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Chapter 7: Classical Mechanics of Special Relativity' Section 7'1: Basic Postulates of the Special T

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Title: Chapter 7: Classical Mechanics of Special Relativity' Section 7'1: Basic Postulates of the Special T


1
(No Transcript)
2
NOTE 2005 is the World Year of Physics! In
2005, there will be a world-wide celebration of
the centennial of Einstein's famous 1905 papers
on Relativity, Brownian Motion, the
Photoelectric Effect (for which he won the Nobel
Prize!). A web page telling you
more lthttp//www.physics2005.org/gt
3
Chapter 7 Special Relativity Sect. 7.1 Basic
Postulates
  • Special Relativity One of 2 major
    (revolutionary!) advances in understanding the
    physical world which happened in the 20th
    Century! Other is Quantum Mechanics of course!
  • Of the 2, Quantum Mechanics is more relevant to
    everyday life also has spawned many more
    physics subfields.
  • However (my personal opinion), Special Relativity
    is by far the most elegant beautiful of the
    2. In a (relatively) simple mathematical
    formalism, it unifies mechanics with EM!
  • The historical reasons Einstein developed it
    the history of its development eventual
    acceptance by physicists are interesting. But
    (due to time) we will discuss this only briefly.
  • The philosophical implications of it, the various
    paradoxes it seems to have, etc. are
    interesting. But (due to time) well discuss this
    only briefly.

4
  • Newtons Laws Are valid only in an Inertial
    Reference Frame Defined by Newtons 1st Law
  • A frame which isnt accelerating with respect to
    the stars.
  • Any frame moving with constant velocity with
    respect to an inertial frame is also an inertial
    frame!
  • Galilean Transformation (Galilean Relativity!)
  • 2 reference frames S, time space
  • coordinates (t,x,y,z) S, time space
  • Coordinates (t,x,y,z). S moving
  • relative to S with const velocity v in the
  • x direction. Figure. Clearly
  • t t , x x - vt, y y, z z
  • ? Galilean Transformation

5
  • Newtons 2nd Law Unchanged by a Galilean
    Transformation (t t, x x - vt, y y, z
    z)
  • F (dp/dt) ? F (dp/dt)
  • Implicit Newtonian assumption t t. In the
    equations of motion, the time t is an independent
    parameter, playing a different role in mechanics
    than the coordinates x, y, z.
  • Newtonian mechanics S is moving
  • relative to S with constant velocity v in the
  • x direction u velocity of a particle in S,
  • u velocity of particle in S. ? u u - v
  • Contrast in Special Relativity,
  • the position coordinates x, y, z time t
  • are on an equal footing.

6
  • Electromagnetic Theory (Maxwells equations)
    Contain a universal constant c The speed of
    Light in Vacuum.
  • This is inconsistent with Newtonian mechanics!
  • Einstein Either Newtonian Mechanics or Maxwells
    equations need to be modified. He modified
    Newtonian Mechanics.
  • ? 2 Basic Postulates of Special Relativity
  • 1. THE POSTULATE OF RELATIVITY
  • The laws of physics are the same to all inertial
    observers. This is the same as Newtonian
    mechanics!
  • 2. THE POSTULATE OF THE CONSTANCY OF THE SPEED
    OF LIGHT The speed of light, c, is independent
    of the motion of its source. A revolutionary
    idea! Requires modifications of mechanics at high
    speeds.

7
  • 2 Basic Postulates
  • 1. RELATIVITY 2. CONSTANT LIGHT SPEED
  • Covariant ? A formulation of physics which
    satisfies 1 2
  • 2. ? The speed of light c is the same in all
    coordinate systems.
  • 1 2 ? Space Time are considered 2 aspects
    (coordinates) of a single Spacetime. A 4d
    geometric framework (Minkowski Space)
  • ? The division of space time is different for
  • different observers. The meaning of
    simultaneity is different for different
    observers. Space time get mixed up in
    transforming from one inertial frame to another.

