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Today we will cover R4 with plenty of chance to ask questions on previous material. The questions may be about problems. Tomorrow will be a final Relativity lecture. Thursday will be a review. The practice test is on the front desk. Chapter R2 problems are due today. R1 and (I hope) R2 problems will be handed back tomorrow. Please turn in meter labs even if they are not completely finished

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Chapter R4 The metric equation

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What is the metric equation? An absolute relationship between the clocks in two different inertial systems In the same way that distance between two points is independent of the coordinate system

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A review of the different types of time Coordinate time (Δt) Measures the time between two events in the same inertial coordinate system Proper time (Δ τ) Depends on the spacetime path taken to get from one event to another. Spacetime interval (Δs) The unique time between two events measured by an inertial clock present at both events This is the time the metric equation calculates.

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A light clock Mirrors L The clock counts the number of times a flash of light bounces back and forth between the two mirrors. One “tick” of the clock happens each time the light hits the bottom mirror. One tick happens every 2L seconds. If L=5 seconds, the clock will tick every 10 seconds.

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An experiment to determine the spacetime interval between events in two inertial systems moving a velocity β with respect to each other. We want to find the relationship between the coordinate time in one system and the spacetime interval. You are the home system The front desk is the other system Event A happens in the home system at point (x 1,y 1,z 1,t 1 ), followed by event B at point (x 2,y 2,z 2,t 2 ),.

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We put a light clock in motion at constant velocity so that it is at the two events. The orientation of the light clock is perpendicular to its motion. L is adjusted so that exactly one tick of the clock corresponds to the time to go from A to B. clock AB

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Worldline of the light in the clock Light flash Δd/2 L L L The distance traveled by the light pulse is: Δd/2 ΔdΔd

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Because in our coordinate system the speed of light = 1, then the coordinate time is just the distance traveled. The space time interval is: Δs = 2L (The time between the flashes) Remember that the coordinate time is that time measured by the students in the room, the space time interval is measured on the “other” moving system containing the clock.

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This means that the coordinate time Δt is given by: Or that: And that the spacetime interval Δs is given by: This is the metric equation!

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Experimental verification - muons Muons are particles created when cosmic rays strike the air at the top of the atmosphere Their half-life in the laboratory is 1.52 μs. Their speed in the atmosphere is 0.994c One detector was placed on the top of a mountain, another at sea level. The vertical distance between them was 6.36 μs.

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The time required for the muons to travel the 6.36 μs distance is Δt= 6.36 μs/.99 = 6.40 μs In this time most of them will have decayed (6.40 μs/1.52 μs) = 4.2 half-lives so only 5.4% should remain. These calculations are done from the standpoint of the Earth. If the clock is on the muon then it will measure Δs and not Δt. Δs is the spacetime interval

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Calculation of Δs (The muon system) Calculate Δs for Δt = 6.40μs and Δd = 6.36μs This is 0.47 half lives so 72% of the muons should live to this points as apposed to 5% as measured in the Earth reference frame. The experiment was within the expected error of the 72% value.

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Comparing spacetime to Euclidian space. In Euclidian space r 2 =x 2 +y 2 (a circle) In spacetime Δs 2 = Δt 2 - Δd 2 (a hyperbola) t Δ t ΔdΔd ΔsΔs x

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Example R4.1: In the home frame two events are separated by a distance of 25 ns, and a time of 52ns. In the other frame the distance between the events is 42 ns. What is the time separation in the other system? Because Because Δs is the same in both systems we can write: Solve this equation to find Δt‘. Δt‘ = 62 ns

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Twin Paradox (part 1) R4.3 Alpha Centauri is 4.3 light years away. A ship makes the trip in 13 years by Earth clocks, how long did the trip take by the clock on the ship? Worldline of Alpha Centauri 4.3 yrs 13 yrs Worldline of spaceship

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Which time are we trying to find? Spacetime (Δs) Put the known values in the above equation and find Δs. Δs = 4.9 years The ship’s clock says the trip took 9.8 years, the Earth’s clock says it took 13 years.

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Exercise R4X.5: If the round trip to Alpha Centauri took 2 years according to a clock on the ship, how long would the Earth’s clock say it took? What do we know? Δ τ = Δs = 1 yr. Δd = 4.3 years Δt =? Δt = 4.4 years

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Problems due Thursday R4B.1, R4B.2, R4B.4

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