Logic of space-time and relativity theory Hajnal Andr´eka, Judit X. Madar´asz and Istv´an N´emeti October 20, 2006 Contents 1 Introduction 3 2 Special relativity 4 2.1 Motivation for special relativistic kinematics in place of New- tonian kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Axiomatization Specrel of special relativity in first-order logic 15 2.4 Characteristic differences between Newtonian and special rel- ativistic kinematics . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Explicit description of all models of Specrel, basic logical investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6 Observer-independentgeometriesinrelativitytheory; duality and definability theory of logic . . . . . . . . . . . . . . . . . 47 2.7 Conceptual analysis and “reverse relativity” . . . . . . . . . . 59 3 General relativistic space-time 60 3.1 Transition to general relativity: accelerated observers in spe- cial relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Einstein’s “locally special relativity principle” . . . . . . . . . 64 3.3 Worldlines of inertial observers and photons in a general rel- ativistic space-time . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 The global grid seen with the eyes of the local grids: general relativistic space-time in metric-tensor field form . . . . . . . 72 3.5 Isomorphisms between general relativistic space-times . . . . 75 3.6 Axiomatization Genrel of general relativistic space-time in first-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . 77 1 4 Black holes, wormholes, timewarp. Distinguished general relativistic space-times 84 4.1 Special relativity as special case of general relativity . . . . . 85 4.2 The Schwarzschild black hole . . . . . . . . . . . . . . . . . . 86 4.3 Double black holes, wormholes . . . . . . . . . . . . . . . . . 100 4.4 Blackholeswithantigravity(i.e.withacosmologicalconstant Λ). Triple black holes . . . . . . . . . . . . . . . . . . . . . . 105 4.5 Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . 107 5 Connections with the literature 109 References 110 Index 117 2 1 Introduction Our goal is to make relativity theory accessible and transparent for any reader with logical background. The reader does not have to “believe” anything. The emphasis is on the logic-based approach to relativity theory. The purpose is giving insights as opposed to mere recipes for calculations. Thereforeproofswillbevisualgeometricones,effortswillbemadetoreplace computational proofs with suggestive drawings. Relativitytheorycomesin(atleast)twoversions, specialrelativity(Ein- stein 1905) and general relativity (Einstein, Hilbert 1915). They differ in scope,thescopeofgeneralrelativityisbroader. Specialrelativityisatheory of motion and light propagation in vacuum far away from any gravitational object. I.e.specialrelativitydoesnotdealwithgravity. Also,specialrelativ- ity is a “prelude” for general relativity, it provides a foundation or starting point for the general theory. General relativity unifies special relativity and the theory of gravitation. In some sense, general relativity is an “exten- sion” of special relativity putting also gravity into the picture. General relativity can be used as a foundation for cosmology, e.g. it is a suitable framework for discussing the (evolution, properties of the) whole universe (expanding or otherwise). Special relativity, on the other hand, is not rich enough for this purpose. General relativity also provides the theory of black holes, wormholes, timewarps etc. Special relativity shows us that there is no such thing as space in itself, instead, a unified space-time exists. This in- separability of space and time becomes more dramatic in general relativity. Namely,generalrelativityshowsusthatgravityisnothingbutthecurvature of space-time. It is extremely difficult, if not impossible, to explain gravity without invoking the curvature (i.e. geometry) of space-time. The crucial pointisthatcurvatureofspaceisnotenough(byfar), itisspace-timewhose curvature explains gravity.1 From a different angle: general relativity is a “geometrization” of much of what we know about the world surrounding us. E.g. it provides a full geometrization of our understanding of gravity and related phenomena like motion and light signals. In Sec. 2 we study special relativity, in Sec. 3 we do the same for general relativity, inSec.4weapplythesoobtainedtoolstoblackholes, wormholes, 1If we took into account the curvature of space only, then apples would no more fall downfromtrees. Gravitationalattractionassuchwoulddisappear. Ontheotherhand,if wekeepthetemporalaspectsofcurvaturebutignorecurvatureofpurespace,thengravity would not disappear, instead, this would cause only minor discrepancies in predicting trajectoriesofveryfastmovingbodies(relativetothesourceofgravity,e.g.theEarthor a black hole). 3 timewarps. The emphasis is on the space-time aspects. In Sec. 5 we briefly discuss the literature. 