## Monday, July 10, 2023

### 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 Tfront, the ball should cover a distance of L plus the distance a platform moves during time Tfront. All this time a ball moves with speed V+S.

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

Since at time Tfront a ball reaches the front edge of a platform, their X-coordinates are the same, which leads us an equation
(V+S)·Tfront = L + V·Tfront
Hence, Tfront = 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 Tback, the ball should cover a distance of L minus the distance a platform moves during time Tback. All this time a ball moves with speed V−S.

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

Since at time Tback a ball reaches the back edge of a platform, their X-coordinates are the same, which leads us an equation
(V−S)·Tback = −L + V·Tback
Hence, Tback = 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·Tfront = L + V·Tfront
−c·Tback = −L + V·Tback
from which follows
Tfront = L/(c−V)
Tback = 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

## Thursday, July 6, 2023

### 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(XA,YA,ZA) and B(XB,YB,ZB)
was equal to
d²(A,B) = (XB−XA)² +
+ (Y
B−YA)² + (ZBZA

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
dβ²(A,B) = (xB−xA)² +
+ (y
B−yA)² + (zBzA)² =
= (X
B−v·T−XA+v·T)² +
+ (Y
B−YA)² + (ZBZA)² =
= dα²(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
xA = γ·(XA − v·TA)
y
A = YA
z
A = ZA

and
xB = γ·(XB − v·TB)
y
B = YB
z
B = ZB

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}.

dα²(A,B) = (XB−XA)² +
+ (Y
B−YA)² + (ZBZA

dβ²(A,B) = (xB−xA)² +
+ (y
B−yA)² + (zBzA

As seen from the coordinate transformation above,
xB−xA =
= γ·
[(XB−v·TB) − (XA−v·TA)] =
= γ·(X
B−XA)−γ·v·(TB−TA)

yB−yA = YB−YA
zB−zA = ZB−ZA

Obviously, unless v=0 and, consequently, γ=1,
xB−xA ≠ XB−XA
and, as a result,
dα²(A,B) ≠ dβ²(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) = (XB−XA)² +
+ (Y
B−YA)² + (ZB−ZA)² +
+ f(T
B−TA)

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
Dα²(A,B) = (XB−XA)² +
+ (Y
B−YA)² + (ZB−ZA)² +
+ k·(T
B−TA

To find unknown coefficient k (if possible, of course), we equate
Dα²(A,B) = Dβ²(A,B)

Considering only X-coordinate and time change from α-frame to β-frame, our equation for k is
(XB−XA)² + k(TB−TA)² =
(x
B−xA)² + k(tB−tA

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.
(XB−XA)² + k(TB−TA)² =
γ²·
[(XB−v·TB) − (XA−v·TA)]² +
+ k·γ²·
[(TB−v·XB/c²) −
− (T
A−v·XA/c²)
]²

Let's simplify the right side of this equation.

It's equal to

γ²·[(XB−XA) − v·(TB−TA)]² +
+ k·γ²·
[(TB−TA) −
− (v/c²)·(X
B−XA)
]² =

= γ²·
[(XB−XA)² −
− 2v·(X
B−XA)·(TB−TA) +
+ v²·(T
B−TA)² +
+ k·(T
B−TA)² −
− 2(k·v/c²)·(X
B−XA)·(TB−TA) +
+ k·(v/c²)²·(X
B−XA
] =

= γ²·
[(1+k·(v/c²)²)(XB−XA
− 2(v+k·v/c²)·(X
B−XA)·(TB−TA)
+ (v²+k)·(T
B−TA
] = Dα²(A,B)

Recall the metric in α-frame is
Dα²(A,B) =
(X
B−XA)² + k·(TB−TA

To satisfy the equation
Dα²(A,B) = Dβ²(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
Dα²(A,B) =
= (X
B−XA)² − c²·(TB−TA

is equal to relativistic distance in α-frame
Dβ²(A,B) =
= (x
B−xA)² − c²·(tB−tA

and, therefore, is relativistic invariant.

## Tuesday, July 4, 2023

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

Notes to a video lecture on UNIZOR.COM

In classic Mechanics that uses Galilean transformation, if an object moves in reference frame β with velocity uβ, 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:
uα = uβ + 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 uβ.
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 uα, 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) = uβ·t
Obviously,
dx(t)/dt = uβ.

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 uα=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 uα as
uα = dX/dT = (dX/dt)/(dT/dt)
and we can easily resolve our problem as follows
dX/dt = (v+uβ )/1−(v/c)²
dT/dt = (1+v·uβ /c²)/1−(v/c)²
dX/dT = (v+uβ )/(1+v·uβ /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 uβ 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.

## Monday, July 3, 2023

### 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
YR = yR = 0
ZR = zR = 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 TR in α-frame. In β-frame its x-coordinate is still −R. The timing tR of this event in β-frame is the time β-frame moves along X-axis by the length R of the rod, that is tR=R/v.

Let's summarize information about event (b):
XR = 0
TB is unknown
x = −R
tR = 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 LαR of a rod moving across with speed v, as perceived by α-observer:
LαR = v·TR = R·√1−(v/c)²