Postulates of Theory of Relativity: UNIZOR.COM - Relativity 4 All - Eins...

Notes to a video lecture on UNIZOR.COM

Postulates of Theory of Relativity

Historically, many concepts, considered absolute, turned out to be relative. For example, when the Earth was considered flat, a concept of vertical was absolute.
When it was discovered that the Earth looks like a sphere, it was understood that a concept of vertical depends on and is related to where you are standing on the surface of Earth.

Nicolaus Copernicus declared that our position in space is not any more special than any other position, it's not absolute, it's not the center of the Universe.
What's important is just our position relative to positions of other objects.

Galileo Galilei declared that the movement of an object is not absolute, but is relative to other objects' movement.
If object A moves relative to object B in some direction, object B at the same time moves relative to object A in the opposite direction.

Theory of Relativity, as many other scientific theories, is based on certain principles that we accept as the basis for further development of a theory.
Obviously, all these might change with time, so what we accept today might be different in the future.

The first principle we accept in the Theory of Relativity is based on comparing the laws of movements in two inertial physical systems with no external to these systems forces involved, when one system moves relatively to the other in a uniform motion along a straight line.

Defining a system of Cartesian coordinates and clock to measure time in each such system allows to express these laws of movement quantitatively as some equations.

The First Postulate of the Theory of Relativity is that all such laws and equations must be identical in these two inertial systems.
There can be no physical experiment that would allow to distinguish one such system from another.
The First Postulate is usually referred to as the Principle of Relativity.

Consequently, if all the laws are the same in these two systems, there is no such notion as absolute rest, because for any such system α there is always another system β uniformly moving relatively to it and, therefore, α is moving relatively to β.

It should be mentioned that the Principle of Relativity does not contradict our intuitive understanding of Nature, and we sometimes experience it in real life. For example, when sitting on a train watching another train slowly passing by, we might not realize for sure which train is moving and which is standing.

The Second Postulate of Theory of Relativity is not as obvious.
Moreover, to establish its validity, numerous very precise experiments were made until, finally, physicists agreed on it.
This principle is related to speed of light and needs a lot of explanation and justification.

First of all, according to Maxwell equations for electromagnetic field, the speed of light should depend only on the properties of the medium. In empty space this speed is about 300,000,000 meters per second, in air it is slightly less, in glass - much less.

What's interesting, when a ray of light goes from one medium to another, like from air to glass, its speed is reduced. But, when it comes out from glass back to air, its speed increases again to air-specific higher speed.
This is not like some material objects going through some obstacle, losing their speed and not recovering it anymore.

Secondly, according to very precise experimental data, the speed of light does not depend on the movement of its source.
Since all inertial systems are moving relatively to each other, the speed of light is the same in all inertial systems.

This is the Second Postulate of Theory of Relativity and this is what differentiates the Theory of Relativity from classical Physics.

Being confirmed many times, the invariance of the speed of light in empty space relative to movement of the light source was accepted by physicists, but no reasonable theoretical explanation for this counterintuitive property of light was offered until Albert Einstein suggested it in his famous article "ON THE ELECTRODYNAMICS OF MOVING BODIES" in 1905.

Einstein suggested that space and time have properties which escaped the attention of physicists working with lower speeds of movement of material objects.

If considered from the classical Physics' viewpoint on space and time, the constant speed of light, apparently, contradicts the Principle of Relativity.

To demonstrate this, consider such an experiment.

You are in the middle of a uniformly moving on rails platform and throw a ball either forward along the movement of a platform or backward, checking the time it takes for a ball to reach a front or back of a platform.


Assume the length from you to both edges of a platform is L, the constant speed of a platform relative to rails is V and the speed you throw a ball relative to rails is S.

Set an X-axis parallel to the movement of a platform with a center of a platform to be at X=0 coordinate.
Then at time T=0 the coordinate of the front edge of a platform is X=L, the back edge is at X=−L.

A ball flying forward has a speed V+S, the one flying backward has speed V−S.
If the time a ball reaches the front of a platform is T_front, the ball should cover a distance of L plus the distance a platform moves during time T_front. All this time a ball moves with speed V+S.

Therefore, the coordinate of a ball at time T_front is
X=(V+S)·T_front
Coordinate of the front edge of a platform at the same moment is X=L+V·T_front.

Since at time T_front a ball reaches the front edge of a platform, their X-coordinates are the same, which leads us an equation
(V+S)·T_front = L + V·T_front
Hence, T_front = L/S
This corresponds to Principle of Relativity because the same result would be, if the platform was not moving relative to rails.

A ball flying backward has a speed V−S.

If the time a ball reaches the front of a platform is T_back, the ball should cover a distance of L minus the distance a platform moves during time T_back. All this time a ball moves with speed V−S.

Therefore, the coordinate of a ball at time T_back is
X=(V−S)·T_back
Coordinate of the back edge of a platform at the same moment is X=−L+V·T_back.

