Einstein

1
Einsteins special relativity and Lorentz
transformation and its consequences

• Einsteins special relativity
• Events and space-time in Relativity
• 3. Proper time and the invariant interval
• Lorentz transformation

2
Einsteins special relativity and Lorentz
transformation and its consequences

• Einsteins special relativity
• Events and space-time in Relativity
• 3. Proper time and the invariant interval
• 4. Lorentz transformation
• 5. Consequences of the Lorentz transformation
• 6. Velocity transformation

3
1. Principle of Relativity by Einstein (1905)
It is based on the following two postulates1)
The laws of physics are the same for all
observers in uniform motion relative to one
another (principle of relativity),
- need a transformation of coordinates which
preserves the laws of physics
2) The speed of light in a vacuum is the same for
all observers, regardless of their relative
motion or of the motion of the source of the
light. http//en.wikipedia.org/wiki/Theory_of_rela
tivity
Observer to whom the car is moving with relative
V the light pulse reaches A before B
Observer in the car the light pulse reaches A
and B at the same time
Simultaneity breaks down ? time cannot be
regarded as a universal entity
- need a different transformation from Galileos
but will converge to it for VltltC
4
2. Events and space-time in Relativity
When and where is the object under our interest?
An Event in Relativity.
--An event is a point defined by (t, x, y, z),
which describes the precise location of a
happening which occurs at a precise point in
space and at a precise time.
--Space-time is often depicted as a
Minkowski diagram.
5
World lines
The world line of an object is the unique path of
that object as it travels through 4-dimensional
space-time. .
World lines of particles/objects at constant
speed are called geodesics.
How can we preserve the speed of light when we
move between frames?
6
3 Proper time and the invariant interval
3.1 invariant interval)
In 3-dimensional EUCLIDIAN space
In coordinate system O
In coordinate system O
In relativity, a similar quantity for pairs of
events, that is frame-independent, or the same
for all observers (i.e. c is constant
Invariant interval we now require
?t is the difference in time
between the events
?r is the difference between the places of
occurrence of the events.
7
3.2 Events, INTERVAL AND THE METRIC
A metric specifies the interval between two events
8
3.3 Proper time (length) the invariant interval
The proper time between two events is the time
experienced by an observer in whose frame the
events take place at the same point in space.
According to the definition of the interval
between two events
, the interval is said to be timelike
--there always is such a frame since positive
interval means
so a frame moving at vector v (?r) /(?t), in
which the events take place at the same point, is
moving at a speed lt c
, the interval is said to be spacelike
--It is still invariant even though there is no
frame in which both events take place at the same
point. (or (c?t)2 lt 0).
--There is no such frame because necessarily it
would have to move faster than the speed of light.
9
Sometimes the proper distance is defined to be
the distance separating two events in the frame
in which they occur at the same time. It only
makes sense if the interval is negative, and it
is related to the interval by
,the interval is said to be light-like or
nulldefining a null geodesic
This is the case in which
Or, in which the two events lie on the worldline
of a photon.
Because the speed of light is the same in all
frames. . an interval equal to zero in one
frame must equal zero in all frames.
The three cases have different causal properties,
which will be discussed later.
10
4. A transformation formula Lorentz
Transformation
4.1 The formula fits into Einsteins two
postulates
We assume that relative transformation equation
for x is the same as the Galileo Trans. except
for a constant multiplier on the right side, i.e,
where ? is a constant which can depend on u and
c but not on the coordinates. (based on Postulate
1) Must be linear to agree with standard
Galilean transformation in low velocity limit
11
4. A transformation formula Lorentz
Transformation
How do we find the ? factor ?
By tracing the propagation of a light wave front
in two different frames, one of which is moving
with a velocity of V along x-axis w.r.t. the
other.
12
Assume a light pulse that starts at the origins
of S and S at t t0
After a time interval the front of the wave moves
It is recorded as
S
(X, t) in S
u
S
and
(X, t) in S
x ct
By Einsteins postulate 2
xct
13
Substituting ct for x and ct for x in eqs. (1)
and (2)
u lt c so ? is always gt 1
Let (3) (4)
When u ltlt c ? 1
14
The relativistic transformation for x and and x
is
If u ltlt c ? 1
Lorentz transf.
Galileo transf.
15
The transformation between t and t can be
derived
For the wave front of light, xct, xct
Divide c into Eq.(1)
Divide c into Eq(2)
The complete relativistic transformation (L.T.)
is
16
Events and space-time in Relativity
Now, time axes are distinct, as shown
below. What is simultaneous in a moving frame is
not simultaneous in the stationary frame. Here
the signal has to be sent later ( t gt 0) from A
.
17
4. 2 The interval of two events under Lorentz
transformation.
For two events, (t1, x1,y1,z1) and (t2,x2,y2,z2),
we define (T, X, Y, Z)
(t1-t2,x1-x2,y1-y2,z1-z2)
then Lorentz transformation becomes
The interval of two events is an invariant under
Lorentz Transformation. For short the interval
is a Lorentz scalar.
18
4.3 Lorentz transfermation in 4-dimensional
formula
The L-T could be formally defined as a genernal
linear transformation that leaves all intervals
between any pair of events unaltered.
Introduce 4-D vector
Here we have introduced
19
L-T can be expressed as