College Physics B - PHY2054C. Special & General Relativity 11/12/2014. My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building.

Transcription

1 Special College - PHY2054C Special & 11/12/2014 My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building

2 Outline Special 1 Special 2 3 4

3 Special Galilean and Light Galilean and electromagnetism do predict different results for observers in different inertial frames: Experiments showed that Maxwell s theory was correct. The speed of light in the vacuum is always c. Galilean relativity for how the speed of light depends on the motion of the source is wrong. Einstein developed theory of relativity: Special. Two Postulates 1 All laws of physics are the same in all inertial reference frames. 2 The speed of light in the vacuum is a constant.

4 Inertial Reference Frames Special The modern definition of an inertial reference is one in which Newton s First Law holds: If a particle moves with a constant velocity, then the reference frame is inertial. Earth s acceleration is small enough that it can be ignored (can be considered an inertial system).

5 Outline Special 1 Special 2 3 4

6 Special Light Clock The two postulates lead to a surprising result concerning the nature of time. A light clock keeps time by using a pulse of light that travels back and forth between two mirrors: The time for the clock to tick once is the time needed for one round trip:

7 Special Moving Light Clock The clock moves with a constant velocity v relative to the ground: From Ted s reference frame, the light pulse travels up and down between the two mirrors: t 0 = 2l/c.

8 Special Moving Light Clock The clock moves with a constant velocity v relative to the ground: From Ted s reference frame, the light pulse travels up and down between the two mirrors: t 0 = 2l/c. Alice sees the light pulse travel a longer distance, but the speed of light is the same for Alice as for Ted. Because of the longer distance, according to Alice the light will take longer to travel between the mirrors.

9 Special Moving Light Clock For Alice, the time for one tick of the clock is: t = t 0 1 v2 c 2 The time for Ted is different from the time for Alice. The operation of the clock depends on the relative motion.

10 Special Special relativity predicts that moving clocks run slow. This effect is called. For typical terrestrial speeds, the difference between t and t 0 is negligible. t 0 is called the proper time: t = t 0 1 v2 c 2 = 1 t 0 (100 mph)2 c 2 t 0 1

11 Special

12 Outline Special 1 Special 2 3 4

13 Special Lorentz γ = 1 1 v 2 /c 2 : L 0 = v t L = v t 0

14 Lorentz Special When measuring the length of the moving meterstick, you do so by noting the positions of the two ends at the same time, according to your clock. However, those two events the two measurements you make do not occur at the same time as seen by the moving observer. In relativity, time is relative, and simultaneity (the idea that two events happen at the same time ) is no longer a well-defined concept.

15 Special Question The short lifetime of muons created in the upper atmosphere of the Earth would not allow them to reach the surface of the Earth unless their lifetime increased by time dilation. From the reference system of the muons, the muons can reach the surface of the Earth because: A Time dilation increases their velocity. B Time dilation increases their energy. C contraction decreases the distance to the Earth. D The relativistic speed of the Earth toward them is added to their velocity.

16 Special Question The short lifetime of muons created in the upper atmosphere of the Earth would not allow them to reach the surface of the Earth unless their lifetime increased by time dilation. From the reference system of the muons, the muons can reach the surface of the Earth because: A Time dilation increases their velocity. B Time dilation increases their energy. C contraction decreases the distance to the Earth. D The relativistic speed of the Earth toward them is added to their velocity.

17 Special Special 1 The speed of light is the maximum possible speed, and it is always measured to have the same value by all observers. 2 There is no absolute frame of reference, and no absolute state of rest. 3 Space and time are not independent, but are unified as spacetime.

18 Special Special Addition of Velocities: v = v 1 + v v 1 v 2 c 2 1 When two velocities are much less than the speed of light, the relativistic addition of velocities gives nearly the same result as the Newtonian equation. Okay for speeds less than 10% of the speed of light! 2 Experiments with particles moving at very high speeds show that the relativistic result is correct.

