Newton’s Laws of Motion

1. The theory of special relativity applies in inertial (i.e., non-accelerating) reference frames. The theory of general relativity generalizes special relativity to include non-inertial reference frames, and provides a unified description of gravity as a geometric property of space and time (spacetime). In this course, we only learn about the theory of special relativity. I encourage you to also take course on the theory of general relativity!

2. Newton’s Laws of Motion • 1st law: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. (The law of inertia) • 2nd law: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. (a = F/m) • 3rd law: For every action, there is an equal and opposite reaction.

3. Newton’s Laws of Motion • 1st law: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. (The law of inertia) • 2nd law: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. (a = F/m) • 3rd law: For every action, there is an equal and opposite reaction.

4. Newton’s Laws of Motion • 1st law: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. (The law of inertia) • 2nd law: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. (a = F/m) • 3rd law: For every action, there is an equal and opposite reaction.

5. Newton’s Laws of Motion • 1st law: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. (The law of inertia) • 2nd law: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. (a = F/m) • 3rd law: For every action, there is an equal and opposite reaction.

6. Newton’s Laws of Motion • In Newtonian (classical) physics, the Principle of Relativity applies: the laws of physics are the same in all reference frames.

7. Newton’s Laws of Motion • In Newtonian (classical) physics, quantities such as displacement, velocity, acceleration, and force are vector quantities, which possess the properties of both magnitude and direction. • When we consider two (or more) components of motion, the above quantities undergo vectorialaddition. or, in vector form vx (Galilean transformation)

8. Learning Objectives • What is the single additional postulate Einstein made that makes the theory of special relativity different from Newtonian (classical) physics?The speed of light is the same in all reference frames. (In Newtonian physics, the speed of light is treated as a vector quantity, and therefore different as measured from different reference frames.) • What are the consequences of applying this additional postulate? E.g., can we still add vectorially physical quantities such as displacement and velocity?1. If the speed of light is constant in all reference frames, length and time must then be different as measured from different reference frames: length contraction and time dilation. (In Newtonian physics, length and time are the same as measured from different reference frames.) • 2. Different events that occur simultaneously in one reference frame need not occur simultaneously in a different reference frame: loss of simultaneity. (In Newtonian physics, different events that occur simultaneously in one refframe also occur simultaneously in a different reference frame.) • Some applications of special relativity on Earth • Some applications of special relativity in astrophysics

9. Learning Objectives • What is the single additional postulate Einstein made that makes the theory of special relativity different from Newtonian (classical) physics?The speed of light is the same in all reference frames. (In Newtonian physics, speed is a vector quantity, and therefore different as measured from different reference frames.) We shall return to the experimental proof in the next lecture. • What are the consequences of applying this additional postulate? E.g., can we still add vectorially physical quantities such as velocity?1. If the speed of light is constant in all reference frames, length and time must then be different as measured from different reference frames: length contraction and time dilation. (In Newtonian physics, length and time are the same as measured from different reference frames.) • 2. Different events that occur simultaneously in one reference frame need not occur simultaneously in a different reference frame: loss of simultaneity. (In Newtonian physics, different events that occur simultaneously in one refframe also occur simultaneously in a different reference frame.) • Some applications of special relativity on Earth • Some applications of special relativity in astrophysics

10. Learning Objectives • What is the single additional postulate Einstein made that makes the theory of special relativity different from Newtonian (classical) physics? The speed of light is the same in all reference frames. (In Newtonian physics, speed is a vector quantity, and therefore different as measured from different reference frames.) We shall return to the experimental proof in the next lecture. • Under this postulate, quantities such as velocities no longer undergo simple vectorial addition! You can see this by setting vx = c, vy =vz =0. or, in vector form (Galilean transformation)

11. Learning Objectives • What is the single additional postulate Einstein made that makes the theory of special relativity different from Newtonian (classical) physics? The speed of light is the same in all reference frames. (In Newtonian physics, speed is a vector quantity, and therefore different as measured from different reference frames.) We shall return to the experimental proof in the next lecture. • Under this postulate, quantities such velocities no longer undergo simple vectorial addition! Instead, as we shall see: (Lorentz transformation)

