General Relativity General Relativity Relativistic Momentum Mass Equivalence Principle

College Physics B

Einstein’s Theories of Relativity Special Relativity Reminder: College Physics B - PHY2054C Time Dilation

Length Contraction Special & General Relativity General Relativity Relativistic Momentum Mass Equivalence Principle

11/12/2014

My Ofﬁce Hours: Tuesday 10:00 AM - Noon 206 Keen Building College Physics B Outline

Einstein’s Theories of Relativity Special Relativity 1 Einstein’s Theories of Relativity Reminder: Time Dilation Special Relativity Length Contraction General 2 Reminder: Time Dilation Relativity Relativistic Momentum Mass Equivalence 3 Length Contraction Principle

4 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Galilean Relativity and Light

Einstein’s Theories of Relativity Galilean Relativity and electromagnetism do predict different Special Relativity results for observers in different inertial frames: Reminder: Time Dilation Experiments showed that Maxwell’s theory was correct. • Length The speed of light in the vacuum is always c. Contraction • General Galilean relativity for how the speed of light depends on Relativity • Relativistic the motion of the source is wrong. Momentum Mass ➜ Equivalence Einstein developed theory of relativity: Special Relativity. Principle 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. College Physics B Inertial Reference Frames

Einstein’s Theories of Relativity Special Relativity

Reminder: Time Dilation

Length Contraction

General Relativity Relativistic Momentum Mass The modern deﬁnition of an inertial reference is one in Equivalence Principle 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). College Physics B Outline

4 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Light Clock

Einstein’s Theories of Relativity The two postulates lead to a surprising result concerning the Special Relativity nature of time. Reminder: Time Dilation Length A light clock keeps time by using a pulse Contraction

General of light that travels back and forth between Relativity two mirrors: Relativistic Momentum Mass The time for the clock to “tick” once is Equivalence • Principle the time needed for one round trip: College Physics B Moving Light Clock

General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Moving Light Clock

Einstein’s Theories of The clock moves with a constant velocity v relative to the Relativity Special Relativity ground: Reminder: From Ted’s reference frame, the light pulse travels up and Time Dilation • Length down between the two mirrors: ∆t 0 = 2l/c. Contraction Alice sees the light pulse travel a longer distance, but the General • Relativity speed of light is the same for Alice as for Ted. Relativistic Momentum Mass ➜ Because of the longer distance, according to Alice the Equivalence Principle light will take longer to travel between the mirrors. College Physics B Moving Light Clock

Einstein’s Theories of Relativity For Alice, the time for one tick of the clock is: Special Relativity Reminder: ∆t Time Dilation ∆t = 0 Length 1 v 2 Contraction q − c2 General Relativity ➜ Relativistic The time for Ted is different from the time for Alice. Momentum Mass Equivalence The operation of the clock depends on the relative motion. Principle College Physics B Time Dilation

Einstein’s Theories of Relativity Special Relativity

Reminder: Time Dilation

Length Contraction

General Relativity Relativistic Special relativity predicts that moving clocks run slow. Momentum Mass Equivalence Principle This effect is called Time Dilation.

For typical terrestrial speeds, the difference between ∆t and ∆t 0

is negligible. ∆t 0 is called the proper time: ∆t ∆t ∆t ∆t = 0 = 0 0 2 (100 mph)2 1 1 v 1 ≈ q − c2 q − c2 College Physics B Time Dilation

Einstein’s Theories of Relativity Special Relativity

Reminder: Time Dilation

Length Contraction

General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Outline

4 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Lorentz Contraction

Einstein’s Theories of Relativity 1 Special Relativity γ = : 2 2 Reminder: 1 v /c Time Dilation p − Length L0 = v∆t = L = v∆t 0 Contraction 6 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Lorentz Contraction

Einstein’s Theories of Relativity Special Relativity

Reminder: When measuring the length of the moving meterstick, Time Dilation you do so by noting the positions of the two ends at Length Contraction the same time, according to your clock. General Relativity Relativistic Momentum However, those two events – the two measurements Mass Equivalence you make – do not occur at the same time as seen by Principle 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. College Physics B Question

Einstein’s Theories of Relativity The short lifetime of muons created in the upper atmosphere Special Relativity of the Earth would not allow them to reach the surface of the Reminder: Time Dilation Earth unless their lifetime increased by time dilation. From Length the reference system of the muons, the muons can reach the Contraction surface of the Earth because: General Relativity Relativistic Momentum A Time dilation increases their velocity. Mass Equivalence Principle B Time dilation increases their energy. C Length contraction decreases the distance to the Earth. D The relativistic speed of the Earth toward them is added to their velocity. College Physics B Question

Einstein’s Theories of Relativity Special Relativity

Reminder: Time Dilation

Length Contraction

General Relativity Relativistic 1 The speed of light is the maximum possible Momentum Mass speed, and it is always measured to have the Equivalence Principle 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. College Physics B Special Relativity

Einstein’s Theories of Relativity Special Relativity

Reminder: Time Dilation

Length Contraction Relativistic Addition of Velocities: General Relativity Relativistic ′ v1 + v2 Momentum v = v1 v2 Mass 1 + 2 Equivalence c Principle

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. College Physics B Outline

4 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Momentum

Einstein’s Theories of Relativity Conservation of momentum is one of the most fundamental Special Relativity conservation rules in physics and is believed to be satisﬁed by Reminder: Time Dilation all the laws of physics, including the theory of special relativity.

