Loading...

PPT – Chapter 7: Classical Mechanics of Special Relativity' Section 7'1: Basic Postulates of the Special T PowerPoint presentation | free to download - id: 1d59a8-ZGQ2N

The Adobe Flash plugin is needed to view this content

(No Transcript)

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

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.

- 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

- 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.

- 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.

- 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.

- 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

- (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

- 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

- (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)

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)

- (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

- (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

- (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.

- (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.

- (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).

- (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.

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

- (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.

- (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).

- 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!

- 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 ?

- 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

- 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

- 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

- 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!

- 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