12. Learning Objectives • What is the single additional postulate Einstein made that makes the theory of special relativity different from Newtonian (classical) physics? The speed of light is the same in all reference frames. (In Newtonian physics, speed is a vector quantity, and therefore different as measured from different reference frames.) We shall return to the experimental proof in the next lecture. • What are the consequences of applying this additional postulate? 1. If the speed of light is constant in all reference frames, length and time must then be different as measured from different reference frames: length contraction and time dilation. (In Newtonian physics, length and time are the same as measured from different reference frames.) • 2. Different events that occur simultaneously in one reference frame need not occur simultaneously in a different reference frame: loss of simultaneity. (In Newtonian physics, different events that occur simultaneously in one reference frame also occur simultaneously in a different reference frame.) • Some applications of special relativity on Earth • Some applications of special relativity in astrophysics

13. Learning Objectives • What is the single additional postulate Einstein made that makes the theory of special relativity different from Newtonian (classical) physics? The speed of light is the same in all reference frames. (In Newtonian physics, speed is a vector quantity, and therefore different as measured from different reference frames.) We shall return to the experimental proof in the next lecture. • What are the consequences of applying this additional postulate? 1. If the speed of light is constant in all reference frames, length and time must then be different as measured from different reference frames: length contraction and time dilation. (In Newtonian physics, length and time are the same as measured from different reference frames.) • 2. Different events that occur simultaneously in one reference frame need not occur simultaneously in a different reference frame: loss of simultaneity. (In Newtonian physics, different events that occur simultaneously in one reference frame also occur simultaneously in a different reference frame.) • Some applications of special relativity on Earth • Some applications of special relativity in astrophysics

14. Length and Time as Absolutes • Consider two people, each having invariant rulers and clocks (rulers that maintain the same length and clocks that tick over at the same rate for both people; that is what we mean by length and time being absolutes). S′ moves to the right at a speed of 1 m/s relative to S. • At time t = 0, both people are located at the same position (along the horizontal axis). At this time, a ball at the same position is kicked to the right at a speed of 10 m/s according to S. • According to S′, at what speed is the ball kicked to the right? t = 0 s S′ 1 m/s 10 m/s S

15. Length and Time as Absolutes • Consider two people, each having invariant rulers and clocks (rulers that maintain the same length and clocks that tick over at the same rate for both people; that is what we mean by length and time being absolutes). S′ moves to the right at a speed of 1 m/s relative to S. • At time t = 0, both people are located at the same position (along the horizontal axis). At this time, a ball at the same position is kicked to the right at a speed of 10 m/s according to S. • According to S′, at what speed is the ball kicked to the right? 10 m/s – 1 m/s = 9 m/s. This common sense treatment of velocity assumes that velocity is not an absolute, but that length and time are absolutes (velocity = length of travel/time of travel). t = 0 s S′ 1 m/s 10 m/s S

16. Length and Time as Absolutes • At time t = 10 s, how far will the ball be located from S according to S ? • 10 m/s × 10 s = 100 m • At time t = 10 s, how far will the ball be located from S′ according to S ? • 100 m – (1 m/s × 10 s) = 100 m – 10 m = 90 m • At time t = 10 s, how far will the ball be located from S′ according to S′? • (10 m/s – 1 m/s) × 10 s = 9 m/s × 10 s = 90 m. Ball velocity = 90 m/10 s = 9 m/s. Velocity is not an absolute! t = 10 s S′ 1 m/s 10 m/s S

17. Time and Velocity as Absolutes • Consider two people, each having invariant clocks (tick over at the same rate for both people; that is what we mean by time being an absolute). S′ moves to the right at a speed of 1 m/s relative to S. • At time t = 0, both people are located at the same position (along the horizontal axis). At this time, a ball at the same position is kicked to the right at a speed of 10 m/s according to both S and S′; that is what we mean by velocity being an absolute. t = 0 s S′ 1 m/s 10 m/s S

18. Time and Velocity as Absolutes • At time t = 10 s, how far will the ball be located from S according to S ? • 10 m/s × 10 s = 100 m • At time t = 10 s, how far will the ball be located from S′ according to S ? • 100 m – (1 m/s – 10 s) = 100 m – 10 m = 90 m • At time t = 10 s, how far will the ball be located from S′ according to S′? • 10 m/s × 10 s = 100 m Contradictory! How can this contradiction be resolved? According to S, length of ruler of S′ must have changed (shrunk by 10%) because space of S′has shrunked. t = 10 s S′ Note: The fact that the moving ruler shrinks in this example is not a proof of length contraction. In this example, we have assumed that time is an absolute. In Special Relativity, time is not an absolute. What this example demonstrates is that, if we assume time and velocity to be absolutes, then length cannot be an absolute. 1 m/s 10 m/s S

