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Towards a Better Understanding of Space-Time Causality

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Towards a Better Understanding of Space-Time Causality Defining Causality Is . Defining Causality: . Algorithmic . The Corresponding . Towards a How to Define Space- . Better Understanding Towards a Working . Discussion and . of Space-Time Causality: This Definition Is . Space-Time Causality . Kolmogorov Complexity and Home Page Causality as a Matter Title Page of Degree JJ II J I 1 2 Vladik Kreinovich and Andres Ortiz Page1 of 14 1Department of Computer Science 2Departments of Mathematical Sciences and Physics Go Back University of Texas at El Paso, El Paso, TX 79968, USA Full Screen [email protected], [email protected] Close Quit Defining Causality Is . Defining Causality: . 1. Defining Causality Is Important Algorithmic . 0 • Causal relation e 4 e between space-time events is one The Corresponding . of the fundamental notions of physics. How to Define Space- . • In Newton's physics, it was assumed that influences can Towards a Working . propagate with an arbitrary speed: Discussion and . This Definition Is . e = (t; x) e0 = (t0; x0) , t ≤ t0: 4 Space-Time Causality . • In Special Relativity, the speeds of all the processes are Home Page limited by the speed of light c: Title Page e = (t; x) e0 = (t0; x0) , c · (t0 − t) ≥ d(x; x0): 4 JJ II • In the General Relativity Theory, the space-time is J I curved, so the causal relation is even more complex. 4 Page2 of 14 • Different theories, in general, make different predic- Go Back tions about the causality 4. • So, to experimentally verify fundamental physical the- Full Screen 0 ories, we need to experimentally check whether e 4 e . Close Quit Defining Causality Is . Defining Causality: . 2. Defining Causality: Challenge Algorithmic . 0 • Intuitively, e 4 e means that: The Corresponding . { what we do in the vicinity of e How to Define Space- . Towards a Working . { changes what we observe at e0. Discussion and . • If we have two (or more) copies of the Universe, then: This Definition Is . { in one copy, we perform some action at e, and Space-Time Causality . { we do not perform this action in the second copy. Home Page 0 Title Page • If the resulting states differ, this would indicate e 4 e : World 1 World 2 JJ II rain∗6e0 e0∗6 no rain J I Page3 of 14 rain dance∗e e∗ no rain dance Go Back • Alas, in reality, we only observe one Universe, in which Full Screen we either perform the action or we do not. Close Quit Defining Causality Is . Defining Causality: . 3. Algorithmic Randomness and Kolmogorov Com- plexity: A Brief Reminder Algorithmic . The Corresponding . • If we flip a coin 1000 times and still get get all heads, How to Define Space- . common sense tells us that this coin is not fair. Towards a Working . • Similarly, if we repeatedly flip a fair coin, we cannot Discussion and . expect a periodic sequence 0101 ::: 01 (500 times). This Definition Is . Space-Time Causality . • Traditional probability theory does not distinguish be- Home Page tween random and non-random sequences. Title Page • Kolmogorov, Solomonoff, Chaitin: a sequence 0 ::: 0 isn't random since it can be printed by a short program. JJ II • In contrast, the shortest way to print a truly random J I sequence is to actually print it bit-by-bit: printf(01. ). Page4 of 14 • Let an integer C > 0 be fixed. We say that a string x Go Back is random if K(x) ≥ len(x) − C, where Full Screen def K(x) = minflen(p): p generates xg: Close Quit Defining Causality Is . Defining Causality: . 4. The Corresponding Notion of Independence Algorithmic . • If y is independent on x, then knowing x does not help The Corresponding . us generate y. How to Define Space- . • If y depends on x, then knowing x helps compute y; Towards a Working . example: Discussion and . This Definition Is . { knowing the locations and velocities x of a mechan- Space-Time Causality . ical system at time t Home Page { helps compute the locations and velocities y at time Title Page t + ∆t. JJ II • Let an integer C > 0 be fixed. We say that a string y is independent of x if K(y j x) ≥ K(y) − C; where J I Page5 of 14 K(y j x) def= minflen(p): p(x) generates yg: Go Back • We say that a string y is dependent on the string x if Full Screen K(y j x) < K(y) − C: Close Quit Defining Causality Is . Defining Causality: . 5. How to Define Space-Time Causality: First Seem- ing Reasonable Idea Algorithmic . The Corresponding . 