1Lecture 8Reminders:Reading prior to this lecture TZD 2.1-2.3Reading for next lecture TZD 2.4-2.9Homework #3 due next WednesdayExam #1 will be Friday, September 25 in class.Today’s topic: Causality and KinematicsTito SalominiGeorge has a set of synchronized clocks in reference frame S, asshown. Gracie is moving to the right past George, and hasher own set of synchronized clocks.Gracie passes George at the event (0,0) in both frames. An observer in George’s frame checks the clock marked (?) at t=0. Compared to George’s clocks, this one reads... -3 -2 -1 0 1 2 3 ...v?GeorgeGracie)()(2xcvttvtxx−=′−=′γγA) Slightly before 3:00 B) Slightly after 3:00C) Exactly 3:00Clicker questionGeorgeThe event has coordinates (-3,0) for George.In Gracie’s frame where the (?) clock is, the time is, a positive quantity. ... -3 -2 -1 0 1 2 3 ...v?GeorgeGracie)()(2xcvttvtxx−=′−=′γγ223))3(0(cvcvtγγ+=−−=′A) Slightly before 3:00 B) Slightly after 3:00Remember the Train?LeiahEvent 1 – firecracker explodesEvent 2 – light reaches detectorDistance between events is htch Δ=Remember What Luke Observes?Event 1 – firecracker explodesEvent 2 – light reaches detectorDistance between events is cΔt’But distance between x-coordinates is Δx’.We can writehcΔt’Δx’()()xtch′Δ−′Δ=222Leia getssince()()xtchΔ−Δ=2220=ΔxSpace-Time IntervalSay the difference between two events is given by21212121tttzzzyyyxxx−=Δ−=Δ−=Δ−=ΔThen the space-time interval()()()()zyxtcsΔ−Δ−Δ−Δ=Δ22222This is a weird kind of “distance” in space-time.In fact, this space-time interval turns out to be invariant, i.e. the same in all inertial reference frames.2Space-TimexctHere is an event in space-time. Any light signal that passes through this event has the dashed world lines. These identify the light cone of this event.Light cones are 45° lines on spacetime diagrams.Space-TimexctThe blue area is the future on this event.The spacetime interval between events A and B isA) PositiveB) NegativeC) zeroAB()()()()zyxtcsΔ−Δ−Δ−Δ=Δ22222Space-TimexctHere is an event in spacetime. The blue area is the future on this event.The pink is its past.CausalityxctIf A precedes B, then Δs2>0.If Δs2>0 in one reference frame, then it’s positive in all reference frames.So, causality is maintainedin special relativity.AB()()()()zyxtcsΔ−Δ−Δ−Δ=Δ22222SpacetimexctHere is an event in spacetime. The yellow area is the elsewhere of the event. No physical signal can travel from the event to its elsewhere. That would imply traveling faster than the speed of light.Axct1AClicker questionStarting from Event 1, which other events could be reached? That is, what other events is it possible to travel to?A. AB. BC. CD. More than one of the aboveE. None of the aboveCBReaching point B requires going faster than the speed of lightReaching point C requires going into the past.3Kinematics)()(2xcvttzzyyvtxx−=′=′=′−=′γγAs mentioned before, since space and time transforms are changing, we should expect that velocity, momentum, etc. will be changing too.This will turn Newton on his head a bit.Velocity... -3 -2 -1 0 1 2 3 ...S... -3 -2 -1 0 1 2 3 ...S’vABAs seen from S, its speed is As seen from S’, its speed istxuΔΔ=txu′Δ′Δ=′An object moves from event A to event B.Velocity Transformation))/(()(2xcvttvxtxuΔ−ΔΔ−Δ=′Δ′Δ=′γγ2/1 cuvvuu−−=′Galilean resultNew in special relativityCancel the γ and divide top and bottom by Δt to get22/1)/(1cuvvutxcvvtx−−=ΔΔ−−ΔΔ=The velocity addition formula in special relativity is:Using the Lorentz transformation for x′ and t′ we get:Note that u and u′ are the velocities of an object in a frame while v is the velocity of one frame to another (S′ velocity relative to S).2/1 cuvvuu−−=′2/1 cvuvuu′++′=Clicker questionA spacecraft travels at speed v=0.5c relative to the Earth. It launches a missile in the forward direction at a speed of 0.5c. How fast is the missile moving relative to Earth?A. 0B. 0.25cC. 0.5cD. 0.8cE. cHave to keep signs straight. Depends on which way you are transforming. Also, the velocities can be positive or negative!Best way to solve these is to figure out if the speeds add or subtract and then use the appropriate formula.Since the missile if fired forward in the spacecraft frame, the spacecraft and missile velocities add in the Earth frame.ccccccccvuvuu 8.025.1/5.05.015.05.0/122==⋅++=′++′=2/1 cuvvuu−−=′2/1 cvuvuu′++′=Velocity addition works with light too!A Spacecraft moving at 0.5c relative to Earth sends out a beam of light in the forward direction. What is the light velocity in the Earth frame?ccccccccvuvuu ==⋅++=′++′=5.15.1/5.015.0/122What about if it sends the light out in the backward direction?ccccccccvuvuu −=−=⋅−+−=′++′=5.05.0/5.015.0/122It works. We get the same speed of light no matter what!Isaac Newton told us that F = m aWhen we apply a force to an object, the object will accelerate (e.g. increase its velocity).One can also think of this equationin terms of momentum transfer per time and velocity change.tvmtpΔΔ=ΔΔ4Suppose we have a 1 kg mass ball that is traveling at v = 2 x 108m/s.If we hit it with a baseball bat and transfer momentum Δp= 2 x 108kg m/s, what happens according to Newton?A) The ball stays at the same velocity.B) The ball accelerates to v = 3 x 108m/sC) The ball accelerates to v = 4 x 108m/sD) The ball decelerates to v = 1 x 108m/sNot Good!Clicker questionEinstein found that he had to re-write all of Newton’s equations of motion (including F=ma).tvmtpΔΔ=ΔΔIf the velocity cannot change anymore, perhaps the mass changes?2200/1 cvvmvmp−==γOne can think of there being a “rest mass” m0(the mass when the object is at rest v=0), and then the mass when moving as:2200/1