Physics 218, Spring April 2004

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Physics 18, Spring 004 16 April 004 Today in Physics 18: forces in relativity Newton s laws in relativity The Minkowski force Relativistic transformation of forces X-ray image of the plsar-driven

1 Physics 18, Spring April 004 Today in Physics 18: forces in relativity Newton s laws in relativity The Minkowski force Relativistic transformation of forces X-ray image of the plsar-driven center of the Crab Nebla in Tars, the remnant of the spernova of 1054 AD (Chandra X- ray observatory image, NASA and Center for Astrophysics) 16 April 004 Physics 18, Spring Newton s laws in relativity We will be interested in learning how to solve force problems in relativity, becase force is ltimately how we relate to the fields Can we still se Newton s laws? (Are those among the laws of physics that are valid in inertial reference frames?) irst law: inertia ( A body in niform motion will remain ) Clearly this is still tre in relativity; otherwise we wold have had more troble last time with momentm and energy conservation Second law: ma This is still tre if one takes m to be the rest mass, and expresses it as d d m v p, 16 April 004 Physics 18, Spring 004 and ses the relativistic momentm thenceforth: m p The easiest way to demonstrate this is to note that mechanical work still increases mechanical energy in relativity, jst as it always has: dp dp d dp W d d d m 16 April 004 Physics 18, Spring (c) University of Rochester 1

2 Physics 18, Spring April 004 m d m d W + 3 ( 1 c ) c m( ) ( ) m c d ( 1 c ) m d d mc 3 ( ) de E final E initial,qed 16 April 004 Physics 18, Spring Third law: for every action and eqal and opposite reaction This clearly doesn t apply in relativity: Sppose two extended objects exert forces () t and () t on each other in some reference frame, so that the third law is satisfied at all times t: () t and () t are simltaneosly applied A observer in a different reference frame wold see those forces applied at different times! Since the objects are not at the same spatial point, events simltaneos in the first frame will not appear so in the second frame, so nless the forces are constant they will not appear eqal and opposite This won t srprise those who remember radiation reaction 16 April 004 Physics 18, Spring The Minkowski (for-)force Since dp, the vales of a force seen from different inertial reference frames are not related simply by a Lorentz transformation, bt instead by a transformation similar to velocity addition However, the vector dp d 1 p K can clearly be part of a for-vector: 0 µ 0 dp d E µ dp K K c Minkowski force 16 April 004 Physics 18, Spring (c) University of Rochester

3 Physics 18, Spring April 004 The scalar prodct of K with itself is therefore Lorentzinvariant (Griff$iths pr$oblem 13$9): µ 0 d E Kµ K ( K ) + K K + c d E 1 de 1 1 d mc c c c 1 c 1 1 mc d m 1 c c 3 ( c ) ( ) a 16 April 004 Physics 18, Spring Compare this to dp d m We worked this ot in the middle of an integral a few pages ago: m d m( a ) 0 c 1 c K c 1 c ( ) ( ) cos θ ; 0 cosθ K c 16 April 004 Physics 18, Spring Ths µ cosθ Kµ K + c ( c ) 1 cos θ Utility: if is measred at rest, an observer in a moving frame will measre 1 ( c ) cos θ 16 April 004 Physics 18, Spring (c) University of Rochester 3

4 Physics 18, Spring April 004 Can we se the Minkowski force to cast Newton s second law? Yes, as it trns ot (Griff$iths pr$oblem 13$8) irst define the for-acceleration in terms of the forvelocity: α µ dη µ d x µ In these terms, the second law is µ µ µ dp d K η µ m mα This bears an odd relationship to the for-velocity itself, as we can see from the varios Lorentz invariants we can constrct: 16 April 004 Physics 18, Spring The inner prodct of the for-velocity with itself trns ot to be constant: µ 0 c ηη µ ( η ) + ηη + c 1 c 1 c And this trns ot to mean that the for-velocity and for-acceleration are orthogonal: d µ µ µ µ d ( ηµ η ) αµ η + ηµ α ηµ α ( c ) 0 ; µ ηµ α 0 Similarly, K µ η µ 0 16 April 004 Physics 18, Spring Alas, real forces are not like the Minkowski force; we still need to derive their transformations To wit: or finite intervals of momentm and time, seen from two inertial reference frames in relative motion along the x axis, E px γ px c py py pz pz x t γ t c 16 April 004 Physics 18, Spring (c) University of Rochester 4

5 Physics 18, Spring April 004 Ths we can, in the limit, get a component of the force: E γ px p x c x lim lim t x γ t c Note that 1 1 dpx 1 dpx x a t t and E t m m are both second order in t, so in the limit the second term in both nmerator and denominator are small compared to the first 16 April 004 Physics 18, Spring Ths, γ px dpx x lim x γ t By the same token py py y lim lim t x γ t c 1 dpy y, γ γ z z similarly γ 16 April 004 Physics 18, Spring More compactly,, 1 γ, where and means parallel and perpendiclar to the direction of relative motion between the two different inertial frames 16 April 004 Physics 18, Spring (c) University of Rochester 5

Cosmic rays. l Some come from the sun (relatively low energy) and some from catastrophic events elsewhere in the galaxy/universe

Cosmic rays. l Some come from the sun (relatively low energy) and some from catastrophic events elsewhere in the galaxy/universe Special relativity The laws of physics are the same in all coordinate systems either at rest or moving at constant speed with respect to one another The speed of light in a vacm has the same vale regardless