Since at time T_back a ball reaches the back edge of a platform, their X-coordinates are the same, which leads us an equation
(V−S)·T_back = −L + V·T_back
Hence, T_back = L/S
This also corresponds to Principle of Relativity because the same result would be, if the platform was not moving relative to rails.

As we see, this classical approach leads to correct results in full agreement with the Principle of Relativity.

But let's repeat the same experiment with a ray of light instead of a ball. The light moves with constant speed c towards positive direction of X-axis and −c towards negative direction of X-axis regardless of the speed of its source.


Therefore, in our two equations for time we have to replace the speed of a ball V+S and V−S with the speed of light c and −c, which leads to these equations
c·T_front = L + V·T_front
−c·T_back = −L + V·T_back
from which follows
T_front = L/(c−V)
T_back = L/(c+V)

Clearly, these two values of time are different, unless a platform is not moving and V=0. The results of our experiment will depend on the speed of a platform.

Therefore, being inside an inertial system, a platform, conducting an experiment with rays of light, we can distinguish between moving platform and platform at rest or two platforms moving with different speeds, which contradicts the Principle of Relativity.

The main achievement of Special Theory of Relativity by Einstein was to offer a model of space and time and their transformation from one inertial system to another that restored the compliance with the Principle of Relativity for systems that involve the light and retained the classical view on them as an approximation, when the speeds of participating objects are not too high.

Based on the First and the Second Postulates, the relativistic laws of transformation of space and time were logically derived by Einstein and confirmed later on by numerous very precise experiments.

While the relativity of space (position and movement) was accepted by physicists since the times of Galileo, the relativity of time was needed to bring the results of experiments with rays of light in compliance with the Principle of Relativity.


Summary

The First Postulate of the Theory of Relativity is that all Physical laws and equations must be identical in all inertial systems
- the Principle of Relativity

The Second Postulate of the Theory of Relativity is that the speed of light is the same in all inertial systems depending only on the medium where light propagates

Relativity Metric: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Relativity Metric


Metric is one of the most important characteristic of space we live in, and it defines a distance between two points.

Prior to Theory of Relativity our space was considered three-dimensional Euclidian and, therefore, the square of a distance between two points
A(X_A,Y_A,Z_A) and B(X_B,Y_B,Z_B)
was equal to
d²(A,B) = (X_B−X_A)² +
+ (Y_B−Y_A)² + (Z_B−Z_A)²

We have analyzed how this distance transformed by Galilean transformation of coordinates from one inertial system into another, moving relative to the first, and came up with its invariance relative to this transformation.

Indeed, the Galilean transformations from inertial reference frame α{X,Y,Z} into inertial reference frame β{x,y,z} that moves relative to α with speed v along its X-axis are
t = T
x = X − v·T
y = Y
z = Z
where T=t is absolute time in both reference frames.

Then the distance between points A and B in β-frame is
d_β²(A,B) = (x_B−x_A)² +
+ (y_B−y_A)² + (z_B−z_A)² =
= (X_B−v·T−X_A+v·T)² +
+ (Y_B−Y_A)² + (Z_B−Z_A)² =
= d_α²(A,B)

With a progress of the Theory of Relativity we came up with Lorentz Transformation of coordinates that takes time into account.
This type of transformation seemed to better correspond to our theories and experimental data, but a concept of length in our traditional sense was no longer invariant (see Length Transformation lecture in this part of a course).

An interesting approach would be to find a different definition of metric in our space that takes time into account and is invariant relative to Lorentz Transformation analogously to classic definition of Euclidean metric being invariant relative to Galilean transformation (see Metric Invariance lecture in the Galilean View part of a course).

Let's attempt to apply Lorentz Transformation to classical definition of Euclidean metric, as mentioned above, and see where it leads us.

Assume, we have two reference frames α{T,X,Y,Z} and β{t,x,y,z}.
Further assume that β-frame moves relative to α-frame along X-axis with constant velocity vector V whose α-coordinates are {v,0,0}.

Then Lorentz transformation of coordinates from α to β is
t = γ·(T − v·X/c²)
x = γ·(X − v·T)
y = Y
z = Z
where γ = 1/√1−v²/c².

Applied this transformation to coordinates of points A and B, we obtain
x_A = γ·(X_A − v·T_A)
y_A = Y_A
z_A = Z_A
and
x_B = γ·(X_B − v·T_B)
y_B = Y_B
z_B = Z_B

Now we are ready to calculate the Euclidean distance between these two points in both reference frames α{T,X,Y,Z} and β{t,x,y,z}.

d_α²(A,B) = (X_B−X_A)² +
+ (Y_B−Y_A)² + (Z_B−Z_A)²

d_β²(A,B) = (x_B−x_A)² +
+ (y_B−y_A)² + (z_B−z_A)²

As seen from the coordinate transformation above,
x_B−x_A =
= γ·[(X_B−v·T_B) − (X_A−v·T_A)] =
= γ·(X_B−X_A)−γ·v·(T_B−T_A)
y_B−y_A = Y_B−Y_A
z_B−z_A = Z_B−Z_A

Obviously, unless v=0 and, consequently, γ=1,
x_B−x_A ≠ X_B−X_A
and, as a result,
d_α²(A,B) ≠ d_β²(A,B)
in a general case of reference frames moving relative to each other.
Lorentz Transformation does not preserve the Euclidean metric from one inertial reference frame to another, unless they are not moving relative to each other.

