Special Relativity I

An inertial frame is one in which a body free of net force keeps a constant velocity, and the Galilean group is the ten-parameter set of coordinate changes — boosts, rotations, spatial displacements and clock shifts — that carries one such frame to another. Use the supplied topic model Y_out = (30-v-w)*te, where Y_out is x″ OF EVENT E IN S″ (m). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. te is Time t of event E, measured in s. Keep v (Boost v of S′ relative to S) = 12 m s⁻¹; w (Further boost w of S″ vs S′) = 8 m s⁻¹ fixed. For this exercise use only 0.871875 s ≤ te ≤ 3.62812 s on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 25.531875 m, with a bounded reading uncertainty ±1.6124 m. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown te by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000275625 s, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Maxwell's equations are not Galilean-covariant: substituting x′ = x − vt, t′ = t into the electromagnetic wave equation adds a cross term the original does not contain, so the equation can hold in its standard form in at most one frame. Use the supplied topic model Y_out = (b+w)^2/max(1-(b+w)^2,0.04), where Y_out is ROUND-TRIP EXCESS β²/(1−β²) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. w is Extra Galilean boost w/c, measured in c. Keep b (Frame speed through the aether v/c) = 0.3 c fixed. For this exercise use only -0.273 c ≤ w ≤ 0.315 c on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.17474173, with a bounded reading uncertainty ±0.026102. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown w by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000588 c, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Einstein synchronisation defines two separated clocks to agree when the outward and return legs of a light signal's round trip between them are assigned equal durations — the ε = ½ choice — which makes the one-way speed of light c by stipulation rather than by measurement. Use the supplied topic model Y_out = 1/(2*max(eps,0.05)), where Y_out is OUTWARD c₊ (c). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. eps is Synchronisation choice ε, dimensionless. Keep sep (Baseline A to B) = 240 m fixed. For this exercise use only 0.26375 ≤ eps ≤ 0.73625 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.90583813 c, with a bounded reading uncertainty ±0.034008 c. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown eps by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00004725, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The Lorentz boost is the unique linear map between inertial frames allowed by homogeneity, isotropy and group closure, fixed up to one universal constant K whose value only experiment supplies. Use the supplied topic model Y_out = 1/sqrt(max(1-k*b*b,0.0001)), where Y_out is γ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. b is Speed β = v/c, dimensionless. Keep k (Constant k = Kc²) = 1 fixed. For this exercise use only 0.145625 ≤ b ≤ 0.854375 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 1.2253951, with a bounded reading uncertainty ±0.032193. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000070875, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.