PHYS 141: Principles of Mechanics

PHYS 141: Principles of Mechanics PART TWO: RELATIVISTIC MECHANICS

I. Basic Principles A. Spacetime • Graphical Depiction: Before we look at how space and time are connected through special relativity, let’s establish how we can describe basic motion: spacetime diagrams. t 1-D motion in time as measured from a Particular Reference Frame x

I. Basic Principles A. Spacetime • Graphical Depiction: Before we look at how space and time are connected through special relativity, let’s establish how we can describe basic motion: spacetime diagrams. t 1-D motion in time as measured from a Particular Reference Frame “FUTURE” x “PRESENT” “PAST”

I. Basic Principles A. Spacetime • Graphical Depiction: Before we look at how space and time are connected through special relativity, let’s establish how we can describe basic motion: spacetime diagrams. t 1-D motion in time as measured from a Particular Reference Frame “SOMEWHERE BEHIND” “SOMEWHERE AHEAD” x “HERE”

I. Basic Principles A. Spacetime a. Event: a particular location and time. t “OVER THERE, THEN” 1-D motion in time as measured from a Particular Reference Frame “event” (x,t) x

I. Basic Principles A. Spacetime a. Event: a particular location and time. t “event 2” (x2,t2) 1-D motion in time as measured from a Particular Reference Frame “event 1” (x1,t1) x

I. Basic Principles A. Spacetime b. World Lines are Spacetime trajectories: How events are ordered. t 1-D motion in time as measured from a Particular Reference Frame “event 2” (x2,t2) “Stay at x = x1 for t2-t1 seconds” “event 1” (x1,t1) x

I. Basic Principles A. Spacetime b. World Lines: How events are ordered. t 1-D motion in time as measured from a Particular Reference Frame “event 1” (x1,t1) “event 2” (x2,t2) x “Instantaneously move From x1 to x2.” Note: this is impossible.

I. Basic Principles A. Spacetime c.Normalized Spacetime diagrams. Let w = ct. Then the slope of a trajectory in spacetime is dw/dx = cdt/dx = c/v, with c = speed of light. w = ct “event 2” (x2,w2) 1-D motion in time as measured from a Particular Reference Frame “event 1” (x1,w1) Speed from E1 to E2 is dw/dx = c/v =1, so v/c = 1. x A

I. Basic Principles A. Spacetime c.Normalized Spacetime diagrams. Let w = ct. Then the slope of a trajectory in spacetime is dw/dx = cdt/dx = c/v, with c = speed of light. w = ct 1-D motion in time as measured from a Particular Reference Frame x A

I. Basic Principles A. Spacetime Normalized Spacetime diagrams. Let w = ct. Then the slope of a trajectory in spacetime is dw/dx = cdt/dx = c/v, with c = speed of light. w=ct 1-D motion in time as measured from a Particular Reference Frame dw/dx = c/v >1, so v/c <1 v/c = 1 dw/dx = c/v <1, so v/c >1 x A

I. Basic Principles A. Spacetime d. Lightcones: Regions of spacetime in which events are connected by paths with speed v/c ≤ 1. w 1-D motion in time as measured from a Particular Reference Frame v/c = 1 “Future light cone” for A v/c = 1 x A “Past light cone” for A

I. Basic Principles A. Spacetime d. Lightcones: Regions of spacetime in which events are connected by paths with speed v/c ≤ 1. w v/c = 1. v/c = 1. “Future light cone” for A Inaccessible to A according to relativity Inaccessible to A according to relativity x “Past light cone” for A

I. Basic Principles B. Galilean Transformations For inertial systems, the Galilean Transformation allows us to translate between frames: all we have to do is (basically) subtract out the motion of primed frame (the train in this case). (x’,y’,z’,t’) Constant speed

I. Basic Principles B. Galilean Transformations

I. Basic Principles B. Galilean Transformations For inertial systems, the Galilean Transformation allows us to translate between frames: all we have to do is (basically) subtract out the motion of primed frame (the train in this case).

I. Basic Principles B. Galilean Transformations • Coordinate Transformations: transformations between two reference frames. Consider a primed reference in motion relative to an unprimed one (let v be along the positive x-direction) Galilean transformations: coordinates for a point P in the moving primed frame as seen from the (unmoving) unprimed frame (and vice versa). y x = x’ + vt’. (I.B.1-4) y = y’. z = z’. t = t’. x

y’ v P x’ I. Basic Principles B. Galilean Transformations • Coordinate Transformations: transformations between two reference frames. Consider a primed reference in motion relative to an unprimed one (let v be along the positive x-direction) Galilean transformations: coordinates for a point P in the moving primed frame as seen from the (unmoving) unprimed frame (and vice versa). x x’ y x = x’ + vt’. y = y’. z = z’. t = t’. x

y’ v P x’ I. Basic Principles B. Galilean Transformations • Coordinate Transformations: transformations between two reference frames. Consider a primed reference in motion relative to an unprimed one (let v be along the positive x-direction) Galilean transformations: coordinates for a point P in the moving primed frame as seen from the (unmoving) unprimed frame (and vice versa). x x’ vt’ y x = x’ + vt’. y = y’. z = z’. t = t’. x