Let's proceed in our quest for a different definition of metric in our space that takes time into account and is invariant relative to Lorentz Transformation.

We change the definition of Euclidean metric d²(A,B) by adding a time-dependent component:
D²(A,B) = (X_B−X_A)² +
+ (Y_B−Y_A)² + (Z_B−Z_A)² +
+ f(T_B−T_A)
where f(T) is some function of time coordinate.

The first and the easiest attempt would be to add a component linearly dependent on a square of time differences similarly to dependency on space coordinates:
f(T) = k·T²
It's reasonable to try to find a factor k that would assure the invariance of Lorentz Transformation. If we succeed, great. If not, we'll try some other solution.

According to our newly proposed metric, the relativistic distance from A to B in α-frame is
D_α²(A,B) = (X_B−X_A)² +
+ (Y_B−Y_A)² + (Z_B−Z_A)² +
+ k·(T_B−T_A)²

To find unknown coefficient k (if possible, of course), we equate
D_α²(A,B) = D_β²(A,B)

Considering only X-coordinate and time change from α-frame to β-frame, our equation for k is
(X_B−X_A)² + k(T_B−T_A)² =
(x_B−x_A)² + k(t_B−t_A)²

If the solution for k is independent of the relative speed v of β-frame moving relative to α-frame and independent of coordinates of points A and B, our problem is solved.

Solving the above equation using Lorentz Transformation from α- to β-frame.
(X_B−X_A)² + k(T_B−T_A)² =
γ²·[(X_B−v·T_B) − (X_A−v·T_A)]² +
+ k·γ²·[(T_B−v·X_B/c²) −
− (T_A−v·X_A/c²)]²

Let's simplify the right side of this equation.

It's equal to

γ²·[(X_B−X_A) − v·(T_B−T_A)]² +
+ k·γ²·[(T_B−T_A) −
− (v/c²)·(X_B−X_A)]² =

= γ²·[(X_B−X_A)² −
− 2v·(X_B−X_A)·(T_B−T_A) +
+ v²·(T_B−T_A)² +
+ k·(T_B−T_A)² −
− 2(k·v/c²)·(X_B−X_A)·(T_B−T_A) +
+ k·(v/c²)²·(X_B−X_A)²] =

= γ²·[(1+k·(v/c²)²)(X_B−X_A)²
− 2(v+k·v/c²)·(X_B−X_A)·(T_B−T_A)
+ (v²+k)·(T_B−T_A)²] = D_α²(A,B)

Recall the metric in α-frame is
D_α²(A,B) =
(X_B−X_A)² + k·(T_B−T_A)²

To satisfy the equation
D_α²(A,B) = D_β²(A,B)
the corresponding coefficients in the increments along each coordinate must be equal: γ²·(1+k·(v/c²)²) = 1
2(v+k·v/c²) = 0
γ²·(v²+k) = k

If our approach is right, the three equations above must have one solution for k that satisfies all of them, and it must not depend on speed v.
Is it possible?

The second equation above can be used first, and the value of k from it is k=−c².
Let's check if this is a solution to the first and the third equations.

The first equation with k=−c² is
γ²·(1−c²·(v/c²)²) = 1
Considering γ² = 1/(1−v²/c²),
the left side is unconditionally equal to 1.

The third equation with k=−c² looks as follows
γ²·(v²−c²) = −c²
Obviously, this is an identity.

It looks like the value k=−c² fits all our criteria and the relativistic distance in α-frame
D_α²(A,B) =
= (X_B−X_A)² − c²·(T_B−T_A)²
is equal to relativistic distance in α-frame
D_β²(A,B) =
= (x_B−x_A)² − c²·(t_B−t_A)²
and, therefore, is relativistic invariant.

Adding Velocities: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Adding Velocities

In classic Mechanics that uses Galilean transformation, if an object moves in reference frame β with velocity u_β, and its β-frame moves relative to another frame α with velocity v, the composition of movements of an object and its frame can be represented in α-frame as a simple vector sum of velocities of an object:
u_α = u_β + v

In Theory of Relativity that uses Lorentz Transformation the situation is a bit more complex.

Consider, as before, two inertial reference frames:
α-frame with coordinates {X,Y,Z} and time T and
β-frame with coordinates {x,y,z} and time t.

At time T=t=0 both reference frames coincide.
Assume that β-frame uniformly moves relatively to α-frame with speed v along α-frame's X-axis, maintaining parallelism of all corresponding axes.

