Information transmittal, time uncertainty and special relativity

2009

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The γ -representation Relativistic identification and control of motion a b s t r a c t Relativistic transformations considered in [Albert Einstein, Zur Elektrodynamik der bewegter Körper. Ann. Phys. 17 (1905) 891-921] are extended for motions with variable speed through the introduction of the γ -representation and continuous piecewise linear approximations which are used for velocity identification on the basis of discrete measurements of the distance function. This demonstrates that Einstein's relativistic transformations originally established to explain some physical phenomena experimentally observed in systems one of which is moving with a constant relative velocity with respect to the other are, in fact, applicable (in some modified form) to systems with variable relative velocity. The method is illustrated in application to the control of a flight to Mars in a spacecraft with comfortable weight conditions on board, same as on Earth, moving with acceleration g = 9.8 m/s 2 at all times except for the lift-off from Earth and soft touchdown on Mars after 42 h flight. Natural time delays due to finite speed of information transmittal are included into relativistic transformations, leading to a 4D space-time frame in the form of a variable affinely connected time-space structure whose dynamics is defined by the relative velocity v(t) = const between the two systems. The results open new avenues in the theoretical and experimental studies of relativistic effects in media moving with variable speed.

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“…It implies that a mapping should exist between the constant parameters p and v = p, cf. Lemma 9.1 in [6].…”

Section: Computation Of the Average Velocities Of (K) As Observed Fromentioning

confidence: 97%

Section: Modification Of Einstein's Transformations For Motions With mentioning

confidence: 98%

Section: Einstein's Derivation Of the Time Transformation [1 Section 3]mentioning

confidence: 99%

Section: Flight To Mars With Normal Weight Conditions In Spacecraftmentioning

confidence: 99%

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Special relativity for motions with variable speed

2009

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“…and trying to satisfy equation (4) identically with respect to t and x , without other assumptions, the reader can find the unique linear relation between a and b resulting in the formula (7) for all three axes, see [3]. Substituting…”

Section: Derivation Of Einstein's Time Transformation [1 Section 3]mentioning

confidence: 99%

Relativistic transformations at variable velocities

2010

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The γ -representation Relativistic identification and control of processes a b s t r a c t Relativistic transformations at constant velocities V and v considered in [A. Einstein, Zur Elektrodynamik der bewegter Körper, Ann. Phys., 17 (1905) 891-921] are modified for the general case of variable speed V (.) of synchronization signal and variable relative velocity v(t) between two frames (K ) and (k) linked through that signal. A differential approach to time synchronization is considered which makes clear that with variable velocities the observed time in the moving frame is represented by a nonlinear functional of those velocities V (.) and v(t) over the segments of trajectory covered by the synchronization (observation) signal, and the difference with respect to Einstein's PDE at constant velocities is in the range of several decimal orders. Moreover, the application of linear transformations at constant velocities to the case of variable velocities may produce illusory effects of a relativistic mirage of expansion. For practical applications, and to preserve the simplicity of relativistic transformations, the averaging approach is developed, and a new form of relativistic transformations is obtained for discretized trajectories with on-line identification of average values for variable relative velocity v(t) and in consideration of a nominal velocity V with possible variations δV (.). An estimation of errors is made through the γ -representation based on direct distance measurements. Natural time delays due to the finite speed of information transmittal are included into relativistic transformations at variable velocities, and equations for software development are presented to support computation of real time trajectories and error estimation. The results open new avenues for theoretical and experimental studies of special relativity in media moving with variable speed and for control of processes by means of actually available signals.

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Unit-free transformations of linear relativity

2010