II. Simultaneity A. Set Up 1. Consider two observers, A&B, stationary with respect to each other and reference frame (w,x). How do we calibrate their identical clocks? w x A B

II. Simultaneity A. Set Up 1. Consider two observers, A&B, stationary with respect to each other and reference frame (w,x). How do we calibrate their identical clocks? w Events are simultaneous (clocks calibrated) By backtracking To the equidistant Spacetime position Between light paths. Events WA and WB are simultaneous in this frame. Note: dotted line indicates same TIME for unprimed frame wA wB x A B

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? w (x,t) x

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? Now consider a frame moving with speed v wrt the original frame. t x=(x’ + vt’) x (x,t)

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? Now consider a frame moving with speed v wrt the original frame. t x=(x’ + v(2t’)) x=(x’ + vt’) x (x,t)

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? Now consider a frame moving with speed v wrt the original frame. t x=(x’ + v(3t’)) x=(x’ + v(2t’)) x=(x’ + vt’) x (x,t)

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? Now consider a frame moving with speed v wrt the original frame. x’ t x=(x’ + v(3t’)) x=(x’ + v(2t’)) x=(x’ + vt’) x (x,t)

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? Now consider a frame moving with speed v wrt the original frame. x’ t x (x,t)

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? Now consider a frame moving with speed v wrt the original frame. x’ t t’=t x (x,t)

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? Now consider a frame moving with speed v wrt the original frame. t x’ x

II. Simultaneity A. Set Up 3. Why? Newtonian addition of speeds. x’ t t’ x

II. Simultaneity A. Set Up 2. What happens to the axes for a Galilean Transformation? Now consider a frame moving with speed v wrt the original frame. t + = x

I. Basic Principles C. The ‘Principle of Relativity’ “The Laws of Mechanics are the same in every inertial frame, and the Galilean Transformation is valid.” Problem: Electromagnetism (1850)

I. Basic Principles D. Postulates of Special Relativity • The speed of light in vacuum is a constant, independent of the motion of the source, the observer, or both. • The Laws of Physics are everywhere the same for inertial frames, and the connection between frames is the Lorentz Transformation. • When v/c is small, then the LT reduces to the GT: x’ = (x - vt) & t’ = t. x’ = (x - vt)/{1 - (v/c)2}1/2. y’ = y. z’ = z. t’ = t - vx/c2)/{1 - (v/c)2}1/2. (I.D.1-4) v P x x’

II. Simultaneity A. Set Up 2. What happens to the axes for a Lorentz Transformation? w (x,w) x

II. Simultaneity A. Set Up w (x’,w’) x

II. Simultaneity A. Set Up w x

II. Simultaneity B. Result 1. Now what is measured by another observer moving with respect to the original frame with speed (v’)? w’ w x A B

II. Simultaneity B. Result 1. Now what is measured by another observer moving with respect to the original frame with speed (v’)? w’ w x’ x A B

II. Simultaneity B. Results 1. Now what is measured by another observer moving with respect to the original frame with speed (v’)? w w’ wB’ x’ x A B

II. Simultaneity B. Results 1. Now what is measured by another observer moving with respect to the original frame with speed (v’)? w’ w Events wA’ and wB’ are simultaneous in the red frame. However, now the events are NOT simultaneous in the unprimed frame. Simultaneity Is frame dependent Note: dotted line indicates same TIME for primed frame wB’ wA’ x A B

II. Simultaneity B. Results 1. Now what is measured by another observer moving with respect to the original frame with speed (v)? w t1 t2 t3 x

II. Simultaneity B. Results 1. Now what is measured by another observer moving with respect to the original frame with speed (v)? w’ w x’ t3’ x

II. Simultaneity B. Results 1. Now what is measured by another observer moving with respect to the original frame with speed (v)? w’ w x’ Example: x = 1 lys, t=0s, v/c = 0.5 x’= (1 lys - 0.5c(0s))/{1-0.52}1/2 = 1.15*(5.5) = 1.15 lys t’ = (0 s - 0.5c(1 ly-s)/c2)/{1-0.52}1/2 =1.15*(-.5) = -0.6 second. t3’ x

II. Simultaneity B. Results 1. Now what is measured by another observer moving with respect to the original frame with speed (v)? w’ w x’ Example: x = 5 lys, t=0s, v/c = 0.5 x’= (5 lys – 0.5c(0s))/{1-0.52}1/2 = 1.15*(5) = 5.8 lys t’ = (0 s – 0.5c(5 lys)/c2)/{1-0.52}1/2 =1.15*(-2.5) = -2.9 second. t3’ x

II. Simultaneity B. Results 1. Now what is measured by another observer moving with respect to the original frame with speed (v’)? w’ w x’ x

PHYS 141: Principles of Mechanics