0 • At first glance, we can check whether e 4 e as follows: How to Define Space- . { First, we perform observations and measurements Towards a Working . in the vicinity of the event e, and get the results x. Discussion and . This Definition Is . { We also perform measurements and observations in the vicinity of the event e0, and produce x0. Space-Time Causality . Home Page { If x0 depends on x, i.e., if K(x0 j x) K(x0), then we claim that e can casually influence e0. Title Page 0 JJ II • If e 4 e , then indeed knowing what happened at e can 0 help us predict what is happening at e . J I • However, the inverse is not necessarily true. Page6 of 14 • We may have x ≈ x0 because both e and e0 are influ- Go Back 00 enced by the same past event e . Full Screen 0 00 • Example: both e and e receive the same signal from e . Close Quit Defining Causality Is . Defining Causality: . 6. Towards a Working Definition of Causality Algorithmic . • According to modern physics, the Universe is quantum The Corresponding . in nature; for microscopic measurements: How to Define Space- . { we cannot predict the exact measurement results, Towards a Working . Discussion and . { we can only predict probabilities of different out- comes; the actual observations are truly random. This Definition Is . Space-Time Causality . • For each space-time event e: Home Page { we can set up such a random-producing experiment Title Page in the small vicinity of e, and JJ II { generate a random sequence re. J I • This random sequence re can affect future results. Page7 of 14 • So, if we know the random sequence re, it may help us predict future observations. Go Back 0 0 Full Screen • Thus, if e 4 e , then for some observations x performed 0 0 0 in the small vicinity of e , we have K(x j re) K(x ). Close Quit Defining Causality Is . Defining Causality: . 7. Discussion and Resulting Definition Algorithmic . 0 The Corresponding . • Reminder: when e 4 e , then the random sequence re can affect the measurement results at e0: How to Define Space- . 0 0 Towards a Working . K(x j re) K(x ): Discussion and . 0 • If e 46 e , then the random sequence re cannot affect This Definition Is . 0 0 0 the measurement results at e : K(x j re) ≈ K(x ). Space-Time Causality . Home Page • So, we arrive at the following semi-formal definition: Title Page { For a space-time event e, let re denote a random sequence generated in the small vicinity of e. JJ II 0 0 { We say e 4 e if for some observations x performed J I 0 in the small vicinity of e , we have Page8 of 14 0 0 K(x j re) K(x ): Go Back • Our definition follows the ideas of casuality as mark Full Screen transmission, with the random sequence as a mark. Close Quit Defining Causality Is . Defining Causality: . 8. This Definition Is Consistent with Physical In- tuition Algorithmic . The Corresponding . 0 • If e 4 e , then we can send all the bits of re from e How to Define Space- . 0 to e . Towards a Working . • The signal x0 received in the vicinity of e0 will thus be Discussion and . This Definition Is . identical to re. Space-Time Causality . • Thus, generating x0 based on r does not require any e Home Page 0 computations at all: K(x j re) = 0. Title Page 0 • Since the sequence x = re is random, we have K(x0) ≥ len(x0) − C: JJ II J I 0 0 • When re = x is sufficiently long (len(x ) > 2C), we have K(x0) ≥ len(x0) − C > 2C − C = C; hence Page9 of 14 0 0 0 0 Go Back 0 = K(x j re) < K(x ) − C and K(x j re) K(x ): • So, our definition is indeed in accordance with the Full Screen physical intuition. Close Quit Defining Causality Is . Defining Causality: . 9. Randomness Is a Matter of Degree Algorithmic . • Reminder: a sequence x is random if K(x) ≥ len(x)−C The Corresponding . for some C > 0. How to Define Space- . Towards a Working . • For a given sequence x, its degree of randomness d(x) can be defined by the smallest integer C for which Discussion and . This Definition Is . K(x) ≥ len(x) − C: Space-Time Causality . Home Page • One can check that this smallest integer is equal to the difference d(x) = len(x) − K(x). Title Page • For random sequences, the degree d(x) is small. JJ II J I • For sequences which are not random, the degree d(x) is large. Page 10 of 14 • In general, the smaller the difference d(x), the more Go Back random is the sequence x. Full Screen Close Quit Defining Causality Is . Defining Causality: . 10. Space-Time Causality Is a Matter of Degree Algorithmic . 0 0 The Corresponding . • Our definition of causality is that K(x j re) < K(x ) − C for some large integer C.
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