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.