Causality in special relativity

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Causality in special relativity --The ladder and
barn paradox -- Lorentz transformation in
Spacetime diagram --Causality -- Nothing can
travel faster than C
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1. The pole and barn paradox
They cannot be both right!!
--Lets assume the doors of the barn are kept
open in usual state, and are designed in such a
way that it can be triggered by passing of the
pole to close and then open again immediately
after, so that the pole can keep a constant
motion passing through the barn.
--The back door of the barn will close when the
front of pole just approaches the back door,
then open again immediately and the front door
will close just when the rear end of pole passes
it and open again immediately.
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Barns frame Pole frame
t0 0 t0 0
Back door closes then opens t1 38.49ns, x10 (10m/0.886c) Back door closes then opens t 1 19.25ns (5m/0.866c)
Front door closes and then opent2 38.49ns, x2 10 Front door closes and then open t2 76.98ns, (20m/0.886c)
pole moves out t3 238.4976.98ns pole moves out t3 19.2576.9896.23ns
10 m
two doors close at different times.
two doors close Simultaneously.
By Lorentz Transformation
The back door closes
The front door closes
--The surprising result is that the back gate is
seen to close earlier, before the front of the
pole reaches it.
--The door closings are not simultaneous in
Poles frame, and they permit the pole to pass
through without hitting either doors.
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2. Lorentz transformation in Spacetime diagram
a) How to set the second RF in the space time
diagram of the first frame?
--The Ct and x cannot point in just any odd
direction because they must be oriented such that
the second observer (S) also measures speed c
for the light pulse.
S
--They have to point in this way shown in blue
lines
b) How to measure with respect to axes which are
not right angles to one another ?
--The diagram in the right shows the set of all
points (in purple) with some particular value of
x.
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c) How is relativity of simultaneity described in
space time diagram?
Increasing speed
d) what happens if the relative speed of the
observers is larger?
The point with x0 covers even more ground in
a given time interval than did the point with
x0. Thus, the ct axis (the set of all points
with x0) is inclined even more towards the
light cone than the ct axis is, as shown in the
above right figure.
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3. Causality
Causality means that cause precedes effect an
ordering in time which every observer agrees
upon.
Q Whether it is possible to change the order of
cause and effect just by viewing two events from
a different frame.
A two events can only be cause and effect if
they can be connected to one another by something
moving at speed less than or equal to the speed
of light.
Two such events are said to be causally connected.
Diagrammatically, event B is causally connected
to event A if B lies within or on the light cone
centered at A
tB gt tA, B occurs after A
Can we perform a Lorentz transformation such
that
tB lt tA, B occurs before A?
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Heres a representative case
t'B gt tA, B occurs after A
In contrast, lets suppose that it were possible
to go into a frame moving faster than light.
X
Then the ct axis would tilt past the light cone,
and the order of events could be reversed (B
could occur at a negative value of t)
C t
t'B lt tA, B occur before A
But it is not true for v gtc.
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4. Nothing can travel faster than C
Its also easy to show that if a body were able
to travel faster than the speed of light, some
observers would observe causality
violations?Effects would precede their causes.
Assume a person is born at event A and travels
faster than the speed of light to event B, where
he dies. His world line is shown in red on the
left
The death occurs after the birth.
Another observer In S
S
Such causality violations are not observed
experimentally. This is evidence that bodies
cannot travel faster than the speed of light
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Relativistic mechanics --Scalars -- 4-vectors
-- 4-D velocity -- 4-momentum, rest mass --
conservation laws -- Collisions -- Photons and
Compton scattering -- Velocity addition
(revisited) and the Doppler shift -- 4-force
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1. Scalars
A scalar is a quantity that is the same in all
reference frames, or for all observers.
It is an invariant number.
E.g.,
But the time interval ?t, or the distance ?x
between two events, or the length l separating
two worldlines are not scalars they do not have
frame-independent values.
2. 4-vectors
This 4-vector defined above is actually a
frame-independent object, although the
components of it are not frame-independent,
because they transform by the Lorentz
transformation.
E.g., in 3-space, the Different observers set
up different coordinate systems and assign
different coordinates to two points C and L, say
Canterbury and London.
--They may assign different coordinates to
the point of the two cites
--They agree on the 3-displacement r separating C
and L., the distance between the two points, etc.
