Relativity IV Non-rel. Modern Art Quiz Lorentz Transformations Relativistic addition of velocities Doppler Effect

History: Special Relativity‘s apparent impact on 20th century art   “In the intellectual atmosphere of 1905 it is not surprising that Einstein and Picasso began exploring new notions of space and time almost coincidentally.….Just as relativity theory overthrew the absolute status of space and time, the cubism of Georges Braque and Picasso dethroned perspective in art.” Anyone want to guess what this is a picture of? Galison et al disputes this claim. Georges Braque, Man with Guitar 2

Special Relativity‘s possible impact on 20th century art Ans: Salvador Dali. 1931 “The persistence of memory.” Who was the artist ? 3

Suppose we have a space ship capable of traveling at 0.5c. Q11.1 Suppose we have a space ship capable of traveling at 0.5c. An astronaut travels to a destination 1 light-year from Earth. When she reaches there, how much time has elapsed according to her clock on the spaceship? D (delta t_0 = delta t/gamma); 0.5**2=0.25 4

Suppose we have a space ship capable of traveling at 0.5c. Q11.1 Suppose we have a space ship capable of traveling at 0.5c. An astronaut travels to a destination 1 light-year from Earth. When she reaches there, how much time has elapsed according to her clock on the spaceship? Takes 2 years to travel 1 light year at 0.5c in frame S D (delta t_0 = delta t/gamma); 0.5**2=0.25 β = 0.5; β2 = 0.25 Δt0 = Δt/γ 5

Q11.2 A crewman measures the length his space ship to be 50 meters long. The space ship passes by Earth at 0.8c with respect to Earth. What is the length of the spaceship according to an observer on Earth ? 50 m 60 m 30 m 23 m C (Lorentz contraction L = L_0/gamma = 50m/gamma gamma=1/sqrt(1-0.8**2)=1.66 50m/1.66=30m 6

Q11.2 A crewman measures the length his space ship to be 50 meters long. The space ship passes by Earth at 0.8c with respect to Earth. What is the length of the spaceship according to an observer on Earth ? 50 m 60 m 30 m 23 m Length contraction 50m/1.66 = 30m C (Lorentz contraction L = L_0/gamma = 50m/gamma gamma=1/sqrt(1-0.8**2)=1.66 50m/1.66=30m γ = 1/sqrt(1-0.8**2) = 1.66 7

Q11.3 A 8

Remember, if one transforms to x,t coordinates, velocity changes sign Q11.3 A Remember, if one transforms to x,t coordinates, velocity changes sign 9

Q11.4 B 10

Q11.4 B 11

The Lorentz transformations Lorentz transformations relate the coordinates and velocities in two inertial reference frames. They are more general than the Galilean transformations and are consistent with the principle of relativity. Galilean transformations. Do not work at relativistic velocities. 12

The Lorentz transformations (“boost along x”) Space and time are intertwined: 4 dimensional “space-time” Intellectual revolution Note the coordinates perpendicular to the “boost“ are unmodified Check the s(i)gn. 13

The Lorentz transformations (“boost along x”) Space and time are intertwined: 4 dimensional “space-time” Intellectual revolution. Note the two equations are related. Time dilation Caution! Don’t confuse these 2 !! 14

How do we calculate a “relativistic boost along y” ? Note the coordinates perpendicular to the “boost“ are unmodified 15

Example using the Lorentz transformations Winning an interstellar race, Mavis pilots her spaceship across a finish line in space at a speed of 0.600 c. A “hooray” message is sent from the back of her ship (event 2) at the instant in her frame of reference that the front of her ship crosses the finish line (event 1). Mavis measures the length of her ship to be 300 m. Stanley is located at the finish line and is at rest relative to it. When and where does Stanley measure events 1 and 2 to occur ? S is Stanley’s frame while S’ is Mavis’ frame Event 1 occurs at x=0, t=0 in S and x’=t’=0 in S’ Event 2 in S’ (Mavis’ frame) occurs at t’=0, x’=-300m Let’s use the Lorentz transformation to find x and t in Stanley’s frame 16