19. Length and Velocity as Absolutes • Consider two people, each having invariant rulers (maintain the same length for both people; that is what we mean by length being an absolute). S′ moves to the right at a speed of 1 m/s relative to S. • At time t = 0, both people are located at the same position (along the horizontal axis). At this time, a ball at the same position is kicked to the right at a speed of 10 m/s according to both S and S′; that is what we mean by velocity being an absolute. t = 0 s S′ 1 m/s 10 m/s S

20. Length and Velocity as Absolutes • How long does it take for ball to reach a length of 100 m from S according to S ? • 100 m / 10 m/s = 10 s • How long does it take for ball to reach a length of 100 m from S′ according to S ? • (>100 m) / (10 m/s) = >10 s • How long does it take for ball to reach a length of 100 m from S′ according to S′? • 100 m / 10 m/s = 10 s Contradictory! How can this contradiction be resolved? According to S, clock of S′ must tick at a slower rate because time for S′passes more slowly. t = 10 s S′ Note: The fact that the moving clock ticks at a slower rate in this example is not a proof of time dilation. In this example, we have assumed that length is an absolute. In Special Relativity, length is not an absolute. What this example demonstrates is that, if we assume length and velocity to be absolutes, then time cannot be an absolute. 1 m/s 10 m/s S

21. Simultaneity in Newtonian Physics • A person stands midway between two flashbulbs, all situated on a platform. Both the platform observer and us are not moving relative to each other. • According to the platform observer, do the two flashbulbs light up simultaneously?

22. Simultaneity in Newtonian Physics • A person stands midway between two flashbulbs, all situated on a platform. Both the platform observer and us are not moving relative to each other. • According to the platform observer, do the two flashbulbs light up simultaneously? Yes

23. Simultaneity in Newtonian Physics • A person stands midway between two flashbulbs, all situated on a platform. Both the platform observer and us are not moving relative to each other. • According to the platform observer, do the two flashbulbs light up simultaneously? Yes • Do we see the two flashbulbs light up simultaneously?

24. Simultaneity in Newtonian Physics • A person stands midway between two flashbulbs, all situated on a platform. Both the platform observer and us are not moving relative to each other. • According to the platform observer, do the two flashbulbs light up simultaneously? Yes • Do we see the two flashbulbs light up simultaneously? Yes. Both we and the platform observer see the two flashbulbs light up simultaneously. Simultaneity is preserved; i.e., both the platform and us agree on the sequence of events.

25. Simultaneity in Newtonian Physics • A person stands midway between two flashbulbs, all situated on a platform. The platform moves at a uniform velocity relative to us. • According to the platform observer, do the two flashbulbs light up simultaneously?

26. Simultaneity in Newtonian Physics • A person stands midway between two flashbulbs, all situated on a platform. Both the platform observer and us are not moving relative to each other. • According to the platform observer, do the two flashbulbs light up simultaneously? Yes

27. Simultaneity in Newtonian Physics • A person stands midway between two flashbulbs, all situated on a platform. Both the platform observer and us are not moving relative to each other. • According to the platform observer, do the two flashbulbs light up simultaneously? Yes • Do we see the two flashbulbs light up simultaneously?

28. Simultaneity in Newtonian Physics • A person stands midway between two flashbulbs, all situated on a platform. Both the platform observer and us are not moving relative to each other. • According to the platform observer, do the two flashbulbs light up simultaneously? Yes • Do we see the two flashbulbs light up simultaneously? Yes. Both we and the platform observer see the two flashbulbs light up simultaneously. Simultaneity is preserved; i.e., both the platform and us agree on the sequence of events.

29. Loss of Simultaneity • The platform is now moving at a uniform velocity relative to us. What if we applied the principle that the speed of light is the same in all reference frames? • According to the platform observer, do the two flashbulbs light up simultaneously?

30. Loss of Simultaneity • The platform is now moving at a uniform velocity relative to us. What if we applied the principle that the speed of light is the same in all reference frames? • According to the platform observer, do the two flashbulbs light up simultaneously? Yes

31. Loss of Simultaneity • The platform is now moving at a uniform velocity relative to us. What if we applied the principle that the speed of light is the same in all reference frames? • According to the platform observer, do the two flashbulbs light up simultaneously? Yes • According to us, do the two flashbulbs light up simultaneously?

32. Loss of Simultaneity • The platform is now moving at a uniform velocity relative to us. What if we applied the principle that the speed of light is the same in all reference frames? • According to the platform observer, do the two flashbulbs light up simultaneously? Yes • According to us, do the two flashbulbs light up simultaneously? No. The platform observer sees the two flashbulbs light up simultaneously, but we do not. Simultaneity is not preserved; i.e., both the platform and us disagree on the sequence of events.

