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Physics 344
Foundations of 21st Century Physics:
Relativistic and Quantum Systems
Instructor: Dr. Mark Haugan
Office: PHYS 282
[email protected]
TA: Dan Hartzler
Office: PHYS 7
[email protected]
Grader: Fan Chen
Office: PHYS 222
[email protected]
Office Hours: If you have questions, just email us to make an
appointment. We enjoy talking about physics!
Help Session: Thursdays 2:00 – 4:00 in PHYS 154
Reading: Chapters 1 through 8 in Six Ideas that Shaped Physics, Unit R.
Exam 1: Wednesday, October 5 at 8:00pm in WTHR 104
The Aberration of Starlight
As is often the case, care in choosing the coordinate system to represent a
physical system can make it easier to predict and explain phenomena. In
relativistic situations, one often has the added difficulty of dealing with two
coordinate systems, S and S’ , at once. We require S’ to be in standard
orientation relative to S so that we can use the standard form of the Lorentz
transformation equations a to analyze situations
S: inertial frame in which the Sun
is essentially at rest throughout
the year.
Earth is moving in the x direction at
speed V = 30 km/sec at t = 0.
Sun
So, S’ in standard orientation and moving at
speed V relative to S has the Earth momentarily
at rest at the origin and is a good approximation
to a coordinate system fixed in an observatory
on Earth near t = t’ = 0.
y
Earth location
at t = 0
z
x
We are interested in observations of a star
made near that time.
The star in question is located on the positive y axis of S, so a photon that arrives at the
Earth’s location at t = 0 has traveled straight down the y axis. The red dots are events
on its worldline. The time and space separations measured between them in S are ∆t21,
∆x21 = 0 and ∆y21 = - c∆t21.
We use the Lorentz transformation equations to determine
the time and space separations between events 1 and 2
measured in S’.
′ = γ∆t21
∆t21
y
1
′ = −γ V ∆t21
∆x21
′ = −c∆t21 y’
∆y21
2
Sun
z
x
We conclude that the star
will be observed in the
direction inclined at an
angle θ toward the x’ axis
and away from the y’ axis
with
tan θ ≈ θ =
1
′
∆y21
2
′
∆x21
′ | γV
| ∆x21
=
≈ 10−4 radians
′ | c
| ∆y21
This is a bit more than 20 seconds of arc and
easily measured by astronomers.
x’
Six months later, the position of this star measured in the Earth’s frame will be inclined at
the same angle but in the opposite direction from the y’ axis.
S: inertial frame in which the Sun
is essentially at rest throughout
the year.
Sun
V
Earth location
at t = 6 months
y
Q1. The direction of a star located on the x axis measured in
the Earth observatory at these two times would be
Earth location
at t = 0
z
x
A) inclined 20 arcseconds away from the x’ direction,
toward y’ direction at t = 0 and away at t = 6 months
B) inclined 20 arcseconds away from the x’ direction,
away from y’ direction at t = 0 and toward at t = 6 months
* C) along x’ axis at both times.
Observations of such yearly variations in the relative directions of stars were first reported
by Bradley in 1728.
Geometry
The examples we examined briefly during recitation yesterday were intended
to remind us that geometry is about spatial properties like lengths of line
segments and angles between them that are independent of any coordinate
system that we might choose to use to represent the segments.
They were also intended to remind us of some of the ways in which we’ve
learned to analyze geometrical situations without choosing a coordinate
system.
In the context of the Euclidean geometry it is extremely useful to connect the
geometrical properties of line segments like QP with the algebraic properties
of directed line segments like the separation or displacement vector that runs
from P to Q.
Q
∆rQP
P
The algebra of such vectors includes the dot
product which allows us to express and work
with geometric properties and relationships like
lengths and angles algebraically, independent
of any specific coordinate system. So, for
example, the squared length of the segment is
simply
2
∆rQP = ∆rQP ⋅ ∆rQP
We can also use this algebra to establish geometrical relationships (theorems)
without ever specifying a coordinate system.
C = B− A
A
θ
B
A ⋅ B = A B cos θ
geometric definition
of the dot product.
For example, the cosine law
(
Even when we use a coordinate system to
represent a situation, this kind of
coordinate-free reasoning is a powerful tool.
