Section 17–3 Past, present, and future

(Affective past / Affective future / Right now)

In this section, Feynman discusses three regions of space-time graph, namely, affective past, affective future, and right now.

1. Affective past:

“Therefore region 2 is sometimes called the affective past, or affecting past; it is the locus of all events which can affect point O in any way (Feynman et al., 1963, section 17–3 Past, present, and future).”

According to Feynman, region 2 is sometimes called the affective past (or affecting past) because it is the locus of all events which can affect point O. Phrased differently, all points of region 2 are in the “past” of O, and anything that happens in this region could affect O. Physics teachers should add that in region 2 (t < 0), the space-time interval between two events O and Q can be expressed as x² < c²t² and the square of the time-like interval (Ds)² is greater than zero. This means that the two events, O and Q, are related causally and they are capable of interacting physically. For example, an observer initially at event Q could move to the position of event O before O occurs. In other words, the observer at Q can influence O (e.g., a transfer of energy) and the two events are separated by a time-like interval that can be causally related.

Feynman elaborates an interesting physical relationship from region 2 (past) to the point O (here and now): a physical object or a signal may move from a point in region 2 to the event O at a speed that is less than the speed of light. For example, an object at Q can move to O at a certain speed less than c, so if this object was in a space ship and moving, it would be located at a point in the past (or region 2). Physics teachers could explain that the principle of causality in special relativity is related to the second postulate. That is, two events O and Q can interact physically, but it occurs while no physical signal can move faster than c. Furthermore, based on the principle of causality, the cause must precede its effect is consistently observed by all inertial observers.

2. Affective future:

“So this is the world whose future can be affected by us, and we may call that the affective future (Feynman et al., 1963, section 17–3 Past, present, and future).”

Feynman explains that region 3 is a region which we can affect from O, for example, we can “hit” things by shooting “bullets” out at speeds less than c. These include spatial points whose future can be affected by us, and he names this region as affective future. We may add that in region 3 (t > 0), the space-time interval between two events O and A (a point in affective future) can be expressed as x² < c²t² and the square of the interval (Ds)² is also time-like. In addition, if the interval is time-like, i.e., (Ds)² > 0, then t_A > t_O remains the same in every inertial frame. This is in contrast to quantum causality in which the causal order of events is not always fixed whereby instantaneous movement is assumed possible.

One may hope Feynman to discuss the principle of causality. Interestingly, the principle of causality as “an effect cannot occur before its cause” is related to the temporal order of events. In Bridgman’s (1927) words, “[i]t appears then, that the fundamental postulate of relativity (that the form of natural laws is the same in all reference systems) demands that the temporal order of events causally connected be the same in all reference systems (p. 87).” Furthermore, the light postulate has the implication that no object or signal can move faster than the speed of light from the perspective of inertial observers. This restriction of speed also makes an assumption about the temporal order of causally connected events (Bridgman, 1927).

3. Right now:

“What we mean by ‘right now’ is a mysterious thing which we cannot define and we cannot affect, but it can affect us later… (Feynman et al., 1963, section 17–3 Past, present, and future).”

Feynman says that the interesting thing about region 1 (the remaining region of space-time) is that we can neither affect it now from O, nor can it affect us now at O, because nothing can move faster than the speed of light. If the sun is exploding “now,” it takes eight minutes for the signal or energy to reach us, and it cannot possibly affect us earlier. However, Feynman did not clearly specify region 1 as “right now,” but it is named by some physicists as “elsewhere” (Thorton & Marion, 2004), “causal present” (Rindler, 2003), or “present” (Resnick, 1968). Generally speaking, O and R are separated by a space-time interval that is space-like, i.e., (Ds)² < 0. The separation between the two events is such that c²t² < x² whereby the influence between two events is limited by the speed of light.

Feynman describes right now as a mysterious thing which we cannot define and affect, but it can affect us later if we had done something far enough in the past. When we look at the star Alpha Centauri, we are seeing the light from the star that was emitted four years ago; we may wonder whether the star still exists “now.” In essence, two events that are separated by a space-like interval (e.g., Alpha Centauri is four light-years away from us) have no absolute time sequence. The “now” is dependent on the reference frame of an inertial observer, that is, the problem of simultaneity is not a unique thing has been elucidated by Einstein. More important, each event in region 1 may be identified as simultaneous with O (here and now) in a certain inertial frame.

Questions for discussion:

1. How would you define the region of affective past?

2. How would you define the region of affective future?

3. How would you define the region of now (or elsewhere)?

The moral of the lesson: two events that are separated by a time-like interval have the same temporal order, and this is different from two events that are separated by space-like intervals that have no absolute order of time.

References:

1. Bridgman, P. W. (1927). The Logic of Modern Physics. New York: Macmillan.

2. 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.

3. Resnick, R. (1968). Introduction to Special Relativity. New York: Wiley.

4. Rindler, W. (2003). Relativity: special, general, and cosmological. Oxford: Oxford University Press.

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

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.