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 space-time / Paths in space-time / Axes of space-time)

In this section, Feynman discusses the geometry of space-time from the perspectives of an analogy, possible paths of an object, and the axes of a space-time diagram.

1. Analogy of space-time:

“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 space-time using the “apparent width” and “apparent depth” of an object that are not its fundamental properties. This is because we get a different width and a different depth if we view the object from a different angle. One may also say that a given depth is a kind of “mixture” of all depth and all width that is similar to the two equations: x′ = xcosθ + ysinθ, y′ = ycosθ − xsinθ. Importantly, the analogy of space-time has its limitations. That is, there are no absolute space and absolute time that are in contrast to relative space-time. In general, the concepts of space and time are interrelated such that there is no “space without time” or “time without space” from the perspectives of observers in different reference frames.

Feynman elaborates that depth and width are just two different aspects of the same thing. Historically, 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).” Minkowski also explains that there is an infinite number of spaces analogously as there is an infinite number of planes in three-dimensional space. In a sense, time and space are analogous to the extent that the space-time interval can be expressed as Sa_i² (to be discussed in the next section). On the other hand, the temporal dimension is different from the spatial dimension if we express the space-time interval as +(ct)²–x² in which they can be distinguished by the different signs.

2. Paths in space-time:

“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).”

According to Feynman, the geometrical entity in which the “blobs” exist by occupying a range of position and occurring during a certain amount of time is called space-time. In addition, a given point (x, y, z, t) in space-time is called an event. Alternatively, we may explain that a Minkowski space-time diagram is a two-dimensional graph that depicts a sequence of events (or world-points) corresponding to the history of an object. We may define a world-point (or an event) as something that occurs at a point in space-time in which it has a definite location and a definite time. Furthermore, the path of an object that is shown in the space-time diagram is also called the world line (Minkowski, 1907).

Feynman describes a stationary object in a space-time diagram as having a certain x, and as time goes on, it has the same x whereby its “path” is a line that is parallel to the t-axis. On the other hand, it is impossible to have a path that is parallel to the x-axis which implies the object exists in everywhere at the same time. (We may interpret the x-axis as a “line of now,” whereas the t-axis as a “line of here.”) However, one may clarify that the t-axis of a space-time diagram is scaled with the speed of light c and labeled by ct such that a light ray’s path is shown as a bisector of the angle between the two axes. This is related to the amazing fact that the Lorentz transformations preserve the quadratic form c²t²-x² and thus, they can be regarded geometrically as a rotation in a four-dimensional space. Poincare should be credited for his geometric interpretation of Lorentz transformations of space-time.

3. Axes of space-time:

“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).”

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. Curiously, Feynman mentions that it is not impossible that the algebraic quantities could be written in terms of cosine and sine, and adds that “but actually they cannot.” However, we can relate x and x′ using the following two forms of equations: x = ax′ + bt′ and x′ = ax - bt. More important, we should not assume one unit of distance on different axes of x and x′ must have the same length (French, 1971). We need to draw the rectangular hyperbola (or calibration curve) as defined by x²−(ct)² =1 to calibrate the unit distance from different axes.

Feynman explains that it is not really possible to think of space-time as a real, ordinary geometry because of a difference in sign in x²−(ct)². It is surprising that he prefers not to deal with the space-time geometry and even explains that this approach does not help much because it is easier to work with the equations. To be precise, the Lorentz transformations equations can be related to hyperbolic geometry and expressed in terms of hyperbolic functions of a real angle. Furthermore, we may draw space-time diagrams to help students develop a geometric understanding of the “time dilation” and “length contraction” without using mathematical equations. Students may also find the graphical presentations of relativity concepts insightful.

Questions for discussion:

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?

The moral of the lesson: a space-diagram is related to the amazing fact that the Lorentz transformations preserve the quadratic form c²t²-x² and the transformation of coordinates can be regarded geometrically as a rotation in a four-dimensional space.

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. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

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

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.