Assume an object moves uniformly in β-frame along its x-axis with speed u_β.
At time t=0 this object is at the origin of β-frame.

Our task is to determine the speed of this object in α-frame u_α, using Lorentz Transformation.

IMPORTANT NOTE:
We deliberately restricted movements only along the X- and x-axes to simplify the calculations.
To expand the results to all three space dimensions is trivial.

The object in β-frame moves according to a simple law
x(t) = u_β·t
Obviously,
dx(t)/dt = u_β.

According to Lorentz transformation,

T =	t + v·x/c² √1−(v/c)²

X =	x + v·t √1−(v/c)²

Our task is to find the speed of an object in α-frame, that is to find u_α=dX/dT.

Using an explicit form of function x(t), the above formulas of Lorentz Transformation are

T =	t + v·uβ·t/c² √1−(v/c)²

X =	uβ·t + v·t √1−(v/c)²

The above expressions are linear functions of time t and their derivative by t would result in constants.

Since
dX/dt = (dX/dT)·(dT/dt),
we can express the u_α as
u_α = dX/dT = (dX/dt)/(dT/dt)
and we can easily resolve our problem as follows
dX/dt = (v+u_β )/√1−(v/c)² 
dT/dt = (1+v·u_β /c²)/√1−(v/c)² 
dX/dT = (v+u_β )/(1+v·u_β /c²)

The final formula is

uα =	v + uβ 1 + v·uβ /c²

The above formula represents the law of addition of velocities in Theory of Relativity.

Obviously, if speeds of an object u_β and of β-frame v are low relative to speed of light c, which is the case under most normal conditions, the above transformation is approximated by the Galilean one quite well.

Length Transformation: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Length Transformation

This lecture continues the previous ones at UNIZOR.COM - Relativity 4 All - Einstein View. We strongly recommend to familiarize yourselves with these lectures prior to studying the material of this one.

As was described in the previous lectures, there is a difference in time perception of the same process by two observers moving relatively to each other.
The proper time, as observed by an observer of the frame, where the process was actually happening, was shorter than the time perceived by an outside observer in some other inertial frame.

In this lecture we address the perception of space in an analogous situation.

Consider, as before, two inertial reference frames:
α-frame with coordinates {X,Y,Z}, time T and an α-observer positioned at the origin of coordinates {0,0,0} and
β-frame with coordinates {x,y,z}, time t and a β-observer in it, who can measure the length of some rigid rod at rest in this system of coordinates.

At time T=t=0 both reference frames coincide.
Assume that β-frame uniformly moves relatively to α-frame with speed v along α-frame's X-axis, maintaining parallelism of all corresponding axes.

A rigid rod is at rest in β-frame, its proper length measured by β-observer is R.
What would its length be from α-observer's perspective?
Would the result depend on orientation of this rod relatively to a trajectory of β-frame in coordinates of α-frame?

Consider two cases:
(a) a rod is perpendicular to a trajectory positioned along Z- and z-axes and
(b) a rod is positioned along X- and x-axes, that is parallel to a trajectory.

We will use the Lorentz Transformation formulas presented in the previous lectures.

The Lorentz transformation from inertial α-frame coordinates {X,Y,Z,T} to inertial β-frame coordinates {x,y,z,t}, when β-frame is uniformly moving with constant speed v along α-frame's X-axis, as derived in earlier lecture of this course, is:

t =	T − v·X/c² √1−(v/c)²

x =	X − v·T √1−(v/c)²

y = Y
z = Z
where c is the speed of light in empty space - the same constant for all inertial reference frames, as postulated in the Theory of Relativity.

The reverse transformation from coordinates {x,y,z,t} of β-frame to coordinates {X,Y,Z,T} of α-frame that moves relative to β-frame with speed −v along x-axis is:

T =	t + v·x/c² √1−(v/c)²

X =	x + v·t √1−(v/c)²

Y = y
Z = z


The case of a rod perpendicular to a trajectory of β-frame's movement

A rod in β-frame is at permanent position for any time moment t stretching perpendicularly to x-axis along z-axis from point
{x,y,z}={0,0,0}
to point
{x,y,z}={0,0,R}
which makes its proper length in β-frame to be always R.

(Click right mouse button to open a bigger picture in a new tab)

Using the transformation of coordinates from β- to α-frame, we obtain the coordinates of this rod in α-frame.

The edge of a rod at β{0,0,0} will have α-coordinates

T0 =

t + v·0/c² √1−(v/c)²

= γ·t

X0 =

0 + v·t √1−(v/c)²

= γ·v·t

Y₀ = y₀ = 0
Z₀ = z₀ = 0
where γ = 1/√1−(v²/c²)

The edge of a rod at β{0,0,R} will have α-coordinates

TR =

t + v·0/c² √1−(v/c)²

= γ·t

XR =

0 + v·t √1−(v/c)²

= γ·v·t

Y_R = y_R = 0
Z_R = z_R = R

From the above expressions for α-coordinates of the edges follows that for any time moment its length in α-frame is the distance from point {X=γ·v·t,Y=0,Z=0} to point {X=γ·v·t,Y=0,Z=R}, which is R.

