Test 1 Review of Concepts of Transformations PDF

# Concepts of Transformations - Test 1 Review - Coordinate system and origin are irrelevant to physics outcomes. - Inertial Reference Frames (invariant physics laws) between two inertial frames. - Galilean vs. Lorentz Transformation (t' = t, v = c) - From Einstein to Galilean/Newtonian Mechanic...

# Concepts of Transformations - Test 1 Review - Coordinate system and origin are irrelevant to physics outcomes. - Inertial Reference Frames (invariant physics laws) between two inertial frames. - Galilean vs. Lorentz Transformation (t' = t, v = c) - From Einstein to Galilean/Newtonian Mechanics (E reduces to G/N when v << c) - Maxwell's Equations Invariant ## Ether, Michelson Experiment, Null Result - The prediction of Maxwell's Equations giving C = 1/ε₀μ₀ universally was, at first seen as an inconsistency between Galilean velocity addition formulas of classical E+M. - To remedy, the idea of the "ether" frame was put forth and immediately accepted as obvious. - A special frame in which the speed of light would be the same in all directions. The medium which carried, supported light waves. A privileged frame for isotropic (interference, diffraction) optical phenomena. ## Prelude to MM experiment - Stellar Aberration (shifting of star's pos. towards the direction of motion and to velocity of orbit) 1700s - Parallax of stars (due to earth's changing position in orbit) 1837 - Concept of Aberration interpreted in the context of ether-based wave theory, indicates that the Earth is moving through the ether. - By mid 1800s physicists such as Fresnel and Airy, where trying to conduct optical-type experiments to quantify any detectable motion of the Earth through the "ether." - Around 1884, Michelson devised the interferometer experiment to "test the other theory (really verify) but found that the amount of fringe shift implied that there was no ether... The Null Result # Postulates of Relativity (Einstein's Special Rel.) - He questioned the assumed existence of an absolute universal time and concepts. - Pulled together all the evidence of the day used to understand the apparent contradiction of Maxwell's Equation (Universality of C), the Null Result of Ether theory, the Lorentz Fitzgerald Length contraction used to explain the null result... - And formulated Special Rel. - **P#1** All inertial frames are equivalent w.r.t all the laws of physics. - **P#2** The speed of light in empty space always has the same value (flies in the face of normal experience) ## Relativity of Time - The key to deriving all of the special relativity concepts and equations. - Time dilation: moving clocks run slow - Δt = γ Δt₀ - Δt₀ = proper time - γ = 1/√(1-v²/c²) (the rate of time passing of clock in the rest frame) - Derived from thought experiment using a single Pythagorean theorem and universality of C. - Evidence for Time Dilation: - Lab perceived lifetimes of atomic clocks in jets - Relativistic unstable particles ## Length Contraction - "Moving yardsticks are perceived to be shorter than 1 meter by outside observers." - L = L₀/γ - L₀ = Proper length (the longest object is perceived to be from within its own rest frame). - This is reciprocal phenomenon to time dilation and allows for consistency between the two concepts. ## Twin Paradox - (Not really a paradox) Why? How is it resolved... - Transformations: - Galilean Transformation (G.T.S.): - x' = x - vt - y' = y - z' = z - t' = t - Lorentz Transformations (L.T.S): - x' = γ (x - vt) - y' = y - z' = z - t' = γ (t - (vx/c²)) - Velocity Addition: - ux' = ux + v - uy' = uy - uz' = uz - ux = ux' - v - uy = uy' - uz = uz' - ux = (ux' +v) / (1+vux'/c²) - uy' = uy / γ (1+vux'/c²) - uz' = uz / γ (1+vux'/c²) ## Doppler Effect - Source approaching or receiver receding from source: f' = √(1+β)/(1-β) f - Source receding from receiver: f' = √(1-β)/(1+β) f - R.M laws: - Must be valid in all inertial frames - Must reduce to the non-rel. (classical) counterparts when v << c - Results must agree with experimental proper mass - Concept of Rest mass and or proper mass — We only speak of mass as measured within the object's own rest frame, and everyone agrees with the value. ## Relativistic mechanics - Rel. Momentum (classically): **p** = m **d**t/d**t**₀ = γm**u** = γ(m₀) **u** = **m**₀**u** / √(1-u²/c²) - Rel. Energy: - E = mc² = rest energy & energy with motion K.E - E = mc² = E₀ - Relativistic Kinetic energy: K = (γ-1) mc² ≠ ½ mu² ## Two useful relations (derived from p = γmu and E=mc²) - **p** = E/c = **u** - E²= (pc)² + (mc²)² ## Conversion of Mass to Energy - Concept of change in rest mass ΔM = M-(m₁+m₂) - ΔM c² = K₁+K₂ = energy released as composite bodies fly a part = work done to push bodies together - Example is Radioactive Decay of Nucleus. - Where M (A) C m² = equivalent "binding" energy - The photon Rocket (Concept... why is it interesting... what is the idea) - P = 1/c √(E²-(mc²)²) - Lorentz Force: F = q(E + **u** x **B**) works for both classical + rel. - Rel. Correspondence between **E** and **B** fields — They are the same phenomena just viewed from different frames of reference. ## Massless Particles - E² = (pc)² + (mc²)² - E² = (pc)² - E = pc - β = 1 - When is non-relativistic mechanics good enough? - For u ≥ 0.1c , need R.M - For K ≥ 0.01E₀ , need R.M ## General Relativity - The equivalence principle: There is no distinction between a uniform gravitational field and the equivalent uniform accel. (**A** = -**g**) - Accelerating reference frames.