Length Contraction According to Newton’s mechanics, a particle with a mass m0

General moving with speed v has a momentum given by: Relativity Relativistic Momentum Mass p = m0 v Equivalence Principle College Physics B Momentum

Length Contraction According to Newton’s mechanics, a particle with a mass m0

General moving with speed v has a momentum given by: Relativity Relativistic Momentum ∆x Mass p = m0 v = m0 Equivalence Principle ∆t0 College Physics B Momentum

Length Contraction According to Newton’s mechanics, a particle with a mass m0

General moving with speed v has a momentum given by: Relativity Relativistic Momentum ∆x Mass p = m0 v = m0 Equivalence Principle ∆t0 From time dilation and length contraction, measurements of both ∆x and ∆t can be different in different inertial frames: ∆x ∆x p = m0 v = m0 = m0 ∆t0 ∆t 1 v 2/c2 p − College Physics B Momentum

Length Contraction According to Newton’s mechanics, a particle with a mass m0

Einstein’s Theories of From time dilation and length contraction, measurements of Relativity Special Relativity both ∆x and ∆t can be different in different inertial frames:

Reminder: Time Dilation ∆x ∆x p = m0 v = m0 = m0 Length ∆t0 ∆t 1 v 2/c2 Contraction p − General m0 Relativity = v Relativistic 2 2 Momentum 1 v /c Mass p − Equivalence Principle College Physics B Mass

Einstein’s Theories of From time dilation and length contraction, measurements of Relativity Special Relativity both ∆x and ∆t can be different in different inertial frames:

Reminder: Time Dilation ∆x ∆x p = m0 v = m0 = m0 Length ∆t0 ∆t 1 v 2/c2 Contraction p − General m0 Relativity = v Relativistic 2 2 Momentum 1 v /c Mass p − Equivalence Principle m 0 = m v with m = m0 = rest mass √1−v 2/c2 College Physics B Mass

Einstein’s Theories of From time dilation and length contraction, measurements of Relativity Special Relativity both ∆x and ∆t can be different in different inertial frames:

Newton’s second law gives mass as constant of proportionality that relates acceleration and force:

F~ = m ~a X 0 At high speeds, though, Newton’s second law breaks down. College Physics B Mass

Einstein’s Theories of Relativity When the speed of the mass is close to the speed of light, Special Relativity the particle responds to a force as if it had a mass larger than Reminder: Time Dilation m0. The same result happens with momentum where at high

Length speeds the particle responds to impulses and forces as if its Contraction mass were larger than m0: General Relativity Relativistic ∆p ∆(m v) Momentum ~ F = = = m0 ~a Mass X Equivalence ∆t ∆t 6 Principle College Physics B Kinetic Energy

W = F ∆x 2 m0c 2 1 2 KE = m0c = m0v 1 v 2/c2 − 6 2 p − This equation implies that mass is a form of energy: E = mc2. College Physics B General Relativity

Einstein’s Theories of Relativity Special Relativity Special relativity is equivalent to Newtonian mechanics Reminder: in describing objects that move much more slowly than Time Dilation

Length the speed of light, but it differs greatly in its predictions Contraction at relativistic velocities. General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B General Relativity

Einstein’s Theories of Relativity Special Relativity Special relativity is equivalent to Newtonian mechanics Reminder: in describing objects that move much more slowly than Time Dilation

Length the speed of light, but it differs greatly in its predictions Contraction at relativistic velocities. General Relativity Relativistic Momentum What about Newton’s law of gravity? Mass ➜ Equivalence Much more complex mathematical problem in the Principle framework of relativity! College Physics B General Relativity

Einstein’s Theories of Relativity Special Relativity Special relativity is equivalent to Newtonian mechanics Reminder: in describing objects that move much more slowly than Time Dilation

In 1915 Einstein said: It is impossible to tell, from within a closed system, whether one is in a gravitational ﬁeld, or accelerating. College Physics B General Relativity

Einstein’s Theories of Relativity Special Relativity

Reminder: Equivalence Principle Time Dilation

Length It impossible to tell, from Contraction within a closed system, if General Relativity one is in a gravitational Relativistic Momentum ﬁeld, or accelerating. Mass Equivalence Principle College Physics B General Relativity

Einstein’s Theories of Relativity Special Relativity

Reminder: Equivalence Principle Time Dilation

Einstein’s Theories of Relativity Special Relativity All matter tends to warp Reminder: spacetime, and in doing Time Dilation

Length so redeﬁnes straight lines Contraction (the path a light beam General Relativity would take). Relativistic Momentum Mass Equivalence Principle A black hole occurs when the “indentation” caused by the mass of the hole becomes inﬁnitely deep. College Physics B Test of General Relativity

Einstein’s Theories of Relativity Special Relativity Einstein noted that light from a star should be deﬂected Reminder: by a measurable amount as it passes the Sun (1915). Time Dilation

Length Contraction

General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Test of General Relativity

Einstein’s Theories of Relativity In 1919, observers led by the British astronomer Sir Special Relativity Arthur Eddington succeeded in measuring the deﬂection Reminder: Time Dilation of starlight during an eclipse. Length Contraction

General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Test of General Relativity

Einstein’s Theories of Relativity Special Relativity

Reminder: Time Dilation

Length Contraction

General Relativity Relativistic Momentum Mass Equivalence Principle