The perceived length of a rod positioned perpendicularly to a trajectory of its movement is the same as its proper length measured in the reference frame, where this rod is at rest.


The case of a rod parallel to a trajectory of β-frame's movement

Assume that in β-frame the rigid rod of the length R is positioned along x-axis from x=−R to x=0.
As β-frame is moving along X-axis of α-frame, the entire length of a rod is passing in-front of the α-observer positioned at the origin of α-frame.


(Click right mouse button to open a bigger picture in a new tab)

Consider two events:
(a) the start of time , when the right end of the rod has coordinates x=0, t=0 in β-frame and X=0, T=0 in α-frame;
(b) the left end of the rod is passing in-front of α-observer, as β-frame moves along X-axis with speed v.

Measuring the time between these two events, as perceived by α-observer, is sufficient to calculate the length of the rod from his viewpoint. All we have to do is to multiply this time by the speed v of β-frame moving relative to α-frame.

The event (a) is, obviously, {X=x=0,Y=y=0,Z=z=0,T=t=0} in both reference frames.
The event (b) occurs when the left end of a rod is passing through point X=0 at some time T_R in α-frame. In β-frame its x-coordinate is still −R. The timing t_R of this event in β-frame is the time β-frame moves along X-axis by the length R of the rod, that is t_R=R/v.

Let's summarize information about event (b):
X_R = 0
T_B is unknown
x = −R
t_R = R/v

The Lorentz Transformation for X-coordinate from β-frame to α-frame is

XR =

−R + v·R/v √1−(v/c)²

= 0

as expected

The Lorentz Transformation for T-coordinate from β-frame to α-frame is

TR =

R/v + v·(−R)/c² √1−(v/c)²

=

=

(R/v)·(1 − v²/c²) √1−(v/c)²

=

= (R/v)·√1−(v/c)² 

Multiplying this timing by the speed of β-frame moving relative to α-frame, we obtain the length L_αR of a rod moving across with speed v, as perceived by α-observer:
L_αR = v·T_R = R·√1−(v/c)² 

Using a traditional symbol
γ = 1/√1−(v²/c²)
(which is greater than 1),
the length, as perceived by α-observer, is
L_αR = R/γ

The length, perceived by an outside observer in some inertial frame, relative to which a rigid rod is moving and is positioned along the trajectory of movement, is shorter than its proper length measured in the frame where this rod is at rest.
.

Time Transformation: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Time Transformation

We have introduced a concept of time dilation in the lecture Time Measuring and, arranging a thought experiment with a light clock, came up with a formula for the perceived time duration of our experiment by an observer outside of a reference frame where this experiment is conducted in terms of the proper time duration measured inside a frame of this experiment.

The formula for time dilation was
T_β = γ·t_β
where
α is some inertial frame with Cartesian coordinates {X,Y,Z}, time T and α-observer in it;
β is another inertial frame with Cartesian coordinates {x,y,z}, time t and β-observer in it;
initially at time T=t=0 α-frame and β-frame coincide;
β-frame is moving with constant speed v relative to α-frame along the X-axis of α, preserving parallelism of corresponding axes;
T_β is a perceived time duration of β-experiment by α-observer;
t_β is the proper time duration of β-experiment as observed by β-observer moving together with β-frame;
γ = 1/√1−(v²/c²);
c is the speed of light in vacuum - a universal constant, the same in all inertial reference frames.

We have also derived the formulas of transformation of Cartesian coordinates and time from one inertial reference frame to another in Lorentz Transformation lecture.
The final form of Lorentz transformation from inertial α-frame coordinates {X,T} to inertial β-frame coordinates {x,t}, when β-frame is uniformly moving with speed v along α's X-axis, in the Special Theory of Relativity is:

x =	X − v·T √1−(v/c)²

t =	T − v·X/c² √1−(v/c)²

If β-frame moves with speed v relative to α-frame along X-axis, we can consider α-frame as moving with speed −v relative to β-frame.
Then the inverse transformation would be

X =	x + v·t √1−(v/c)²

T =	t + v·x/c² √1−(v/c)²

Armed with the above Lorentz Transformation formulas, we can attempt to derive the formula for time dilation more mathematically.
Assume some experiment is conducted at point x=x₀ in β-frame.
The beginning of this experiment is at time t_beg and it ends at time t_end, so the duration of an experiment is
Δt = t_end − t_beg

The α-observer sees this experiment begins at point X_beg and time T_beg, ending at point X_end and time T_end.

