Relativity and Astrophysics Lecture 25 Terry Herter. Momenergy Momentum-energy 4-vector Magnitude & components Invariance Low velocity limit

Transcription

1 Mo Mo Relativity and Astrophysics Lecture 5 Terry Herter Outline Mo Moentu- 4-vector Magnitude & coponents Invariance Low velocity liit Concept Suary Reading Spacetie Physics: Chapter 7 Hoework: (due Wed. /04/09) 6-4, 6-7, 7.3, and 7.9 A90-5 Mo A90-5

2 Mo The Power of It All The conservation law all us to solve for quantities without knowing the details We don t have to know how the objects defor in the sticking case We don t need to know about the details for the collisions at all (for the copletely inelastic and elastic cases) In cases where this a potential we can include this in the conservation equations. For instance, using Newton s law of gravity we can write a conservation of equation which relates velocity to distance fro the arth we don t have to solve the details of the acceleration For a vertically falling object we have F GM r V GM r v GM ro r Where the object starts at r o with zero velocity. Note that we lose inforation. We don t know how long it takes to travel over this distance just the speed at the end. A90-5 Mo 3 Mo Mo Relativity cobines oentu and into a single concept, oentu- (or o) This quantity is conserved in a collision Mo proportional to ass Consider different ass pebbles hitting a windshield Mo is a directed quantity It atters which direction the pebble coe fro Mo is a 4-vector xpect space and tie coponents due to the unity of spacetie (three spatial parts and one tie part_ Space part represent oentu, tie part represents Points in the direction of a particles spacetie displaceent Mo is reckoned using proper tie for a particle Mo is independent of reference frae o ass spacetie displaceent proper tie for that displaceent Looks like Newtonian oentu but odified for instein s relativity A90-5 Mo 4 A90-5

3 Mo Magnitude of Mo Don t confuse a 4-vector with its agnitude The proper tie is the agnitude of the spacetie displaceent The fraction is a unit 4-vector pointing in the direction of the worldline of the particle The agnitude of o is its ass. tie Mo (oentu = 0) Mo oentu Mo space oentu A90-5 Mo 5 Coponents of Mo Let t stand for proper tie then the coponents of o are d dx p x d dy p y d Let s look at its agnitude agnitude p p p dx dy dz d Or ore copactly written agnitude of o arrow p which is just he equation for a hyperbola in spacetie again. At right is a plot of the o 4-vector for a single particle observed in 5 different inertial reference fraes. x dz p z d x-oentu A90-5 Mo 6 y z d d A90-5 3

4 Mo Moentu: Space Part Consider a particle oving along the x-axis with a velocity v in the lab frae The displaceent of the particle is x = vt, or for sall displaceents, dx = v. The proper tie is: d dx / v / / v / v / So that the relativistic expressions for and oentu are dx dx & p d x d d v x For low velocities the oentu expression becoes very close to the Newtonian value A90-5 Mo 7 Moentu Units Relating velocities (diensionless vs. entional units) v v c For a oentu in diensionless units p Newton v p v For oentu in entional units p Newton p Newton c vc v Valid for low speed Valid for low speed p pc vc v Convert fro oentu in units of ass to entional units by ultiplying by c, the speed of light. A90-5 Mo 8 A90-5 4

5 Mo nergy: Tie Part For a particle oving along the x-axis with a velocity v in the lab frae the is d Which can be copare with the Newtonian expression (using K as the sybol for kinetic ) K v How does the relativistic expression for copare with the Newtonian for kinetic? At low velocities, v = 0, we have Which is called the of the particle. Rest is siply the ass The relativistic does not go to zero like the K! So to define a kinetic above and beyond a particles we have K A90-5 Mo 9 nergy Units Note that if we divide the oentu and we get the speed of the particle & p v p v d To ert in units of ass to in entional units we have c c The (and perhaps the ost faous equation in physics) is The kinetic is K At low speeds (v << ), we have c c c Particle at v / ~ v v K Newton v c v Valid at low speed A90-5 Mo 0 A90-5 5

6 Mo Saple Proble 7- (pg. 0) Consider a 3 kg ass object which oves 8 eters in the x direction in 0 eters of tie. What is its and oentu? What is its? What is its kinetic? Copare this to the Newtonian K. Verify the velocity equals its oentu divided by its. The speed is x v t The and oentu are kg 5 kg & p v 3 kg The is 3 kg The relativistic and Newtonian kinetic energies are K kg & & v The correct relativistic result is quite a bit larger that the Newtonian prediction, and in fact the correct result grows with out liit as v approaches (ore on next slide). Finally the velocity can be recovered fro v p / 4 / K Newton 0.5 v kg 3 kg kg A90-5 Mo nergy in the low-velocity liit In ters of oentu the expression for looks like p / p At low velocities p/ is sall, and we can use the usual expansion to get p x n nx corrections corrections Suppose (p/) = 0., then the approxiate and exact forula give respectively approxiate exact p p.05 / /.. 00 The correction is negative and very sall: correction = The approxiate ter squared roughly gives the error fro the exact result, e.g. a agnitude of 0% iplies the error is ~ (0.) = 0.0 = % A90-5 Mo A90-5 6

7 Mo Low-velocity liit (cont d) In ters of velocity the expression for looks like / v v / At low velocities v is sall, and we can use the usual expansion to get x n nx corrections v corrections Suppose v = 0.9, then the approxiate and exact forula give respectively approxiate ~ v exact / v 0.8 / The approxiate ter squared roughly gives the error fro the exact result, e.g. a agnitude of 0% iplies the error is ~ (0.) = 0.0 = % The correction is positive and very sall: correction = A90-5 Mo 3 K: Correct vs. Newtonian.0 0 (K/unit ass) relativistic Newtonian (% error in K_Newt) % error in Newtonian K speed speed The correct (relativistic) result for the kinetic is K The correct K increases without bound as v approaches while the Newtonian result approaches 0.5 (for a kg ass) Plots: The left hand plot shows the correct vs. the low velocity Newtonian approxiation (which extrapolates incorrectly to high velocities). The right hand plot shows the percentage error in the Newtonian result copared to the correct one. A90-5 Mo 4 A90-5 7