19 Outline Special 1 Special 2 3 4

20 Special Conservation of momentum is one of the most fundamental conservation rules in physics and is believed to be satisfied by all the laws of physics, including the theory of special relativity. According to Newton s mechanics, a particle with a mass m 0 moving with speed v has a momentum given by: p = m 0 v

21 Special Conservation of momentum is one of the most fundamental conservation rules in physics and is believed to be satisfied by all the laws of physics, including the theory of special relativity. According to Newton s mechanics, a particle with a mass m 0 moving with speed v has a momentum given by: p = m 0 v = m 0 x t 0

22 Special Conservation of momentum is one of the most fundamental conservation rules in physics and is believed to be satisfied by all the laws of physics, including the theory of special relativity. According to Newton s mechanics, a particle with a mass m 0 moving with speed v has a momentum given by: p = m 0 v = m 0 x t 0 From time dilation and length contraction, measurements of both x and t can be different in different inertial frames: x x p = m 0 v = m 0 = m 0 t 0 t 1 v 2 /c 2

23 Special Conservation of momentum is one of the most fundamental conservation rules in physics and is believed to be satisfied by all the laws of physics, including the theory of special relativity. According to Newton s mechanics, a particle with a mass m 0 moving with speed v has a momentum given by: p = m 0 v = m 0 x t 0 From time dilation and length contraction, measurements of both x and t can be different in different inertial frames: p = m 0 v = m 0 x t 0 = m 0 = m 0 v 1 v 2 /c 2 x t 1 v 2 /c 2

24 Special From time dilation and length contraction, measurements of both x and t can be different in different inertial frames: p = m 0 v = m 0 x t 0 = m 0 x t 1 v 2 /c 2 = m 0 1 v 2 /c 2 v

25 Special From time dilation and length contraction, measurements of both x and t can be different in different inertial frames: p = m 0 v = m 0 x t 0 = m 0 x t 1 v 2 /c 2 = m 0 1 v 2 /c 2 v = m v with m = m 0 1 v 2 /c 2 m 0 = rest mass

26 Special From time dilation and length contraction, measurements of both x and t can be different in different inertial frames: p = m 0 v = m 0 x t 0 = m 0 x t 1 v 2 /c 2 = m 0 1 v 2 /c 2 v = m v with m = m 0 1 v 2 /c 2 m 0 = rest mass Newton s second law gives mass as constant of proportionality that relates acceleration and force: F = m0 a At high speeds, though, Newton s second law breaks down.

27 Special When the speed of the mass is close to the speed of light, the particle responds to a force as if it had a mass larger than m 0. The same result happens with momentum where at high speeds the particle responds to impulses and forces as if its mass were larger than m 0 : F = p t = (m v) t m 0 a

28 Special Kinetic Energy When the speed of the mass is close to the speed of light, the particle responds to a force as if it had a mass larger than m 0. The same result happens with momentum where at high speeds the particle responds to impulses and forces as if its mass were larger than m 0 : F = p t = (m v) t m 0 a Kinetic Energy (non-trivial, requires calculus): W = F x KE = m 0 c 2 1 v 2 /c 2 m 0c m 0v 2 This equation implies that mass is a form of energy: E = mc 2.

29 Special Special relativity is equivalent to Newtonian mechanics in describing objects that move much more slowly than the speed of light, but it differs greatly in its predictions at relativistic velocities.

30 Special Special relativity is equivalent to Newtonian mechanics in describing objects that move much more slowly than the speed of light, but it differs greatly in its predictions at relativistic velocities. What about Newton s law of gravity? Much more complex mathematical problem in the framework of relativity!

31 Special Special relativity is equivalent to Newtonian mechanics in describing objects that move much more slowly than the speed of light, but it differs greatly in its predictions at relativistic velocities. What about Newton s law of gravity? Much more complex mathematical problem in the framework of relativity! In 1915 Einstein said: It is impossible to tell, from within a closed system, whether one is in a gravitational field, or accelerating.

32 Special It impossible to tell, from within a closed system, if one is in a gravitational field, or accelerating.

33 Special It impossible to tell, from within a closed system, if one is in a gravitational field, or accelerating. Unavoidable conclusion: Spacetime is curved.

34 Special Curved Space All matter tends to warp spacetime, and in doing so redefines straight lines (the path a light beam would take). A black hole occurs when the indentation caused by the mass of the hole becomes infinitely deep.

35 Special Test of Einstein noted that light from a star should be deflected by a measurable amount as it passes the Sun (1915).

36 Special Test of In 1919, observers led by the British astronomer Sir Arthur Eddington succeeded in measuring the deflection of starlight during an eclipse.

37 Test of Special