For example,
A = cos θ xˆ + sin θ yˆ
)(
)
2 | C | = C ⋅C = B − A ⋅ B − A = B ⋅ B − 2A⋅ B + A⋅ A
2
2
=| B | −2 | A || B | cos θ + | A |
y
unit
circle
A
θ
B = cos ϕ xˆ − sin ϕ yˆ
so,
A ⋅ B = cos(θ + ϕ ) = (cos θ xˆ + sin θ yˆ ) ⋅ (cos ϕ xˆ − sin ϕ yˆ )
= cos θ cos ϕ − sin θ sin ϕ
ϕ
x
B
since
xˆ ⋅ xˆ = 1
xˆ ⋅ yˆ = 0
yˆ ⋅ yˆ = 1
Spacetime Geometry
Since physics, like geometry, happens whether or not we choose a coordinate
system to represent a physical, or geometrical, situation, we expect to be able
to reason in coordinate-free, therefore, geometrical, ways about relativistic
physics.
Actually, we have been using the
fact that directed line segments in
spacetime diagrams “add” head-totail and can be multiplied by scalars
like vectors all along. Let’s simply
be explicit about it by introducing
dimensionless unit vectors parallel
to an inertial coordinate system’s
time and space axes so that we
⇒
can write
∆sQP = ∆tQPtˆ + ∆xQP xˆ
t
Q
⇒
∆tQP
4-vector
separation
∆sQP
∆tQP tˆ
P
∆xQP xˆ
∆xQP
x
Consider two such spacetime separation 4-vectors
2
t
⇒
⇒
∆s20 = ∆t20tˆ + ∆x20 xˆ
⇒
∆s20
⇒
Their difference is another separation 4-vector
⇒
⇒
∆s10
⇒
∆s21 = ∆t21tˆ + ∆x21 xˆ = ∆s20 − ∆s10 = ( ∆t20tˆ + ∆x20 xˆ ) − ( ∆t10tˆ + ∆x10 xˆ )
= ( ∆t20 − ∆t10 ) tˆ + ( ∆x20 − ∆x10 ) xˆ
⇒
∆s21 = ∆s20 − ∆s10
⇒
∆s10 = ∆t10tˆ + ∆x10 xˆ
⇒
1
0
x
so, the familiar algebra of vector addition and multiplication by scalars correctly
expresses the way that coordinate differences represent separations between
pairs of events.
It turns out that we can introduce a dot product* that correctly expresses the
the relationship between the squared interval separating a pair of events and
the coordinate differences that represent their spacetime separation.
⇒
⇒
tˆ ⋅ tˆ = 1
∆s10 ⋅ ∆s10 = ( ∆t10tˆ + ∆x10 xˆ ) ⋅ ( ∆t10tˆ + ∆x10 xˆ )
since
xˆ ⋅ tˆ = 0
2
2
2
2
ˆ
ˆ
ˆ
= ∆t10t ⋅ t + 2∆t10 ∆x10 xˆ ⋅ t + ∆x10 xˆ ⋅ xˆ = ∆t10 − ∆x10
xˆ ⋅ xˆ = −1
* A mathematician would refer to this as a psuedo-dot or psuedo-inner product because of the – signs.
The minus sign on the right simply expresses the fact that the dimensionless
unit vector in the x direction points in a spacelike direction, as do the unit
vectors in the y and z directions.
tˆ ⋅ tˆ = 1
⇒
⇒
∆s10 ⋅ ∆s10 = ( ∆t10tˆ + ∆x10 xˆ ) ⋅ ( ∆t10tˆ + ∆x10 xˆ )
since
xˆ ⋅ tˆ = 0
= ∆t102 tˆ ⋅ tˆ + 2∆t10 ∆x10 xˆ ⋅ tˆ + ∆x102 xˆ ⋅ xˆ = ∆t102 − ∆x102
xˆ ⋅ xˆ = −1
The geometric interpretation of this spacetime dot product is confirmed by the
fact that the dot product of two 4-vectors is an invariant.
Consider,
2
2
2
∆s21
= ∆t21
− ∆x21
= (∆t20 − ∆t10 ) 2 − (∆x20 − ∆x10 ) 2
2
t
⇒
⇒
⇒
∆s21 = ∆s20 − ∆s10
⇒
2
= (∆t20
− 2∆t20 ∆t10 + ∆t102 )
2
− (∆x20
− 2∆x20 ∆x10 + ∆x102 )
2
2
= ∆t20
− ∆x20
∆s20
− 2 ( ∆t20 ∆t10 − ∆x20 ∆x10 )
⇒
∆s10
1
+ ∆t102 − ∆x102
0
⇒
x
⇒
⇒
⇒
≡ ∆s − 2 ∆s20 ⋅ ∆s10 + ∆s102
2
20
The dot product ∆s20 ⋅ ∆s10 = ( ∆t20 ∆t10 − ∆x20 ∆x10 ) must be an invariant, i.e., frameindependent quantity, because the intervals between event pairs are.