8
  • Event ? A point in 4d spacetime.
  • To make all 4 dimensions have the same units,
    define the time dimension as ct.
  • The square of distance between events A
    (ct1,x1,y1,z1)
  • B (ct2,x2,y2,z2) in 4d spacetime
  • (?s)2 ? c2(t2-t1)2 - (x2-x1)2 - (y2-y1)2 -
    (z2-z1)2
  • or (?s)2 ? c2(?t)2 - (?x)2 - (?y)2 - (?z)2
    (1)
  • Note the different signs of time space coords!
  • Now, go to differential distances in spacetime
  • (1) ? (ds)2 ? c2(dt)2 - (dx)2 - (dy)2 - (dz)2
    (2)
  • A body moving at v (dx)2 (dy)2 (dz)2
    (dr)2 v2(dt)2
  • ? (ds)2 c2- v2(dt)2 gt 0
  • Bodies, moving at v lt c, have (ds)2 gt 0

9
  • (ds)2 ? c2(dt)2 - (dx)2 - (dy)2 - (dz)2
    (2)
  • (ds)2 gt 0 ? A timelike interval
  • (ds)2 lt 0 ? A spacelike interval
  • (ds)2 0 ? A lightlike or null interval
  • For all observers, objects which travel with v lt
    c have (ds)2 gt 0 ? Such objects are called
    tardyons.
  • Einsteins theory the Lorentz Transformation
  • ? The maximum velocity allowed is v c. However,
    in science fiction, can have v gt c. If v gt c,
    (ds)2 lt 0
  • ? Such objects are called tachyons.
  • A 4d spacetime with an interval defined by (2) ?
    Minkowski Space

10
  • The interval between 2 events (a distance in 4d
    Minkowski space) is a geometric quantity.
  • ? It is invariant on transformation
  • from one inertial frame S to
  • another, S moving relative to
  • S with constant velocity v
  • (ds)2 ? (ds)2 (3)
  • ? (ds)2 ? The invariant spacetime interval.
  • (3) ? The transformation between S S must
    involve the relative velocity v in both space
    time parts. Or Space Time get mixed up on
    this transformation! Simultaneity has different
    meanings for an observer in S an observer in S

11
  • (ds)2 ? (ds)2 (3)
  • Relatively simple consequences of (3)
  • 1. Time Dilation
  • S ? Lab frame, S ? moving frame
  • (3) ? Time interval dt measured in the lab
  • frame is different from the time interval dt
    measured in the moving frame. ? To distinguish
    them Time measured in the rest (not moving!)
    frame of a body (Sif the body moves with v in
    the lab frame) ? Proper time ? t. Time measured
    in the lab frame (S) ? Lab time ? t. For a body
    moving with v
  • In S, (ds)2 c2(dt)2 ,
  • In S, (ds)2 ? c2(dt)2 - (dx)2 - (dy)2 - (dz)2
  • c2(dt)2 - (dr)2 c2(dt)2 -
    v2(dt)2 c2(dt)21-(v2)/(c2)

12
Time Dilation
  • (ds)2 ? (ds)2 (3)
  • A body moving with v
  • In S, (ds)2 c2(dt)2 ,
  • In S, (ds)2 c2(dt)21-(v2)/(c2)
  • Using these in (3)
  • ? c2(dt)2 c2(dt)21-(v2)/(c2)
  • Or dt ? ?dt (4)
  • where ? ? 1/1 - ß2½ ? 1 - ß2-½ , ß ? (v/c)
  • (4) ? dt lt dt ? Time dilation
  • ? Moving clocks (appear to) run slow(ly)

13
  • (ds)2 ? (ds)2 (3)
  • Relatively simple consequences of (3)
  • 2. Simultaneity is relative!
  • Suppose 2 events occur simultaneously in
  • S (? the lab frame), but at different space
  • points (on the x axis, for simplicity). Do they
    occur simultaneously in S (? the moving frame)?
  • In S, dt 0, dy dz 0, dx ? 0. ? In S, (ds)2
    - (dx)2
  • In S, (invoking the Lorentz transformation
    ahead of time) dydz0, ? In S, (ds)2
    c2(dt)2 - (dx)2
  • (3) ? - (dx)2 c2(dt)2 - (dx)2 Or c2(dt)2
    (dx)2 - (dx)2 (invoking the Lorentz transform
    ahead of time) (dx)2 ?2(dx)2
  • ? c2(dt)2 ?2 -1 (dx)2 Or (algebra) c dt
    ?ßdx
  • ? The 2 events are not simultaneous in S

14
  • (ds)2 ? (ds)2 (3)
  • Relatively simple consequences of (3)
  • 3. Length Contraction
  • Consider a thin object, moving with v to x
  • in S. Let S be attached to the moving object.
    Instantaneous
  • measurement of length. In S dt 0. For an
    infinitely thin object
  • dy dz 0. ? In S, (ds)2 - (dx)2 In S,
    (invoking the Lorentz transformation ahead of
    time) dydz0,
  • ? In S, (ds)2 c2(dt)2 - (dx)2 (3) ?
    -(dx)2 c2(dt)2 - (dx)2 Or (dx)2 c2(dt)2
    (dx)2 (invoking the Lorentz transform ahead of
    time using results just obtained) c2(dt)2
    ?2ß2(dx)2
  • ? (Algebra) (dx)2 ?2(dx)2 Or dx ?dx. For
    finite length L ?L or L (L)?-1 lt L
  • Lorentz-Fitzgerald Length Contraction