According to Lorentz Transformation, the coordinates of β-experiment for α-observer are

Xbeg =

x0 + v·tbeg √1−(v/c)²

Tbeg =

tbeg + v·x0/c² √1−(v/c)²

Xend =

x0 + v·tend √1−(v/c)²

Tend =

tend + v·x0/c² √1−(v/c)²

Now we can calculate the duration of our experiment as perceived by α-observer.
ΔT = T_end − T_beg =

=	tend−tbeg √1−(v/c)²

= Δt·γ
This is exactly the same formula obtained in lecture Time Measuring through a thought experiment with a light clock.

Incidentally, while in β-frame our experiment was conducted at a single point x₀ on x-axis, an observer in α-frame sees it at moving along X-axis for a distance
ΔX = X_end − X_beg =

=	v·(tend − tbeg) √1−(v/c)²

= v·Δt·γ

Lorentz Transformation: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Lorentz Transformation

As in the previous lecture about time dilation, consider two inertial reference frames:
frame α with Cartesian coordinates X, Y, Z and time T
and frame β with Cartesian coordinates x, y, z and time t.

So, all capital coordinates (X, Y, Z) and time (T) will be related to α-frame, events and experiments occurring in it will be referred to as α-events and α-experiments, and observer at rest in it will be referred to as α-observer.
Similarly, all lower case coordinates (x, y, z) and time (t) will be related to β-frame, β-events, β-experiments and β-observer.

Assume that at time T=t=0 both frames coincide, that is their origins are at the same point in space and their corresponding axes overlap, but frame β is moving relative to α along the X-axis with constant speed v, so the velocity vector of β-frame in α-coordinates has components {v,0,0}.

Since the movement is relative, we can say at the same time that frame α is moving relative to β along its x-axis with speed −v, so the velocity vector of α-frame in β-coordinates has components {−v,0,0}.

We still rely on two main principles:
(a) Principle of Relativity that states that all the Physics laws must be the same, if expressed quantitatively, using corresponding coordinates and time, in all inertial reference frames.
(b) Speed of light is constant and the same in all inertial reference frames in empty space, we will use a symbol c for this universal constant.

Our current task is purely mathematical - express the transformation of space and time coordinates from one inertial reference frame to another that moves relative to the first, preserving the integrity of the above principles.

At this point we would like to remind that this type of task is not unusual in Math.
One of the elementary tasks in coordinate geometry was to transform the Cartesian coordinates if XY-axes are rotated by some angle around the origin.
In this case the transformation from coordinate system {X,Y} to {x,y}, where axes are rotated by angle φ around origin is
x = X·cos(φ) − Y·sin(φ)
y = X·sin(φ) + Y·cos(φ)

Consider now our case of the transformation of space and time coordinates between two inertial reference frames moving relatively to each other as described above:
α-frame {X,T} and
β-frame {x,t} that uniformly moves relatively to α-frame along the X-axis with constant speed v, preserving the parallelism of corresponding axes, provided at time T=t=0 both reference frames coincide.

Since the direction of β-frame movement is along X-axis, coordinates y and z will be the same as, correspondingly, Y and Z. This simplifies our task, reducing it to transformation of only two coordinates - X in space and T in time into x and t.

We will look for a simple linear transformation from {X,T} system to {x,t} system of the form
x = p·X + q·T
t = r·X + s·T
where p, q, r and s are four unknown coefficients of transformation, which we are going to determine.

We should not add any constants into above transformations since coordinates (X=0,T=0) should transform into (x=0,t=0).

Equation 1
Since β-frame moves along X-axis of α-frame with speed v, its origin of space coordinate (point x=0) must at any moment of α-time T be on a distance v·T from the origin of coordinates of a stationary α-frame.
Hence, if X=v·T then x=0 for any T.
From this and the first transformation equation
x = p·X + q·T
we derive:
0 = p·v·T + q·T or
0 = (p·v + q)·T.

Since this equality is true for any time moment T,
p·v + q = 0
and, unconditionally,

q = −p·v

This is the first equation for our unknown coefficients.

Equation 2
Consider a ray of light issued at time T=t=0 from the point of coinciding origins of both reference frames in the direction of positive coordinates X and x.
Since the speed of light c is the same in both systems {X,T} and {x,t}, according to Principle of Relativity, an equation of the motion of the front of the light wave in the α-frame must be X=c·T and in the β-frame it is x=c·t.
Therefore, if X=c·T, then x=c·t.
Put X=c·T into both equations of transformation of coordinates.
We get
x = p·c·T + q·T and
t = r·c·T + s·T
Substitute these expressions into x=c·t:
p·c·T + q·T = r·c²·T + s·c·T.
Reduce by T,

p·c + q = r·c² + s·c

This is the second equation for unknown coefficients.

