Extra notes on rela,vity. Wade Naylor

Transcription

1 Extra notes on rela,vity Wade Naylor

2 Over 105 years since Einstein s Special theory of relativity A. Einstein,

3 The postulates of special relativity 1. The principle of relativity (Galileo) states that 1. The laws of physics are the same to all observers (in all inertial reference frames) 2. Universality of the speed of light c: 1. c = miles per second or meters per second! From these 2 postulates we found that for moving objects lengths contract clocks slow down mass and energy are related

4 Q2. Newton s law of gravitation states gravity is due to action at a distance (F=GmM/r 2 ). But if light takes 8 minutes to go around the sun and nothing travels faster than light; how can the Earth know how to act? A. gravity is not a force B. because of free fall C. gravity interacts faster than light D. the first two above E. none of the above 2012 Pearson Education, Inc.

5 Q2. Newton s law of gravitation states gravity is due to action at a distance (F=GmM/r 2 ). But if light takes 8 minutes to go around the sun and nothing travels faster than light; how can the Earth know how to act? A. gravity is not a force B. because of free fall C. gravity interacts faster than light D. the first two above E. none of the above 2012 Pearson Education, Inc.

6 General Relativity

7 Q1. What is the reason for the free fall of an astronaut in space? A. There is no gravity in space. B. This is a consequence of Newton s third law. C. The astronaut has no normal (upward) force acting upon him. D. Two of the above three statements are correct. E. All of the first three statements are correct Pearson Education, Inc.

8 Q1. What is the reason for the free fall of an astronaut in space? A. There is no gravity in space. B. This is a consequence of Newton s third law. C. The astronaut has no normal (upward) force acting upon him. D. Two of the above three statements are correct. E. All of the first three statements are correct Pearson Education, Inc.

9 Free fall means what it says Astronauts always feel sick at first when in space for this very reason! Copyright 2012 Pearson Education Inc.

10 Last week s ques,on 1. Can you see a problem with A Journey to VEGA discussion? 2. How is it that some Quasars/galaxies have been observed with redshihs z > 8.2 (high redshih is z>0.1)? 1+z = obs emit = r c + v c v

11 Galilean Transformations t = t y = y z = z x = x + vt or x = x vt y y v z x z x Inertial Frame A frame of reference that is moving at constant velocity Any other frame moving uniformly (constant velocity) with respect to an inertial frame is also inertial Only non-accelerated frames are inertial!

12 No Ether ) speed of light constant Consider the Earth going around the sun? Ether v = 30 km/s or km/h Sun Earth As the Earth goes around the sun we expect to see the speed of light change as it moves with or against the Ether!!!

13 Michelson-Morley Experiment The Michelson-Morley experiment consisted of an interferometer which measurers the time taken for light to travel along the two arms (see picture) The light beams as they travel in different directions would be expected to interfere if light travels in the ETHER, because they will take different times to reach the light detector Taken from

14 Galileo & Newton versus Einstein? The concept of Now is very different Future All observer agree on now Future Now Elsewhere Now Elsewhere Past Future of B B Common Future Future of A A Past In Einstein s special theory of relativity there is no concept of now or simultaneity; Only relative simultaneity Past of B Common Past Past of A

15 Subtleties? Different observers disagree on NOW Observer O uses two clocks to measure s single clock, and vice versa ) disagreement on NOW There is only relative simultaneity Spacetime diagrams help a great deal!

16 Asymmetry relative simultaneity Fig:

17 Twin paradox Terence stays on Earth while Stella makes a 14 yr. round trip into space; 7 yr. outward journey. Terence Stella V=24/25=0.96 = 1 v 2 = 1 (0.96) 2 = 0.28 Assuming that Stella is moving, then Terence sees Stella s proper time Δτ as t = 1 v 2 = = 50yrs

18 However, can t Stella argue that the Earth was traveling with respect to her ship? Terence? Stella V=24/25=0.96 Usual answer: SR does not say that all frames of references are equivalent, only inertial frames! Stella must accelerate to v=0.96 then change direction and then slow down to v=0 back at Earth.

19 Spacetime diagrams However, SR allows for infinite accelerations and we can assume that Stella instantaneously changes direction (No GR required)! For Stella, as she changes her frame she sees time jump from A to C As v increases the jump becomes larger because lines of simultaneity get steeper! v = 0.5/c Fig: Twin_paradox

20 t D C 25yrs Terence Stella returns B Stella leaves At faster speeds this jump gets larger! Note that Stella only covers a very small part of the spacetime of Terence: Terence = Δ PBD Stella = Δ PBA + Δ BCD A P Terence s line of simultaneity x For Stella, Terence s time is PA = 2 yrs., AC=46 yrs., CD=2 yrs. For Terence, Stella s time is PB=7 yrs., BD=7 yrs.