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With each 4-displacement we can associate a
scalar the interval (s)2 along the vector. The
interval associated with the above defined
4-vector is
Because of the similarity of this expression to
that of the dot product between 3-vectors in
three dimensions, we also denote this interval by
a dot product and also by
and we will sometimes refer to this as the
magnitude or length of the 4-vector.
--We can generalize this dot product to a dot
product between any two 4-vectors
--When frames are changed, 4-displacement
transform according to the Lorentz
transformation, and obeys associativity over
addition and commutativity
ii) A 4-vector multiplied or divided by a scalar
is another 4-vector
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3. 4-velocity
In 3-dimensional space, 3-velocity is defined by
where ?t is the time it takes the object in
question to go the 3-displacement ? r.
Can we put the 4-displacement in place of the
3-displacement r so that we have
However, this in itself won't do, because we are
dividing a 4-vector by a non-scalar (time
intervals are not scalars) the quotient will not
transform according to the Lorentz transformation.
The fix is to replace ?t by the proper time ??
corresponding to the interval of the
4-displacement the 4-velocity is then
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Although it is unpleasant to do so, we often
write 4-vectors as two-component objects with the
rest component a single number and the second a
3-vector. In this notation
--What is the magnitude of
Let us change into the frame in which the object
in question is at rest.
It is a scalar so it must have this value in all
frames. You can also show this by calculating the
dot product of
You may find this a little strange. Some
particles move quickly, some slowly, but for all
particles, the magnitude of the 4-velocity is c.
But this is not strange, because we need the
magnitude to be a scalar, the same in all frames.
If you change frames, some of the particles that
were moving quickly before now move slowly, and
some of them are stopped altogether. Speeds
(magnitudes of 3-velocities) are relative the
magnitude of the 4-velocity has to be invariant.
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4. 4-momentum, rest mass and conservation laws
--Under this definition, the mass must be a
scalar if the 4-momentum is going to be a
4-vector.
--The mass m of an object as far as we are
concerned is its rest mass, or the mass we would
measure if we were at rest with respect to the
object.
--Again, by switching into the rest frame of the
particle, or by calculationg the magnitude we
find that 4-momentum, we can show
As with 4-velocity, it is strange but true that
the magnitude of the 4-momentum does not depend
on speed.
Why introduce all these 4-vectors, and in
particular the 4-momentum?
--all the laws of physics must be same in all
uniformly moving reference frames
--only scalars and 4-vectors are truly
frame-independent, relativistically invariant
conservation of momentum must take a slightly
different form.
--In all interactions, collisions and decays of
objects, the total 4-momentum is conserved (of
course we dont consider any external force here).
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--Furthermore,
You better forget any other expressions you
learned for E or p in non-relativistic mechanics.
We get a relation between m, E and
which, after multiplication by c2 and
rearrangement becomes
This is the famous equation of Einstein's, which
becomes
when the particle is at rest
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In the low-speed limit
i.e., the momentum has the classical form, and
the energy is just Einstein's famous mc2 plus the
classical kinetic energy mv2/2. But remember,
these formulae only apply when v ltlt c.
5. Conservation laws
For a single particle 4-momenum before an
action that after
For a multi-particle system
Summed over All the 4-momenta of all the
components of the whole system before interaction
Summed over all the 4-momenta of all the
components of the whole system after interaction
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5. Collisions
In non-relativistic mechanics collisions divide
into two classes
elastic
inelastic
energy and 3-momentum are conserved.
only 3-momentum is conserved
In relativistic mechanics 4-momentum, and in
particular the time component or energy, is
conserved in all collisions
No distinction is made between elastic and
inelastic collisions.
Before the collision
After the collision
Non-relativiistic theory gives M2m, v v/2
By conservation of 4-momentum before and after
collsion, which means that the two 4-vectors are
equal, component by component,
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The ratio of these two components should provide
v/c
--the mass M of the final product is greater
than the sum of the masses of its progenitors, 2m.
--So the non-relativistic answers are incorrect,
Q Where does the extra rest mass come from?
A The answer is energy.
In this classically inelastic collision, some of
the kinetic energy is lost.