15
  • (ds)2 ? (ds)2 (3)
  • (3) ? Spacetime is naturally divided into 4
    regions. For an arbitrary event A at x y z
    t 0, we can see this by looking at the light
    cone of the event. Figure.
  • the z spatial dimension is suppressed.
  • Light cone set of (ct,x,y) traced
  • out by light emitted from ct x
  • y 0 or by light that reaches
  • x y 0 at ct 0. The past
  • the future are inside the light cone.

16
  • (ds)2 ? (ds)2 (3)
  • Consider event B at time tB such that (dsAB)2 gt 0
    (timelike). (3) ? All inertial observers agree on
    the time order of events A B. We can always
    choose a frame where A B have the same space
    coordinates. If tB lt tA 0
  • in one inertial frame, will be so in all
  • inertial frames. ? This region is called
  • THE PAST.
  • Similarly, consider event C at time tC
  • such that (dsAC)2 gt 0, (3) ? All inertial
  • observers agree on the time order of events A
    C. If tCgttA 0
  • in one inertial frame, it will be so in all
    inertial frames.
  • ? This region is called THE FUTURE.

17
  • (ds)2 ? (ds)2 (3)
  • Consider an event D at time tD such that (dsAD)2
    lt 0 (spacelike). (3) ? There exists an inertial
    frame in which the time ordering of tA tD are
    reversed or even made equal.
  • ? This region is called
  • THE ELSEWHERE
  • or THE ELSEWHEN. In the region
  • in which D is located, there exists an
  • inertial frame with its origin at event
  • A in which D A occur at the same time but in
    which D is somewhere else (elsewhere) than the
    location of A. There also exist frames in which D
    occurs before A frames in which D occurs after
    A (elsewhen).

18
  • (ds)2 ? (ds)2 (3)
  • The light cone obviously separates the
    past-future from the elsewhere (elsewhen). On the
    light cone, (ds)2 0. Light cone a set of
    spacetime points from which emitted light could
    reach A
  • (at origin) those points from which
  • light emitted from event A could
  • reach.
  • Any interval between the origin a
  • point inside the light cone is timelike (ds)2 gt
    0. Any interval between the origin a point
    outside the light cone is spacelike (ds)2 lt 0.

19
Sect. 7.2 Lorentz Transformation
  • Lorentz Transformation A derivation (not in
    the text!)
  • Introduce new notation x0 ? ct, x1 ? x,
  • x2 ? y, x3 ? z. Lab frame S inertial frame
  • S, moving with velocity v along x axis.
  • We had (ds)2 ? (ds)2 . Assume that this also
  • holds for finite distances (?s)2 ? (?s)2 or
  • (in the new notation)
  • (?x0)2- (?x1)2 - (?x2)2 - (?x3)2 (?x0)2 -
    (?x1)2 - (?x2)2 - (?x3)2
  • Assume, at time t 0, the 2 origins coincide.
  • ? ?xµ xµ ?xµ xµ (µ 0,1,2,3)
  • ? (x0)2- (x1)2 - (x2)2 - (x3)2 (x0)2 - (x1)2
    - (x2)2 -(x3)2

20
  • (x0)2- (x1)2 - (x2)2 - (x3)2 (x0)2 - (x1)2 -
    (x2)2 -(x3)2 (1)
  • Want a transformation relating xµ xµ.
  • Assume the transformation is LINEAR
  • xµ ? ?µLµ?x? (2)
  • Lµ? to be determined
  • (2) Mathematically identical (in 4d
  • spacetime) to the form for a rotation in 3d
    space.
  • We could write (2) in matrix form as x ? L?x
  • Where L is a 4x4 matrix x, x are 4d column
    vectors. We can prove that L is symmetric acts
    mathematically as an orthogonal matrix in 4d
    spacetime.