2 Special relativity Inthissection,amongothers,wegiveafirst-orderlogic(FOL)axiomsystem for special relativity such that we use only a handful of simple, streamlined axioms. In our approach, axiomatization is not the end of the story, but rather the beginning. Namely: axiomatizations of relativity are not ends in themselves(goals), instead, theyareonlytools. Ourgoalsaretoobtainsim- ple, transparent, easy-to-communicate insights into the nature of relativity, to get a deeper understanding of relativity, to simplify it, to provide a foun- dation for it. Another aim is to make relativity theory accessible for many people (as fully as possible). Further, we intend to analyze the logical struc- ture of the theory: which assumptions are responsible for which predictions; what happens if we weaken/fine-tune the assumptions, what we could have done differently. We seek insights, a deeper understanding. We could call this approach “reverse relativity” in analogy with “reverse mathematics”. 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics Why should we replace Newtonian Kinematics with such an exotic or coun- ter-common-sensetheoryasspecialrelativity? TheNewtoniantheoryproved very successful for 200 years. By now, the Newtonian picture of motion has becomethe same as the current common-sense picture of motion. Hence the question is why we have to throw away our common-sense understanding of motion.2 The answer is that there are several independently good reasons for replacing the Newtonian worldview with relativity. These reasons are really good and decisive ones. They are so compelling, that any one of them wouldbesufficientforjustifyingandmotivatingourreplacingtheNewtonian worldview with relativity. We will mention a few of these reasons, but for simplicity of presentation, we will base this work on a fixed one of these reasons, namely on the outcome of the Michelson-Morley experiment. We will call this outcome of the Michelson-Morley experiment the Light Axiom. There are deeper, more philosophical reasons for replacing the Newtonian 2A second, equally justified question would ask why exactly those postulates/axioms are assumed in relativity which we will assume. We will deal with both questions. 4 worldview with relativity theory, which might convince readers who are not experimentallyminded,i.e.whoarenoteasilyconvincedbymerefactsabout how results of certain experiments turned out. These philosophical reasons (under the name “principles of relativity”) are intimately intertwined with issues which were significantly present through the last 2500 years of the history of our culture; see p. 63 and [Bar89]. We now turn to the Light Axiom which will play a central role in this work. The first test of the Light Axiom was the Michelson-Morley exper- iment in 1887 and it has been tested extremely many times and in many radically different ways ever since. As a consequence, the Light Axiom has been generally accepted. An informal, intuitive formulation of the axiom is the following. (Later we will present this axiom in more formal, more precise terms, too, see AxPh in Sec. 2.3.) Light Axiom: The speed of light is finite and direction independent, in the worldview of any inertial observer. Inotherwords,theLightAxiommeansthefollowing. Imaginea(huge)space- ship drifting through outer space in inertial motion. (Inertial here means that the rockets of the spaceship are switched off, and that the spaceship is not spinning.) Assume a scientist in this inertial spaceship is making exper- iments. The claim is that if the scientist measures the speed of light, he will find that this speed is the same in all directions and that it is finite. It is essential here that this is claimed to hold for all possible inertial spaceships irrespective of their velocities relative to the Earth or the Sun or the center of our galaxy or whatever reference system would be chosen. The point is that no matter which inertial spaceship we choose, the speed of light in that spaceship is independent of the direction in which it was measured, i.e. it is “isotropic”. In the technical language what we called “inertial spaceship” above is called an inertial reference frame, and the scientist in the spaceship making the experiments is called an “observer”. Later “observer” and reference frame tend to be identified.3 Let us notice that the Light Axiom is surprising, it is in sharp contrast with common-sense. Namely, common-sense says that if we send out a light signal from Earth, and a spaceship is racing with this light signal moving with almost the speed of the signal in the same direction as the signal does, thenthevelocityofthesignalrelativetothespaceshipshould beverysmall. 3However,itisgoodtokeepinmindthatsomethought-experimentsarecarriedoutby a team of observers (and if the members of this team do not move relative to each other then they are called, for simplicity, a single observer). 