Equation 3
Consider a ray of light issued at time T=t=0 from the point of coinciding origins of both reference frames in the direction of negative coordinates X and x.
Repeat the logic of a previous paragraph for the light moving now in the opposite direction with a speed −c.
Therefore, if X=−c·T, then x=−c·t.
Put X=−c·T into both equations of transformation of coordinates.
We get
x = −p·c·T + q·T
t = −r·c·T + s·T
Substitute these expressions into x=−c·t:
−p·c·T + q·T = r·c²·T − s·c·T.
Reduce by T,

−p·c + q = r·c² − s·c

This is the third equation for unknown coefficients.

System of Equations
So, this is the system of three linear equations for four unknown coefficients of transformation p, q, r, s:
(a) q = −p·v
(b) p·c + q = r·c² + s·c
(c) −p·c + q = r·c² − s·c
Usually, we need four equations for four unknown, but additional considerations will help to resolve this situation.

Solution
From (b) and (c), adding and subtracting these equations, we get:
2q = 2r·c², therefore q = r·c²
2p·c = 2s·c, therefore p = s

Now the system of equations is
q = −p·v
q = r·c²
p = s

Let's express q, r and s in terms of only one unknown p and known constants c and v.
q = −p·v
r = q/c² = −p·v/c²
s = p

The original system of equations now looks like
x = p·X − p·v·T
t = −p·(v/c²)·X + p·T

In a matrix form the transformation from α-frame {X,T} to β-frame {x,t} looks like

x

t

=

p	−p·v
−p·v/c²	p

·

X

T

Symmetrically, the motion of system {X,T} relative to system {x,t} is a uniform motion with speed −v along x-axis.
That means that expression of {X,T} coordinates in terms of {x,t} should look similar to above with the only difference of the sign of the speed v
X = p·x + p·v·t
T = p·(v/c²)·x + p·t

In a matrix form the transformation from β-frame {x,t} to α-frame {X,T} looks like

X

T

=

p	p·v
p·v/c²	p

·

x

t

Since the latter matrix of transformation is an inverse of the one that represents transformation from α-frame {X,T} to β-frame {x,t}, the product of these two matrices must produce a unit matrix.

1	0
0	1

Therefore, multiplying two matrices of transformation, we should have satisfied the following equations for all four elements of the unit matrix
p² − p²·v²/c² = 1
p²·v − p²·v = 0
−p²·v²/c² + p²·v²/c² = 0
−p²·v²/c² + p² = 1

The first equation is the same as the fourth and produce the value for coefficient p
p²·(1 − v²/c²) = 1 and
p = 1/√1−v²/c² .

The second and the third equations above are identities that we can disregard.

Consequently, the values for other coefficients of transformation of coordinates are
q = −p·v = −v/√1−v²/c² 
r = −p·v/c² = (−v/c²)/√1−v²/c² 
s = p = 1/√1−v²/c² 

Traditionally, factor v/c is replaced with Greek letter β, which results in formulas:
x = (1/√1−β² )·X +
+ (−v/√1−β² )·T
t = ((−v/c²)/√1−β² )·X +
+ (1/√1−β² )·T

One more simplification is usually done by introducing Lorentz factor γ equaled to 1/√1−β² :
x = γX − γv·T = γ(X − v·T)
t =−γv·X/c²+γT = γ(T−v·X/c²)

The final form of Lorentz transformation from inertial α-frame coordinates {X,T} to inertial β-frame coordinates {x,t}, when β-frame is uniformly moving with speed v along α's X-axis, in the Special Theory of Relativity is:

x =	X − v·T √1−(v/c)²

t =	T − v·X/c² √1−(v/c)²

where c is the speed of light in empty space - the same constant for all inertial reference frames.

The final transformation looks exactly like in Einstein's article On the Electrodynamics of Moving Bodies (1905), but the derivation is mathematically rigorous based on two postulates - Principle of Relativity and the universality of the constant speed of light in all inertial reference frames in vacuum.

Time Measuring: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Time Measuring

We are planning to conduct a few thought experiments. For this we need some simple clock to measure time between events in different inertial reference frames.

Consider the following clock construction.
Take a rigid rod, its edges are points A and B.
At edge point A we install the source of light and a sensor of light.
At edge point B we install a mirror that reflects a beam of light back to its origin.

We define the unit of time as the time from a moment we send a beam of light from point A to point B till we receive back a reflected beam back at point A.
This will be our light clock that we can use to measure any time interval.


As confirmed by numerous experiments and follows from the Maxwell equations, the speed of light c is exactly the same in any inertial reference frame in empty space.
That's what makes this clock perfect for time measurement.

According to the Principle of Relativity, which we accept as an axiom, any observer in one inertial reference frame, who measured a unit of time by this light clock, can go to another inertial reference frame, take this clock with him, and the interval of time measured by this clock there would be exactly the same.

If some process, like boiling a pot of water or growing corn, in one inertial frame takes certain time, according to this light clock, exactly the same process would take exactly the same amount of time in any other inertial reference frame using the same clock locally to that frame.

Now consider two inertial reference frames:
frame α with Cartesian coordinates X, Y, Z and time T
and frame β with Cartesian coordinates x, y, z and time t.