33. Learning Objectives • What is the single additional postulate Einstein made that makes the theory of special relativity different from Newtonian (classical) physics? The speed of light is the same in all reference frames. (In Newtonian physics, speed is a vector quantity, and therefore different as measured from different reference frames.) We shall return to the experimental proof in the next lecture. • What are the consequences of applying this additional postulate? 1. If the speed of light is constant in all reference frames, length and time must then be different as measured from different reference frames: length contraction and time dilation. (In Newtonian physics, length and time are the same as measured from different reference frames.) • 2. Different events that occur simultaneously in one reference frame need not occur simultaneously in a different reference frame: loss of simultaneity. (In Newtonian physics, different events that occur simultaneously in one reference frame also occur simultaneously in a different reference frame.) • Some applications of special relativity on Earth • Some applications of special relativity in astrophysics

34. Special Relativity on the Earth • The effects of special relativity, which only becomes appreciable at velocities approaching the speed of light, are usually unfamiliar/unappreciated in daily life. • Because speeds on Earth are usually much smaller than the speed of light, measurements with extremely high precision are usually required to detect the effects of special relativity on the Earth. In 1971, a classic test of (both special and general) relativity was performed by Joseph C. Hafele and Richard E. Keating, who took four caesium-beam atomic clocks aboard commercial airliners and flew twice around the world, first eastward, then westward, and compared the clocks against those of the United States Naval Observatory.

35. Special Relativity on the Earth • Because flying planes are accelerating in the Earth’s gravitational potential, there is an effect due to general relativity (GR). For flight paths along the equator, the predicted GR terms are the same for planes flying eastwards or westwards. Commercial aircrafts, of course, follow their individual flight paths. • They found that the eastward-flying clocks lost time, whereas the westward flying clocks gained time, relative to clocks on the Earth’s surface. Why?

36. Special Relativity on the Earth • Using the center of the Earth as the reference frame, eastward-flying clocks move at the highest velocity, and therefore suffers most from time dilation. The clock at the Earth’s surface moves at a higher velocity than westward-flying clocks, and hence suffers more time dilation than the latter. • Considering only the effect of special relativity, why do eastward-flying clocks lose more time than is gained by westward-flying clocks? Assignment question

37. Special Relativity on the Earth • How many of you have GPS (Global Positioning Satellite) navigation on your cell phone? How many of you know how this system works?

38. Special Relativity on the Earth • Each GPS satellite transmits data that indicates its location and the current time. All GPS satellites synchronize operations so that these repeating signals are transmitted at the same instant, requiring accurate clocks on each satellite corrected for the effects of (special and general) relativity. The signals arrive at a GPS receiver at slightly different times because some satellites are farther away than others. The distance to the GPS satellites can be determined by estimating the amount of time it takes for their signals to reach the receiver. When the receiver estimates the distance to at least four GPS satellites, it can calculate its position in three dimensions. • A GPS receiver "knows" it is located somewhere on the surface of an imaginary sphere centered at the satellite. It then determines the sizes of several spheres, one for each satellite. The receiver is located where these spheres intersect.

39. Special Relativity on the Earth • There are a few situations on Earth where objects with velocities approaching the speed of light are encountered. One is in particle accelerators, where protons and/or electrons are accelerated to velocities close to the speed of light. • In Newtonian physics, to double the velocity of a particle, how much energy would we have to impart on a particle with a kinetic energy E? Large Hadron Collider • In special relativity, the answer depends on the velocity of the particle.

40. Special Relativity on the Earth • When cosmic rays – charged particles comprising ~89% protons, ~10% helium nuclei, ~1% nuclei of heavier elements, and ~1% electrons travelling close to the speed of light, accelerated possibly by active supermassive black holes, supernova explosions, or stellar flares – strike the Earth’s atmosphere, they collide with molecules (mainly nitrogen and oxygen) to produce a cascade of subatomic and elementary particles known as an air shower. • A cosmic ray particle striking the Earth’s atmosphere produces a subatomic particle pion (π), which is unstable and decays to produce an elementary particle called a muon (μ). Only a small fraction of the elementary particles produced in an air shower is shown in this figure.

