that the squared interval s² = 0 means that the “time part of interval is equal to the space part” (t² = x²+y²+z²) and it is known as a light-like interval. Mathematically, it can also be represented as c²t² = x² and |x/t| = c that holds in all inertial frames.

Questions for discussion:

1. How would you explain that the space-time interval is invariant?

2. Would you express the space-time interval as c²t²−x²−y²−z² or t²−x²−y²−z²?

3. How would you explain the signs of space-time intervals?

The moral of the lesson: the space-time interval c²t²−x²−y²−z² remains invariant after the transformation and it can be classified as space-like (x²+y²+z² > c²t²), time-like (c²t² > x²+y²+z²), and light-like (c²t² = x²+y²+z²).

References:

1. 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.

2. Taylor, E. F. & Wheeler, J. A. (1992). Spacetime Physics (2^nd Edition). New York: W. H. Freeman and Co.

3. Thornton, S. T. & Marion, J. B. (2004). Classical Dynamics of Particles and Systems (5^th Edition). Belmont, CA: Thomson Learning-Brooks/Cole.

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.

Section 16–5 Relativistic energy

(Energy has mass / Mass has energy / Rest energy)

In this section, Feynman discusses the concept of “energy has mass” and “mass has energy,” as well as the rest energy that may not be known in a physical process.

1. Energy has mass:

“… the excess mass of the composite object is equal to the kinetic energy brought in. This means, of course, that energy has inertia (Feynman et al., 1963, section 16–5 Relativistic energy).”

According to Feynman, if a proton and a neutron are “stuck together” (and can still be “seen”), the total mass M should be to 2m_w instead of 2m₀. This is because the excess mass of the composite object is due to the kinetic energy and it means that energy has inertia. In chapter 7, Feynman explains that “anything which has energy has mass—mass in the sense that it is attracted gravitationally (Feynman et al., 1963).” In the last chapter, Feynman has also shown that moving gas molecules are heavier because of the kinetic energy. Similarly, in volume II, Feynman states that “the energy E₀ has the relativistic mass E₀/c² the photon has a mass (not rest mass) ℏω₀/c² and is ‘attracted’ by the earth (Feynman et al., 1964, section 42–6 The speed of clocks in a gravitational field).” It is essentially related to the principle of equivalence of energy and mass.

According to Wilczek (2005), “[s]tated as m = E/c², Einstein’s law suggests the possibility of explaining mass in terms of energy. That is a good thing to do because in modern physics energy is a more basic concept than mass (p. 863).” Feynman elaborates that if two particles join together and produce potential energy or any other form of energy, then the mass of the composite object is equivalent to the total energy that has been put in. In other words, the conservation of mass is equivalent to the conservation of energy and there is no strictly inelastic collision in the special theory of relativity. An inelastic collision is a collision in which the total kinetic energy of the two colliding particles is not the same after the collision as it was before; the so-called loss of kinetic energy may appear as part of the mass of the composite object.

2. Mass has energy:

“How much energy will they have given to the material when they have stopped? Each will give an amount (m_w−m₀)c²… (Feynman et al., 1963, section 16–5 Relativistic energy).”

The concept of “mass has energy” can be shown by the famous equation E = Dmc². Feynman explains that the total energy released can be calculated using the equation (m_w−m₀)c² and some energy is left in the material as thermal energy, potential energy, or other forms of energy. Furthermore, Einstein was thinking more about the origin of inertia or mass instead of making bombs. One may clarify this using Wilczek’s (2005) words: “[t]he usual way of writing the equation, E = mc², suggests the possibility of obtaining large amounts of energy by converting small amounts of mass. It brings to mind the possibilities of nuclear reactors or bombs… Actually, Einstein’s original paper does not contain the equation E = mc², but rather m = E/c² (p. 863).”

Feynman used the equation E = mc² to estimate the energy liberated under fission in an atomic bomb. Historically, the energy that should be liberated when an atom of uranium undergoes fission was estimated before the first direct test of atomic bomb. Interestingly, Feynman adds that if Einstein’s formula had not worked, they would have measured it anyway. One may be surprised that Feynman used the word “they.” In his autobiography, Feynman (1997) says that “we decided that the big problem -- which was to figure out exactly what happened during the bomb’s implosion, so you can figure out exactly how much energy was released and so on -- required much more calculating than we were capable of. A clever fellow by the name of Stanley Frankel realized that it could possibly be done on IBM machines… (p. 125).”

3. Rest energy:

“…we do not have to know what things are made of inside; we cannot and need not identify, inside a particle, which of the energy is rest energy of the parts into which it is going to disintegrate (Feynman et al., 1963, section 16–5 Relativistic energy).”

Feynman discusses the question of whether we could always add the rest energy m₀c² to the kinetic energy to determine the total energy of an object is mc². In a sense, this is possible if we are sure of the component pieces of rest mass m₀ inside a composite object that has a mass of M. Generally speaking, we may not be always sure of (or cannot “see”) the parts inside, for example, a K-meson may disintegrate into two pions or three pions. The K-mesons are also known as kaons that helped to understand the problem of parity violation (and CP violation). Feynman cites this example possibly because he and Gell-Mann (1958) developed a theory of weak interactions to explain the parity violation.

Although Feynman promotes the concept of relativistic mass, he ends the chapter by providing two equations that only include rest mass: E²−p²c² = m₀²c⁴ and Pc = Ev/c. The two equations are useful because we can use them to find the velocity v, momentum P, or the total energy E of an object. One may argue that the two equations are also useful because they do not require the concept of relativistic mass and are applicable to photons. However, Feynman did not seem to be aware of particle physicists that prefer the concept of invariant mass. Physicists that oppose the use of relativistic mass may cite the last two sentences of this chapter and explain that relativistic mass is rarely used (as mentioned by Feynman).

Questions for discussion:

1. How would you explain that “energy has mass” or “energy has inertia”?

2. How would you explain that “mass has energy” or “how energy is released”?

3. Would you cite the last two sentences of the chapter to explain that the relativistic mass is useless?

The moral of the lesson: “energy has mass” and “mass has energy,” but it is debatable whether the concept of relativistic mass is useful.  

References:

1. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

2. Feynman, R. P., & Gell-Mann, M. (1958). Theory of the Fermi interaction. Physical Review, 109(1), 193-198.

3. 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.

4. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

5. Wilczek, F. (2005). Nobel Lecture: Asymptotic freedom: From paradox to paradigm. Reviews of Modern Physics, 77(3), 857-870.