So, all capital coordinates (X, Y, Z) and time (T) will be related to α-frame, events occurring in it (α-events) and observer at rest in it (α-observer).
Similarly, all lower case coordinates (x, y, z) and time (t) will be related to β-frame, events occurring in it (β-events) and observer at rest in it (β-observer).

Assume that at time T=t=0 both frames coincide, that is their origins are at the same point in space and their axes overlap, but frame β is moving relative to α along the X-axis with speed v, so the velocity vector of β-frame in α-frame coordinates has components {v,0,0}.

Since the movement is relative, we can say at the same time that frame α is moving relative to β along the X-axis with speed −v, so the velocity vector of α-frame in β-frame coordinates has components {−v,0,0}.

Consider identical light clocks in both α- and β-frames arranged between local in each frame points A{0,0,0} and B{0,0,R}, where R is any real number. The first point is an origin of a corresponding frame and the second point is on the Z-axis on a distance R from the origin.

Since the distance between these points in each frame is R, the interval of time between emitting a ray of light from the origin and receiving a reflected ray back at the origin (with the total distance covered by light ray equaled to 2R and the speed of light c) equals in each frame for a local observer at rest in each corresponding frame to
T_α = t_β = 2R/c

Besides this local observation from within each frame, consider how the same time interval is evaluated by an observer in the other reference frame.

The interval of time for a ray of light covering in α-frame the distance from the origin of coordinates to a mirror and back can be analyzed by an observer at rest in β-frame (t_α) and an interval of time for a ray of light moving in β-frame along the analogous path can be analyzed by an observer at rest in α-frame (T_β).
These remote observations are not necessarily produce the same measurements of time interval as observed locally.

Let's see now how an observer in α-frame {X,Y,Z,T} evaluates the time interval of a round trip of a ray of light in the β-frame {x,y,z,t} moving with speed v along α's X-axis.

Assume, at some moment, when β's origin is at α coordinate {X₀,0,0}, the light was emitted by the β light clock from its origin.
From a β-observer viewpoint the ray emitted by a light clock from β point {0,0,0} goes vertically along its Z-axis, reaches the point {0,0,R}, is reflected back and received at the β's origin after t_β=2R/c interval of time.

However, an observer at α who analyses the events in β-frame thinks differently.
Since β-frame is moving relatively to α-frame, from an α-observer viewpoint the ray that should reach the mirror in β-frame and reflect back should go along trajectory as on the following picture

The α-observer concludes that, while the ray moves from β's origin at point X₀ in α-frame coordinates towards the mirror Q, the mirror moves further to the right in α-frame with speed v, and, while the reflected ray moves back to the β's origin, this origin moves even more to the right to point X₁ in α-frame coordinates.

Let's calculate the time T_β that should pass on α-observer's clock during the time the ray in β-frame moves to a mirror and back (index β signifies the frame where an event takes place, in this case we are talking about β-event).

The distance from X₀ to X₁ is the distance β-frame moves during time T_β on α-observer's clock with uniform speed v.
During the same time T_β the light with uniform speed c covers the distance from β-frame's origin at point X₀ in α-frame up to the mirror at point Q and back to β-frame's origin that by that time has moved with uniform speed v to point X₁ in α-frame.

Therefore,
X₀X₁ = v·T_β
X₀Q + QX₁ = c·T_β
PQ = R
X₀P = PX₁ = ½·v·T_β
X₀Q = QX₁ = ½·c·T_β
R² + (½·v·T_β)² = (½·c·T_β)²
4R² = −(v·T_β)² + (c·T_β)²

But the β-observer's clock would show that this same process took t_β local time, which is the time light covers the distance from point β{0,0,0} straight up to point β{0,0,R} and back, covering the distance of 2R with speed c.
Therefore,
2R = c·t_β

Comparing the above expressions, we come with the following
4R² = −(v·T_β)² + (c·T_β)² =
= (c·t_β)²

From this we can derive the relationship between time measurement T_β of a β-event in α-frame by a remote α-observer and time measurement t_β of the same β-event in β-frame by local β-observer.
−(v·T_β)² + (c·T_β)² = (c·t_β)²

From this follows
(1−(v²/c²))·T_β² = t_β²

Traditionally, physicists use symbol γ to express this relationships between the timing in both reference frames
γ = 1/√1−(v²/c²)

Then the above relationship takes the form of
T_β = γ·t_β

As you see, a remote evaluation of a time interval of an event is always greater than local.
This property of time is symmetrical.
The α-event takes longer, if evaluated from the β-frame than locally in the α-frame.
t_α = γ·T_α
In some way it's similar to a situation when two people look at each other from a distance. Each person sees the other smaller than that person really is.

This relationship between measuring a time interval of certain event by a local observer in some inertial frame and the time of the same event measured by an observer in another inertial frame moving relative to the first one is called time dilation.

TIME IS NOT ABSOLUTE