Section 15–7 Four-vectors

(Four-position / Rotation of space-time / Four-momentum)

In this section, Feynman briefly discusses the position four-vectors, rotation of space-time, and four-momentum.

1. Four-position:

“… we expect that there will be vectors with four components, three of which are like the components of an ordinary vector, and with these will be associated with a fourth component, which is the analog of the time part (Feynman et al., 1963, section 15–7 Four-vectors).”

According to Feynman, we can extend the concept of vectors that have only space components by including a time component. The new concept may be known as position four-vectors that have four components: three of which are the components of a position vector bounded in a three-dimensional space and a fourth component that is associated with time. It means that space and time are inter-related to the extent that space should not be independently defined without time, whereas time should not be independently defined without space. However, it is debatable whether the definition of space-time in terms of four-vector is merely a convention. In other words, one may argue that space-time is arbitrarily defined as a matter of convenience depending on how a physicist formulates a theory of space and time.

A four-vector is a set of four real (or complex) numbers that can be transformed by the Lorentz transformation equations. In his seminal paper titled space and time, Minkowski (1907) writes that “I will call a point in space at a given time, i.e. a system of values x, y, z, t a worldpoint. The manifold of all possible systems of values x, y, z, t will be called the world (p. 112).” Currently, physicists prefer the term event instead of worldpoint. This idealized point in Minkowski spacetime has a time and spatial position that can be represented by a position four-vectors: R = (ct, r) where c is the speed of light and r is a three-dimensional position vector. Alternatively, some authors may adopt Poincare’s (1906) imaginary time dimension ict that is used as a matter of convention.

2. Rotation in space-time:

“… Lorentz transformation is analogous to a rotation, only it is a “rotation” in space and time, which appears to be a strange concept (Feynman et al., 1963, section 15–7 Four-vectors).”

Feynman explains that Lorentz transformation equations help to determine a new x′ which is a mixture of x and t (or a new t′ is a mixture of t and x). Mathematically, the Lorentz transformation of x and t can be visualized as a rotation of space and time in a space-time diagram. It can be further shown that the mathematical equation x′²+y′²+z′²−c²t′² = x²+y²+z²−c²t² is invariant for different values x, y, z, t in all inertial frames of reference. Physics teachers may clarify that Minkowski did not unify space and time such that they are equivalent to each other just like the equation E = mc² that means mass-energy equivalence. Although space and time can be transformed into different values, the invariant quantity x²+y²+z²−c²t² can be related to the postulate of the absolute world (Minkowski, 1907).

The theory of special relativity is not only about the relative space and time that can be calculated using Lorentz transformation. In addition, the interval x²+y²+z²−c²t² is invariant for all inertial frames of reference. Thus, Minkowski (1907) explains that “I think the word relativity postulate used for the requirement of invariance under the group G_c is very feeble. Since the meaning of the postulate is that through the phenomena only the four-dimensional world in space and time is given, but the projection in space and in time can still be made with certain freedom, I want to give this affirmation rather the name the postulate of the absolute world (p. 117).” Perhaps special relativity should be named as “theory of variance (relativity) and invariance.” Although Einstein has destroyed absolute space and absolute time, Minkowski has helped to develop the concept of invariant (absolute) space-time interval.

3. Four-momentum:

“… the transformation gives three space parts that are like ordinary momentum components, and a fourth component, the time part, which is the energy (Feynman et al., 1963, section 15–7 Four-vectors).”

Feynman briefly mentions that four-momentum will be analyzed further in the next chapters. However, we can simply apply the idea of four-vectors to momentum such that the Lorentz transformation gives three components of linear momentum (space parts) and a fourth component (time part) which is the energy. Mathematically, we can multiply the four-velocity (or velocity 4-vector) by the rest (invariant) mass to obtain the four-momentum, P ≡ mV = (E, p). Using this approach, the law of conservation of energy and momentum appears as a single law (or conservation of four-momentum). In a sense, momentum is not completely defined without the concept of energy, whereas energy is not completely defined without the concept of momentum.

One may prefer Feynman to explain how direct experimental observations support the special theory of relativity. This can be achieved by many convincing experiments that are related to the law of conservation of energy and momentum. On the other hand, it is necessary for the law of conservation of momentum to include the time component (energy) in order to have the property of Lorentz invariance. That is, we need the law of conservation of energy to merge with the law of conservation of momentum to form invariant four-vectors in the geometry of space-time. Essentially, the energy and momentum are conserved in an interaction from the perspective of observers in all inertial frames of reference.

Questions for discussion:

1. How would you define the position four-vectors?

2. How would you explain that the Lorentz transformation is analogous to a “rotation” in space and time?

3. How would you define the four-momentum?

The moral of the lesson: the Lorentz transformation of x and t of a moving object as viewed by an inertial observer can be visualized as a rotation of space and time in a Minkowski space-time diagram.

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

3. Poincaré, H. (1906). Sur la dynamique de l’electron. Rendiconti del Circolo Matematico di Palermo, 21, 129-176.