5 Hence, one would think that the astronaut in the spaceship will observe the motion of the light signal as very slow. With the same kind of reasoning, the astronaut should observe light signals moving in the opposite direction very fast. But the Light Axiom states that light moves with the same speed in all directions for the astronaut in the spaceship, too. Hence, the Light Axiom flies in the face of common-sense. This gives us a hint/promise that very interesting, surprising things might be in the making. See also Fig. 18 on p. 40. In fact, if we add the Light Axiom to Newtonian Kinematics, then we obtain a logical contradiction. I.e. (Newtonian Kinematics + Light Axiom) is an inconsistent theory in the usual sense of logic as we will outline soon (cf. Prop. 2.1). Seeing this contradiction, Einstein did the natural thing. He − weakened Newtonian Kinematics (NK for short) to a weaker theory NK − such that NK became consistent with the Light Axiom. Then the theory − (NK + Light Axiom) became known as Special Relativistic Kinematics (SRK for short). We will study this theory under the name Specrel to 0 be introduced in a logical language soon. We represent the above outlined process by the following diagram: (NK + Light Axiom) leads to Contradiction (!) ⇓ − NK gets replaced with the weaker NK ⇓ − (NK +Light Axiom) receives the name Special Relativity (SRK). SRK is consistent (this will be proved in Cor. 2.2, p.43). Toseetheaboveprocessmoreclearly,letusinvokeapossibleaxiomatization of NK, still on the intuitive level. Preparation for NK: If we want to represent motion of “particles” or “bodies” or “mass-points”, it is natural to use a 4-dimensional Cartesian co- ordinatesystemR R R R(whereRisthesetofrealnumbers), withone × × × timedimensiontandthreespacedimensionsx,y,z. Athree-dimensionalpart ofthisisdepictedinFig.1. Thetime-axistisdrawnvertically. Representing the motion of a body, say b, in a 4-coordinate system can be done by spec- ifying a function f which to each time instance t R tells us the space ∈ coordinates x,y,z where the body b is found at time t. Hence a function f : Time Space specifies motion of a particle in this sense. The function → f representing motion of b is called the worldline or lifeline of b. Fig. 1 represents motion of bodies, in this spirit. Besides the coordinate axes, we have represented the worldlines of inertial bodies b ,b and b in Fig. 1. The 1 2 3 6 t b b 2 3 b 1 point p x y Figure 1: A space-time diagram. Wordlines of bodies b ,b ,b represent 1 2 3 motion. (Coordinate z is not indicated in the figure.) b is motionless and 3 b moves faster than b . 1 2 straight line labeled by b is the worldline of b . The slope of the worldline 1 1 of b is greater than that of b which means that b moves faster than b 1 2 1 2 does. The worldline of the third body b is parallel with the time-axis, this 3 means that b is motionless. Bodies b and b meet at space-time point 3 1 2 p = t,x,y,z . Such a meeting (of two or more bodies) is called an event. h i We will extensively refer to such 4-dimensional coordinate systems and such worldlines of bodies and events. The axioms of NK are summarized as (i)-(v) below. (i) Each observer “lives” in a 4-coordinate system as described above. The observer in his own coordinate system is motionless in the origin, i.e. his worldline is the time-axis.4 (ii) Inertial motion is straight: Let o be an arbitrary inertial observer and let b be an inertial body. Then in o’s 4-coordinate system the worldline of b is a straight line. I.e. in an inertial observer’s worldview or 4-coordinate system all worldlines of inertial bodies appear as straight lines. As we said, an observer in his metaphorical “spaceship” is inertial if his rocketsareturnedoffandthespaceshipisnotspinning. Inspecialrelativity, 4It is sufficient to assume that his worldline is parallel with the time-axis. 7 we discuss only inertial motion, hence in our axiomatization the adjective “inertial” could be omitted. (Of course, then we need a general claim that only inertial things/objects will be studied.) (iii) Motion is permitted: In the worldview or 4-coordinate system of any inertial observer it is possible to move through any point p in any direction with any finite speed. (iv) Anytwoobservers“observe”thesameevents. I.e.ifaccordingtoo 1 bodies b and b have met, then the same is true in the 4-coordinate system 1 2 of any o . We postulate the same for triple meetings e.g. of b ,b ,b . 