21
  • (x0)2- (x1)2 - (x2)2 - (x3)2 (x0)2 - (x1)2 -
    (x2)2 -(x3)2 (1)
  • xµ ? ?µLµ?x?
    (2)
  • Now, invoke some PHYSICAL REASONING
  • The motion (velocity v) is along the x axis. Any
    physically reasonable transformation will not mix
    up x,y,z (if the motion is parallel to x that
    is, it involves no 3d rotation!).
  • ? y y , z z or x2 x2, x3 x3
  • ? (1) becomes (x0)2 - (x1)2 (x0)2 - (x1)2
    (3)
  • Also L22 L33 1. All others are zero except
    L00, L11, L01, L10. Further, assume that the
    transformation is symmetric. ? Lµ? L?µ (this
    is not necessary, but it simplifies math. Also,
    after the fact we find that it is symmetric).

22
  • Under these conditions, (2) becomes
  • x0 L00 x0 L01 x1 (2a)
  • x1 L01 x0 L11 x1 (2b)
  • (x0)2 - (x1)2 (x0)2 - (x1)2
    (3)
  • (2a), (2b), (3) After algebra we get
  • (L00)2 - (L01)2 1 (4a) (L11)2 -
    (L01)2 1 (4b)
  • (L00 - L11)L01 0 (4c)
  • (4a), (4b), (4c) This looks like 3 equations 3
    unknowns. However, it turns out that solving will
    give only 2 of the 3 unknowns (the 3rd equation
    is redundant!).
  • ? We need one more equation!

23
  • To get this equation, consider the origin of the
    S system
  • at time t in the S system. (Assume, at time
  • t 0, the 2 origins coincide.) Express it in
  • the S system ? At x1 0, (2b) gives
  • 0 L01 x0 L11 x1
  • We also know x vt or x1 ßx0
  • Combining gives L01 - ßL11 (5)
  • Along with (L00)2 - (L01)2 1 (4a)
  • (L11)2 - (L01)2 1 (4b)
    (L00 - L11)L01 0 (4c)
  • This finally gives L11 ? 1/1 - ß2½ 1
    - (v2/c2)-½
  • and (algebra) L01 - ß?
    , L00 ?

24
  • Putting this together, The Lorentz Transformation
    (for v x) x0 ?(x0 - ßx1), x2 x2
  • x1 ?(x1 - ßx0), x3 x3
  • The inverse Transformation (for v x)
  • x0 ?(x0 ßx1), x2 x2
  • x1 ?(x1 ßx0), x3 x3
  • In terms of ct,x,y,z The Lorentz Transformation
    is
  • ct ?(ct - ßx) (t ?t - (ß/c)x)
  • x ?(x - ßct), y y, z z, ß (v/c)
  • This reduces to the Galilean transformation for v
    ltltc
  • ß ltlt 1, ? ? 1 ? x x - vt, t t, y y,
    z z

25
  • Lorentz Transformation (for v x) in terms of a
    transformation (rotation) matrix in 4d
    spacetime (a rotation in the x0-x1 plane)
  • x ? L?x
  • Or
  • x0 ? -ß? 0 0
    x0
  • x2 -?ß ? 0 0
    x1
  • x3 0 0 1 0
    x2
  • x4 0 0 0 1
    x3

26
  • The generalization to arbitrary orientation of
    velocity v is straightforward but tedious!
  • The Lorentz Transformation (for general
    orientation of v)
  • ct ?(ct - ß?r),
  • r r ß-2(ß?r)(? -1)ß - ?ctß
  • In terms of the transformation (rotation)
    matrix in 4d spacetime x ? L?x

27
  • Briefly back to the Lorentz Transformation (v
    x)
  • x ? L?x , L ? A Lorentz boost or A
    boost
  • Sometimes its convenient to parameterize the
    transformation in terms of a boost parameter or
    rapidity ?. Define
  • ß ? tanh(?) ? ? 1 - ß2-½
    cosh(?), ß? sinh(?) Then
  • x0 cosh(?) -sinh(?) 0 0
    x0
  • x2 -sinh(?) cosh(?) 0 0
    x1
  • x3 0 0
    1 0 x2
  • x4 0 0
    0 1 x3
  • ? x0 x0 cosh(?) - x1 sinh(?), x1 -x0
    sinh(?) x1 cosh(?)
  • Should reminds you of a rotation in a plane, but
    we have hyperbolic instead of trigonometric
    functions. From complex variable theory ?
    imaginary rotation angle!

28
  • These transformations map the origins of S S
    to (0,0,0,0).
  • L a rotation in 4d spacetime.
  • A more general transformation is
  • ? The Poincaré Transformation
  • Rotation L in 4d spacetime translation a
  • x ? L?x a
  • If a 0 ? Homogeneous Lorentz
    Transformation
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