41. Special Relativity on the Earth • When cosmic rays – charged particles comprising ~89% protons, ~10% helium nuclei, ~1% nuclei of heavier elements, and ~1% electrons travelling close to the speed of light, accelerated possibly by active supermassive black holes, supernova explosions, or stellar flares – strike the Earth’s atmosphere, they collide with molecules (mainly nitrogen and oxygen) to produce a cascade of subatomic and elementary particles known as an air shower. • A cosmic ray particle striking the Earth’s atmosphere produces a subatomic particle pion (π), which is unstable and decays to produce an elementary particle called a muon (μ). • Muonshave the same charge as but are ~200 times more massive than the electron. • Muons also are unstable, and decay after an average lifetime τ = 2.20 μs as measured at rest in the laboratory. According to an observer at rest at the Earth’s surface, do muons produced in an air shower have an average lifetime shorter, longer, or the same as those measured at rest in a laboratory?

42. Special Relativity on the Earth • When cosmic rays – charged particles comprising ~89% protons, ~10% helium nuclei, ~1% nuclei of heavier elements, and ~1% electrons travelling close to the speed of light, accelerated possibly by active supermassive black holes, supernova explosions, or stellar flares – strike the Earth’s atmosphere, they collide with molecules (mainly nitrogen and oxygen) to produce a cascade of subatomic particles known as an air shower. • A cosmic ray particle striking the Earth’s atmosphere produces a pion (π), which is unstable and decays to produce a particle called a muon (μ). • Muons have the same charge as but are ~200 times more massive than the electron. • Muons also are unstable, and decay after an average lifetime τ = 2.20 μs as measured at rest in the laboratory. Muons produced in an air shower, however, are measured by an observer on the Earth to have an average lifetime of τ = 22.5 μs, an effect of time dilation.

43. Learning Objectives • What is the single additional postulate Einstein made that makes the theory of special relativity different from Newtonian (classical) physics? The speed of light is the same in all reference frames. (In Newtonian physics, speed is a vector quantity, and therefore different as measured from different reference frames.) We shall return to the experimental proof in the next lecture. • What are the consequences of applying this additional postulate? 1. If the speed of light is constant in all reference frames, length and time must then be different as measured from different reference frames: length contraction and time dilation. (In Newtonian physics, length and time are the same as measured from different reference frames.) • 2. Different events that occur simultaneously in one reference frame need not occur simultaneously in a different reference frame: loss of simultaneity. (In Newtonian physics, different events that occur simultaneously in one reference frame also occur simultaneously in a different reference frame.) • Some applications of special relativity on Earth • Some applications of special relativity in astrophysics

44. Special Relativity in Astrophysics • Astrophysical situations where the effects of special relativity are appreciable: - light from stars and gas clouds in our Galaxy that are moving at relatively high velocities with respect to Earth is Doppler shifted due to a combination of time dilation and geometrical effects - the light from galaxies, all of which are moving at relatively high velocities with respect to our Galaxy due to their peculiar velocities as well as the expansion of the Universe, is Doppler shifted

45. Special Relativity in Astrophysics • Astrophysical situations where the effects of special relativity are appreciable: - events that occur over a given duration in galaxies, especially in more distant galaxies receding at higher velocities due to the expansion of the Universe, have a different duration as measured by an observer on the Earth due to the effect of time dilation γ-ray burst in distant galaxy

46. Special Relativity in Astrophysics • Astrophysical situations where the effects of special relativity are appreciable: - jets, whose emission is produced by electrons travelling near the speed of light and spiraling in magnetic fields, from active supermassive black holes. Centaurus A in optical (b&w) and radio (color)

47. Special Relativity in Astrophysics • Astrophysical situations where the effects of special relativity are appreciable: - jets, whose emission is produced by electrons travelling near the speed of light and spiraling in magnetic fields, from active supermassive black holes. Centaurus A in optical (b&w) and radio (color)

48. Special Relativity in Astrophysics • Astrophysical situations where the effects of special relativity are appreciable: - jets, whose emission is produced by electrons travelling near the speed of light and spiraling in magnetic fields, from active supermassive black holes. Jets sometimes appear to be one-sided, an effect usually of relativistic beaming. Virgo cluster Optical jet from M87 M87

49. Special Relativity in Astrophysics • Astrophysical situations where the effects of special relativity are appreciable: - jets, whose emission is produced by electrons travelling near the speed of light and spiraling in magnetic fields, from active supermassive black holes. Jets sometimes appear to be one-sided, an effect usually of relativistic beaming. Radio emission from the quasar 3C 175 lobes jet Location of central supermassive black hole and center of host galaxy