2 1 2 3 (v) Absolute time: Any two observers agree about the amount of time elapsed between two events. (Hence temporal relationships are absolute.) So, now, NK is defined as the theory axiomatized by (i)-(v) above. It is easy to see that (NK + Light Axiom) is inconsistent. Einstein’s idea was to check which ones of (i)-(v) are responsible for contradicting the Light Axiom and to throw away or weaken the “guilty” axioms of NK. We will see that (v) is guilty and that part of (iii) is suspicious. Hence we throw away − − (v) and weaken (iii) to a safer form (iii ) where (iii ) is the following. − (iii ) Slower-than-light motion is possible: in the worldview of any inertial observer, through any point in any direction it is possible to move with any speed slower than that of light (here, light-speed is understood as measured at that place and in that direction where we want to move). In the formal part we will carefully study whether all of these modifica- tions are really needed and to what extent (cf. Thm.s 2.3, 2.5). We define − NK as the remaining theory: − − NK := (i),(ii),(iii ),(iv) { } and Special Relativistic Kinematic is defined as − SRK := (NK +Light Axiom). The formalized version of this SRK will appear later as the theory Specrel . We will prove that Specrel is consistent (i.e. contradiction- 0 0 free) and will study its properties. Therefore SRK is also consistent, since, as we said, Specrel is a formalized version of SRK. Actually, the whole 0 process of arriving from NK and the Light Axiom (or some alternative for the latter) to SRK will be subjected to logic-based conceptual analysis in Sec. 2.5. Before turning to formalizing (and studying) Special Relativity SRK in logic, let us prove (informally only) on the present level of precision why 8 absolute time (i.e. axiom (v)) is excluded by the Light Axiom, or more pre- cisely, it is excluded if we want to keep a fragment of our intuitive picture of the world, i.e. if we want to keep (i), (ii), (iv) of NK. We will prove: − (NK + Light Axiom) Negation of (v), ⊢ where we use turnstile “ ” as the symbol of logical provability or deriv- ⊢ ability. I.e. A B means that from statement A one can prove, rigorously, ⊢ statement B. Actually, we will prove something stronger and stranger from the Light − Axiom (and a fragment of NK ). We will prove that the time elapsed be- tween two events may be different for different observers even in the special case when this elapsed time is zero for one of the observers. I.e. the very question whether two events happened at the same time or not will de- pend on the observer: two events A and B may happen at the same time for me, while event A happened much later than event B for the Mar- tian in his spaceship. We will refer to this phenomenon by saying that “simultaneity is not absolute”. Moreover, we will see later (Cor. 2.1) that the temporal order of some events may be switched: event A may precede event B for me, while for the Martian in his spaceship, event B precedes event A. ′ We say that events e and e are simultaneous for observer O if in O’s ′ coordinate system the two events e,e happen at the same time. Proposition2.1. (Simultaneityisnotabsolute)AssumeSRK.Movingclocks getoutofsynchronism, i.e.: AssumethataspaceshipS isinuniformmotion ′ relative to another one, say E, and assume that two events e,e happen simultaneously at the rear and at the nose of the spaceship S according to ′ the spaceship S. Then e and e take place at different times in E’s coordinate system. ′ I.e., thetimeelapsedbetweeneande iszeroas“seen”fromthespaceship ′ S, but the time elapsed between e and e is nonzero as “seen” from E. See Fig. 3. Intuitive proof. Assume that we are in spaceship E, and let us call E “Earth”. Assume that spaceship S – let us call it “Spaceship” – moves away from us in a uniform motion with, say, 0.9 light-speed. The captain of Spaceship positions his brothers called Rear, Middle, and Nose at the rear, middle and nose of the spaceship, respectively, and asks Rear and Nose to switch on their flashlights towards Middle exactly at the same time. Then 9 the light signals (photons5) Ph1 and Ph2 from the two flashlights arrive to Middle at the same time, because Middle is exactly in the middle of the spaceship, and because the speed of Ph1 sent by Rear is the same as the speed of Ph2 sent by Nose (by the Light Axiom). See Fig. 2. time Ph1 Ph2 space Figure 2: Seen from Spaceship, the two light-signals (i.e. photons) Ph1 and Ph2aresentoutatthesametime, andmeetinthemiddle. Thisisindicated by the clocks at the rear and at the nose of the spaceship. Notice that time in this figure is running upwards! I.e., this figure is similar to drawings in cartoons in that a sequence of scenes is represented in it. However, here the temporal order of the scenes is switched: the scene at the bottom took place earliest. The reason for this convention is our seeking compatibility with the usual space-time diagrams like Fig. 1. How do we see all this from the Earth? We see that Rear and Nose send 5Weusetheword“photon”asasynonymforlightsignal. Ittacitlyreferstothecorpus- cularconceptionoflight. Inthisworkwedonotneedthequantum-mechanicaldefinition of photons. (That will be needed only in the final, as yet nonexistent, generalization of general relativity called quantum gravity.) 10