21 Why is Stella is surprised that Terence has aged? Bad spacetime coordinates! Consider an example in 2D Euclidean space y D Analogy taken from Schutz s book Down to bad coordinates? θ C A B x Imagine measuring the line AD in x-y frame, but at point B you rotate the axes by an angle θ to frame Clearly then you would begin at point C and measure CD Total will be AB+CD AD For Stella to realize this fact she must keep smb on the outward journey for (see previous page): AD/0.28 = 48/0.28 =171yrs!

22 Imagine Stella and Terence send laser light pulses to each other every second ) f e =1 Stella sees more blue-shifted light Replace v by v for blueshifts 1 + v/c 1 v/c red-shift 1 v/c 1 + v/c Terence see more red-shifted light Thus, Terence ages more! Still confused? Terence to Stella Stella to Terence Fig:

23 Last week s ques,on 1. Can you see a problem with A Journey to VEGA discussion? 2. How is it that some Quasars/galaxies have been observed with redshihs z > 8.2 (high relaovisoc redshih is z>0.1)? 1+z = obs emit = r c + v c v

24 Homework: Consider Paradoxes? Twin Paradox Time dilation Barn-pole Paradox Lorentz Contraction Relativity of simultaneity

25 Appendix: Spacetime diagrams Draw t-axis against x -axis A 45 degree line is the light cone (speed of light = c [=1]) tan = v An constant velocity inertial object (no acceleration) An accelerating non-inertial object is a curved line Note that two inertial frames O with coordinates (t,x) and O with coordinates (t,x ) are related as in the following cartoon: a -a

26 Appendix: The interval (metric) The spacetime interval is defined as s 2 = t 2 + x 2 + y 2 + z 2 = t 2 + r 2 and it can be shown that the interval is invariant: s 2 = s 2 for any two di erent frames O and Ō Important definitions s 2 < 0 timelike s 2 > 0 spacelike s 2 = 0 null or lightlike

27 Appendix: Relativistic Doppler effect Light source (f e ) t e = λ e Next wave meets at time delay c e v c = 1 (1 v/c)f e = c e f e v Stella However, due to time dilation Stella will measure the time between waves as t o = t e 1 v 2 /c 2 = 1 v2 /c 2 (1 v/c)f e = 1 f o Thus, Stella observes frequency f o = 1 v/c f e v/c v f e v c c 0 Non-relativistic limit

28 Lorentz length contraction Length contracted pole/ladder Length contracted garage/barn Ref: These lead to P.T.O. 28

29 Lorentz contraction paradoxes? Various kinds have been devised We shall look at barn-pole (or ladder-garage) type paradoxes l S =20m Terence Stella v=0.8c Barn b T =15m Key point is that length and time are linked so length contraction leads to time dilation and hence relative simultaneity

30 Barn-Pole: double door variation Problem is only with concept of NOW, there is only relative simultaneity As we can see Stella and Terence disagree on the times when both doors are actually open and shut! Ref: Barn (Terence s) frame Pole (Stella s) frame

31 Double door spacetime diagram Blue and red bands show the barn & pole spacetime, respectively. Front of the pole hits back of barn at event A. D is the point where the end of the pole enters the barn AB is simultaneous in barn frame so this will be what the barn sees as the pole length at the time of event A and thus, the pole fits in the barn However, from the point of view of the pole, AC is the pole length and thus, the back of the pole is outside the barn. The above diagram is in the rest frame of the barn, with x and t being the barn frame. The pole frame is for a person sitting on the front of the pole (axes x and t ). Ref: Ladder_paradox

32 Barn-pole: single door variation Consider a 20m pole which an Olympic athlete (Stella) runs with at speed v=0.8c into a barn of length 15m? l S =20m Terence Stella v=0.8c Barn b T =15m Pole fitting into length contracted barn. Ref: Finite transmission speed (=c) of the shock wave prevents the pole from behaving rigidly and thus, Stella and Terence disagree on the time the door shuts; however, both agree that the door does shut!

33 Single door spacetime diagram In barn frame rod stops simultaneously all along its length. Barn frame sees the ladder as AB, but the pole frame sees the pole as AC. When the back of the pole enters the garage at point D, it has not yet felt the effects of the impact. Spacetime diagram when one of the doors remains shut: Ref: en.wikipedia.org/wiki/ladder_paradox According to someone at rest with respect to the back of the pole, the front of the ladder will be at point E and will see the ladder as DE. The length in the pole frame is not the same as CA which is the rest length of the pole before impact. (See previous slide.)

34 References and final comment References and final comment John Baez s web page for many useful discussions on physics math.ucr.edu/home/baez/physics/ go to SR and twin paradox Wikipedia has many nice diagrams Both of these web cites discuss a myriad of paradoxes in SR including the Barn-pole paradox, e.g., Ladder_paradox For criticism of Rindler s Man in grate paradox see Even over 105 years later, SR still causes much debate and sometimes controversy! However, this is only our Newtonian view of the universe! J