What Have Spacetime, Shape and Symmetry to Do with Thermodynamics? Jim McGovern School of Mechanical and Transport Engineering, Dublin Institute of Technology, Ireland, [email protected] ABSTRACT: Some highly speculative and serendipitous ideas that might relate thermodynamics, spacetime, shape and symmetry are brought together. A hypothetical spacetime comprising a pointwise 4-colour rhombohedral lattice with a fixed metric is considered. If there were absolute symmetry between distance and time it is suggested that the velocity of light would be an integer conversion factor between the units of these quantities. In the context of such outlandish hypotheses the answer to the question posed would be ‘everything’. Keywords: thermodynamics, spacetime, shape, symmetry 1. INTRODUCTION have been outside my domain as a Mechanical Engineer who, though in an I have always found there to be some academic post, is expected to be practically- very disquieting terms and concepts in minded. thermodynamics; these are topics at the end For this paper on the above topic I have of short ‘what-is?’ or ‘why?’ chains starting organised some of the notes that I have with just about anything and ending with made, some of the references I have names such as force, mass, energy, consulted and some of the things that I have temperature, volume, time, action, distance, tried out. direction. Generally as an engineer I got on with the tasks at hand. As a teacher I have told my students there are certain things that 2. SPACETIME, SHAPE AND we don’t understand, but take as given SYMMETRY starting points from which we can develop and apply practical scientific and 2.1 Spacetime engineering techniques that allow us to We live in a 3-D spatial world and in our understand to a significant degree how simple experience and our natural reference things work and therefrom to predict how frame we can be at the same place at different physical arrangements will behave different times so, in all, it seems that and so solve engineering problems or come spacetime is 4-D. Therefore it seems natural up with more-or-less ingenious or new that four co-ordinates should be required to designs and devices. describe the location of an event in But the frustrating ‘why questions’ a spacetime. little below the surface of what is Time and distance are fundamentally scientifically understood have drawn me equivalent and indistinguishable. Time is relentlessly to chase things that really should embedded in distance and distance is embedded in time. The choice of the 4-D quantum. This may well be the Planck reference frame should be arbitrary. distance (or time). Physical reality is independent of any change in the orientation (i.e. rotation), just 2.2 Shape as it is independent of any change of the If spacetime is a point lattice with a position (i.e. translation) of the reference metric then it has shape. Notwithstanding frame. A general change of the position and the distortions of spacetime when velocities orientation of the reference frame is called a and gravitation are involved, four arbitrary spacetime transformation and involves points in spacetime can form a line, a changes in the coordinates of events with surface or a volume. Irrespective of the respect to the reference frame. It may also arbitrary number of points in discrete involve changes in any parameters that spacetime that might be selected arbitrarily, depend on those coordinates. Of course, the selection has a describable shape. Of certain parameters that depend on the course any everyday object like a hat that spacetime coordinates may yet be invariant exists in a region of spacetime has an under spacetime transformations. immensely complex shape. This ‘shape’ In spite of Einstein’s realisations that includes all events of the hat for which it mass and other parameters depend on can, in principle, be identified as a hat. velocity and that spacetime itself can be curved and distorted, there is a remarkable 2.3 Symmetry metrical consistency that can be observed Symmetry is sameness when objects are throughout visible spacetime. Fundamental viewed at different spacetime coordinates or entities such as atoms, neutrinos, quarks or when an object is viewed from different photons exist in innumerable quantities and spacetime coordinates. Hermann Weyl’s ‘measure’ the same for each type. It also book [3] on the topic included the statement seems that Planck’s constant, the Planck of a general principle (p. 125) ‘If conditions length and the Planck time are expressions which uniquely determine their effect of the universality of the metric. possess certain symmetries, then the effect Quantum field theory [1], quantum will exhibit the same symmetry’ and the chromodynamics theory [2] and remark (p. 126) ‘As far as I can see, all a observations in particle physics must priori statements in physics have their origin somehow be precisely consistent with in symmetry.’ cosmological observations and with Symmetry underlies quantum mechanics Einstein’s theories of special and general generally, quantum electrodynamics, relativity. History shows that good theories quantum chromodynamics, the Standard evolve and become reconciled, thus Model of particle physics, general and revealing deeper beauty. An eventual special relativity and the observed structure ‘theory of everything’ will be expected to of the universe at all scales. explain why the current theories do not entirely fit together over the huge range of 2.4 Lineland, 1-D Spacetime; or is it 2-D? orders of magnitude of the observations that Edwin Abbott [4] through his monograph are being made. ‘Flatland’ has stimulated much thought on The absolute precision of nature’s the nature of dimensions and space. metrics points to a discrete nature of Thinking about a ‘world’ of only one spacetime, ‘the vacuum’. Atoms of the same dimension (‘Lineland’) or of two (Flatland) type have the same size. Different single- proves useful in appreciating the universe in atom clocks keep the same time. This which we actually find ourselves (which is suggests that spacetime is a lattice, based on normally thought of as having three ‘space’ a fundamental metric. There is a smallest dimensions). A key observation that can be made on consideration of Abbott’s depiction each interior node was a joint or gusset for of a hypothetical one-dimensional world is twelve struts [5], [6]. Exactly the same that an object or a pattern could not be configuration of nodes was described by inverted by means of a displacement alone. Richard Buckminster Fuller as the Isotropic An object such as a lower case ‘i’ that points Vector Matrix [7]. This configuration upwards in one-dimensional space could provides the basis of a volume-occupying never become an object that points lattice (conventionally described as a 3-D downwards by means of a displacement. lattice). Fig. 2 illustrates an interior node Fig. 1(a) shows the letter ‘i’ and an identical and twelve nodes that surround it. letter displaced below the first. A reference point is shown mid-way between the two letters, but there is no symmetry about it. Figure 1(b) shows the letter ‘i’ and an upside-down letter ‘i’. Again there is a reference point between them and there is symmetry about that point. Here the reference point is an ‘axis of symmetry’. Several issues arise from consideration of such one-dimensional cases. One of these is the number of dimensions: perhaps the up- direction and the down-direction constitute two dimensions, not one. If the up-direction and the down-direction are taken as two Figure 2: Node configuration of the Bell different (but collinear) dimensions, which Fuller lattice. each have a unique direction, then an upright ‘i’ in each dimension, as shown in The Bell Fuller lattice formed by Fig. 1(b), constitutes a symmetrical pair of replication of the node pattern shown in the letter ‘i’. Fig. 2 has quite amazing symmetry properties. For instance, it includes all three of the only planar configurations of discrete points that are regular (or symmetric) [8]. These are the square tessellation of the plane, Fig. 3, the triangular, Fig. 4, and the hexagonal or honeycomb tessellation, Fig. 5. The central node in Fig. 2 is included in three distinct planes of the square tessellation and four distinct planes of the triangular tessellation, which also includes the honeycomb tessellation. Figure 1: The letter ‘i’ in 1-D space (Lineland) 3. THE BELL FULLER LATTICE Alexander Graham Bell devised a Figure 3: Square tessellation of the plane. cuboctahedral truss made up of struts where Figure 4: Triangular tessellation. Figure 7: A nucleated tetrahedron on the Bell Fuller lattice. Fig. 7 illustrates a nucleated tetrahedron of edge length 12 constructed on the Bell Fuller lattice. In this diagram there is a Figure 5: Honeycomb tessellation. nucleated cuboctahedron shown at the centre of volume of the tetrahedron. All Fig. 6 illustrates a nucleated cube on the these lattice diagrams were prepared with Bell Fuller lattice. Here the term ‘nucleated’ the aid of a computer program called means that there is a node at the centre and Springdance [9] and were rendered using there are nodes at each of the vertices of the POV-Ray [10]. cube. This cube contains a nucleated regular If spacetime were a lattice of discrete octahedron, whose vertices are the centre points then this beautiful lattice, the Bell points of the six faces of the cube. It also Fuller lattice, would be a strong candidate to contains two intersecting, nucleated, regular represent its shape. tetrahedra whose vertices are the vertices of the cube. These are the most compact nucleated regular tetrahedra that can be constructed on the lattice. It can be seen from this diagram that nucleated cubes based on the Bell Fuller lattice are both face-centred and body-centred.