But total energy is conserved. Even in classical
mechanics the energy is not actually lost, it is
just converted into other forms, like heat in
the ball, or rotational energy of the final
product, or in vibrational waves or sound
travelling through the material of the ball.
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Strange as it may sound, this internal energy
actually increases the mass of the product of
the collision in relativistic mechanics.
The consequences of this are strange. For
example, a brick becomes more massive when one
heats it up. Or, a tourist becomes less massive
as he or she burns calories climbing the steps of
the Effiel Tower.
All these statements are true, but it is
important to remember that the effect is very
very small unless the internal energy of the
object in question is on the same order as mc2.
For a brick of 1 kg, mc2 is 1020 Joules, or 3
1013 kWh, a household energy consumption over
about ten billion years (roughly the age of the
Universe!)
For this reason, macroscopic objects (like bricks
or balls of putty) cannot possibly be put into
states of relativistic motion in Earth-bound
experiments. Only subatomic and atomic particles
can be accelerated to relativistic speeds, and
even these require huge machines (accelerators)
with huge power supplies.
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6. Photons and Compton scattering
6.1 properties of photon
i) Can something have zero rest mass?
Substitute Epc into v p c2/E c
So massless particles would always have to travel
at v c, the speed of light. Strange??
Photons, or particles of light, have zero rest
mass, and this is why they always travel at the
speed of light.
ii) The magnitude of a photon's 4-momentum
but this does not mean that the components are
all zero.
--The time component squared, E2/ c2, is exactly
cancelled out by the sum of the space components
squared,
--Thus the photon may be massless, but it carries
momentum and energy, and it should obey the law
of conservation of 4-momentum.
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6.2 Compton scattering.
The idea of the experiment is to beam photons of
known momentum Q at a target of stationary
electrons,and measure the momenta Qof the
scattered photons as a function of scattering
angle.
We therefore want to derive an expression for Q
as a function of ?.
Before the collision the 4-momenta of the photon
and electron are
after they are
The conservation law is
For all photons
and for all electrons
Also, in this case
And
Equation (a) becomes
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But by conservation of energy, ( ?-1)mc is just
Q-Q, and (a - b)/ab is just 1/b-1/a, so we have
what we are looking for
This prediction of special relativity was
confirmed in a beautiful experiment by Compton
(1923) and has been reconfirmed many times since
by undergraduates in physics lab courses.
In addition to providing quantitative
confirmation of relativistic mechanics, this
experimental result is a demonstration of the
fact that photons, though massless, carry
momentum and energy.
Quantum mechanics tells
The energy E of a photon is related to its wave
frequency by E h?
Then
so we can rewrite the Compton scattering equation
in its traditional form
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7. Particle decay and pair production
7.1 Particle decay
An elementary particle of rest mass M decays from
rest into a photon and a new particle of rest
mass M/2, Find its velocity.
For 3-momentum conservation, the particle moves
in x direction, and the photon moves in x
direction.
By momentum conservation
(2) Into (1)
Solve for u
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7.2 Pair production (gamma photon can not be
converted to e- and e
Show that the following pair production cannot
occur without involvement of other particles.
Let m be the rest mass of electrons and u, v the
3-velocities of electron and positron.
Sub. (3) into (2)
Sub. (3) into (1)
Compare (4) and (5)
For ux and vx lt c(6) can not be satisfied
Pair production needs an additional particle to
carry off some momentum.
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8 Velocity addition (revisited) and the Doppler
shift
8 .1 Velocity addition revisited
In S, a particle of mass m moves in the
x-direction at speed vx, so its 4-momentum is
In S moving at speed v, the 4-momentum of the
particle
This is a much simpler derivation than that found
before.
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8.2 Photon makes a angle from x axis)
?v
?
Equate each component on both side
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Doppler effect from
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Aberration of light from
Light rays emitted by source in S
Light rays observed in S
When v is very large so that ?0.9, and cos?obs
0.9, ?obs 26?
http//www.anu.edu.au/Physics/Savage/TEE/site/tee/
learning/aberration/aberration.html
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9. 4-force
We recall the 4-velocity and 4-momentum are
defined in terms of derivatives with respect to
proper time rather than coordinate time t . The
definitions are
Also, if the rest mass m of the object in
question is a constant (not true if the object in
question is doing work, because then it must be
using up some of its rest energy!),