Example using the Lorentz transformations S is Stanley’s frame while S’ is Mavis’ frame Event 1 occurs at x=0, t=0 in S and x’=t’=0 in S’ Event 2 in S’ (Mavis’ frame) occurs at t’=0, x’=-300m Let’s use the Lorentz transformation to find x and t in Stanley’s frame [but be careful, let’s change x’ x, t’t and therefore u(-u)] Note that in Mavis’ frame the two events are simultaneous (so simultaneity has broken down in this example). Note that in Mavis’ frame the two events are simultaneous (so simultaneity has broken down in this example). 17

Relativistic addition of velocities (take differentials) Start from Lorentz transformations. 18

Relativistic addition of velocities cont’d This gives velocities in S’ in terms of S where S’ is moving at velocity u with respect to S. Question: How do we get velocities in S in terms of velocities in S’ ? Ans: interchange primed and unprimed velocities and change u to –u (why ?) 19

Relativistic addition of velocities Question: What happens if vx=c ? Ans: vx’=c (according to Einstein’s second postulate). Let’s check if this really works ✔ Question: what happens if u<<c ? 20

Relativistic addition of velocities example (part a) A spaceship moving away from earth at 0.900c fires a robot probe at 0.700c relative to the spaceship. What is the probe’s velocity relative to the earth ? Let S and S’ be the reference frames of the Earth and the Spaceship. Their relative velocity of S’ and S are u=0.900c. Note the sgn I get v_x= 0.9816 21

Relativistic addition of velocities example (part b) A scoutship is sent to catch up at 0.950 relative to the earth. What is the velocity of the scoutship relative to the spaceship ? Let S and S’ be the reference frames of the Scoutship and the Spaceship. Their relative velocity of S’ and S are u=0.900c. Note the sgn I get 0.3448c 22

Bank Robbery! The notorious intergalactic bank robbers, known as the Warp Drive Kids, has hit the First Alpha Centari bank, and is trying to make a getaway. They are cruising to the Dark Sector at their maximum speed of 0.8c Fortunately a police star cruiser was already en route to the Dark Sector at its maximum speed of 0.3c. Therefore it launched an interceptor at 0.6c. Assuming both start equidistant from the Dark Sector, will intercept happen before the bad guys can go dark? I get v_x= 0.9816 According to Galileo? Yes No Dead heat Impossible to know 23

Bank Robbery! The notorious intergalactic bank robbers, known as the Warp Drive Kids, has hit the First Alpha Centari bank, and is trying to make a getaway. They are cruising to the Dark Sector at their maximum speed of 0.8c Fortunately a police star cruiser was already en route to the Dark Sector at its maximum speed of 0.3c. Therefore it launched an interceptor at 0.6c. Assuming both start equidistant from the Dark Sector, will intercept happen before the bad guys can go dark? I get v_x= 0.9816 According to Galileo? Yes No Dead heat Impossible to know 0.6c+0.3c = 0.9c 24

Bank Robbery! The notorious intergalactic bank robbers, known as the Warp Drive Kids, has hit the First Alpha Centari bank, and is trying to make a getaway. They are cruising to the Dark Sector at their maximum speed of 0.8c Fortunately a police star cruiser was already en route to the Dark Sector at its maximum speed of 0.3c. Therefore it launched an interceptor at 0.6c. Assuming both start equidistant from the Dark Sector, will intercept happen before the bad guys can go dark? I get v_x= 0.9816 According to Einstein? Yes No Dead heat 25

Bank Robbery! The notorious intergalactic bank robbers, known as the Warp Drive Kids, has hit the First Alpha Centari bank, and is trying to make a getaway. They are cruising to the Dark Sector at their maximum speed of 0.8c Fortunately a police star cruiser was already en route to the Dark Sector at its maximum speed of 0.3c. Therefore it launched an interceptor at 0.6c. Assuming both start equidistant from the Dark Sector, will intercept happen before the bad guys can go dark? I get v_x= 0.9816 Vtotal = u+vintercept/(1+u.vintercept) Yes No Dead heat Vtotal = 0.76c Vtotal = (0.3c)+(0.6c)/(1+ (0.3c).(0.6c)) 26

Salzburg, Austria Read 37.9 General Relativity