Timelike Displacement 4-Vectors and Particle 4-Velocity
We will examine the relationship between the dimensionless unit vectors of
an inertial coordinate system S and those of a system S’ in standard orientation
relative to S in detail next time.
However, because we know how coordinate differences representing the
spacetime separations between pairs of events transform, using SR units,
∆t ′ = γ ( ∆t − V ∆x )
∆x′ = γ (∆x − V ∆t )
∆y′ = ∆y
∆z′ = ∆z
we know how things have to work out.
The 4-vector representing the spacetime separation between a pair of events
is not invariant in the way that the interval between the events is, but the
4-vector is a new kind of frame-independent (geometrical) object that will be
represented by different components in different frames,
⇒
∆s = ∆ttˆ + ∆xxˆ + ∆yyˆ + ∆zzˆ = ∆t ′tˆ′ + ∆x′xˆ′ + ∆y′yˆ ′ + ∆z′zˆ′
Considering the separation 4-vector for two events on the worldline of a particle
and using what we’ve learned about proper time, we can introduce a frame
independent way to deal with particle motion.
As we’ve done before, we consider pairs
of events on a particle’s world line close
enough so that the particle’s velocity
does not change appreciably in the
time-like interval. This means that the
proper time lapse measured between
the events by a clock moving with the
particle is simply the invariant timelike
interval between the events.
1  ∆x21 2 ∆y21 2 ∆z21 2 
) +(
) +(
) ∆t21
(
c 2  ∆t21
∆t21
∆t21 
∆t
∆τ 21 = 1 − v 2 / c 2 ∆t21 = 21
γ
∆τ 21 = 1 −
This diagram is drawn from the perspective
of a frame S and in that frame the particle’s
displacement 4-vector is represented as
2
⇒
∆t21
∆s21
1
∆x21
⇒
∆s21 = ∆t21tˆ + ∆x21 xˆ + ∆y21 yˆ + ∆z21 zˆ
We can use it and the proper time interval to
construct a 4-vector that provides a frame-independent way to work
with the particle’s motion.
The particle’s 4-velocity vector is simply the displacement 4-vector rescaled by
dividing it by the corresponding invariant proper time interval. So,
⇒
∆s21 = ∆t21tˆ + ∆x21 xˆ + ∆y21 yˆ + ∆z21 zˆ
t
and
⇒
event 2 on worldline
⇒
∆τ 21 = ∆s21⋅ ∆s21
⇒
= 1 − ( vx2 − v y2 − vz2 )∆t21 ≡
1
γ
∆s21
∆t21
∆t21
∆x21
event 1
on worldline
yields
⇒
⇒
u≡
∆s21 ∆t21 ˆ ∆x21
∆y
∆z
=
t+
xˆ + 21 yˆ + 21 zˆ
∆τ 21 ∆τ 21
∆τ 21
∆τ 21
∆τ 21
= γ tˆ + γ vx xˆ + γ vy yˆ + γ vz zˆ
particle worldline
x
⇒
⇒
Notice that because ∆τ 21 = ∆ s21⋅ ∆ s21 particle 4-velocity is a dimensionless
unit vector tangent to the particle’s worldline,
⇒
⇒ ⇒
u⋅ u ≡
⇒
⇒
⇒
∆s21 ∆s21 ∆s21⋅ ∆s21
⋅
=
=1
2
∆τ 21 ∆τ 21
∆τ 21
Particle 4-velocity is a useful, frame-independent measure of particle motion
because its components in any frame S are simply related to the components
of the particle’s velocity measured in that frame. Since
⇒
∆s21 ∆t21 ˆ ∆x21
∆y21
∆z21
ˆ
ˆ
u≡
=
t+
x+
y+
zˆ
∆τ 21 ∆τ 21
∆τ 21
∆τ 21
∆τ 21
= γ tˆ + γ vx xˆ + γ vy yˆ + γ vz zˆ
⇒
we have vx = ux / ut , vy = uy / ut and vz = uz / ut .
Also, because a particle’s 4-velocity is simply its displacement 4-vector
rescaled by a scalar, the Lorentz transformation equations transform the
components of the 4-velocity from a frame S to S’ in the same way that they
transform components of the displacement 4-vector, in SR units,
ut ' = γ ( ut − Vu x )
u x′ = γ (u x − Vut )
u y′ = u y
u z′ = u z
which is much simpler transformation equations for velocity components that
we derived earlier.