RELATIVISTIC ENERGY-MOMENTUM OF A BODY WITH A FINITE VOLUME TADAS K. NAKAMURA Center for Arts and Science, Fukui Prefectural University, 4-1-1 Kenjojima, Matsuoka, Yoshida 910-1195, Fukui, Japan (E-mail: tadas@fpu.ac.jp) (Received: 31 October 2005; Accepted in final form: 12 April 2006) Abstract. The problem of energy-momentum in a body with a finite volume has been causing confu- sion in the theory of relativity, especially in relativistic thermodynamics. Its correct understanding has been given since the early years of relativity, however, erroneous misunderstandings are still found in papers and textbooks to this date. The present paper introduces a simple paradox to demonstrate the problem, and gives a brief review on a way to handle the energy-momentum correctly. Keywords: relativistic thermodynamics, energy-momentum 1. Introduction The theory of relativistic thermodynamics is not so simple as supposed at the beginning of the theory of relativity (Einstein, 1907; Planck, 1908). There has been a heated controversy with complicated confusions in the 1960s (see Stuart et al., 1970, for a review). Numerous papers have been published, each of which proposes its own version of formulation, claiming this is the theory. One of the main problems in this controversy is the definition of thermodynamical quantities, such as energy or temperature. For example, Lansberg and Jones (1967) has a table with 20 different sets of definitions. In these years there was a confusion in defining the total energy-momentum of a body with finite volume. Some people regarded the energy-momentum of a body as a four vector, i.e., a frame-independent physical entity. This is erroneous because the volume of a body depends on reference frames. Usually the volume of a body is defined as the cross section of the body’s world tube at a specific time viewed from a specific reference frame. Therefore, the total energy-momentum in a volume defined in one reference frame is not the same physical entity as that in another reference frame, in other words, they are distinct four vectors that do not connect each other by the Lorentz transform. Gamba (1967) has argued that this problem causes one of the major confusion in the controversy of relativistic thermodynamics. He states that the problem of finite volume had been well understood from the very beginning of relativity (e.g., Fermi, 1923), nevertheless, the similar misunderstanding can be found in a number of Space Science Reviews (2006) 122: 271–278 DOI: 10.1007/s11214-006-8186-y C  Springer 2006