## Sunday, September 24, 2023

### Relativistic Momentum: UNIZOR.COM - Relativity 4 All - Conservation

Notes to a video lecture on UNIZOR.COM

Relativistic Momentum

One of the fundamental laws of Newtonian mechanics is the Second Law describing the relationship between a vector of force F, inertial mass m and a vector of acceleration a
F = m·a
where inertial mass m is considered a constant characteristic of an object independent of time, space or motion.

Since a vector of acceleration a, by definition, is the first derivative of a vector of speed v by time t, the same law can be written as
F = m·dv/dt = d(m·v)/dt

The product of mass and a vector of speed is a vector of momentum of motion p=m·v, so the same Second Law can be written as
F = dp/dt

The Conservation Law of Momentum, based on uniformity of space, states that the conservation of momentum is universal and should be maintained through transformation from one inertial frame to another (see the previous lecture Noether Theorem in this course).
That's why it makes sense to consider the latter expression of the Newton's Second Law in terms of a derivative of the momentum of motion as the most appropriate form.

It makes sense to discuss the physical meaning of a momentum and its expression as a product of mass and velocity.
As seen from the last formula, a force is a rate of change of a momentum. The greater the change of momentum - the greater force that caused its change.

From another perspective, the momentum can be viewed as a degree of resistance to the change of motion.

Consider an object uniformly moving along a straight line and some external force that tries to change this motion.

It is intuitively acceptable that an object with greater mass resists the changes to its movement stronger and, therefore, requires stronger force to achieve similar change in motion.
Similarly, the faster an object moves - the more difficult to change its trajectory, and we can reasonably assume that an object with higher speed of movement requires stronger force to changes its movement.

The purpose of this lecture is to analyze the transformation of momentum from one inertial frame to another using the Law of Conservation of Momentum as a tool.

We do know how relativistic speed is transformed (see lectures Einstein View - Adding X-Velocitis and Einstein View - Adding Y-,Z-Velocitis in this course).
The Law of Conservation of Momentum will help us to determine how an object's momentum changes when viewed from different reference frames.

Let's analyze the ideal elastic collision of two identical spherical objects as shown on the following pictures describing this collision in the reference frame β{x,y,z,t} at rest relative to the point of collision (the origin of coordinates) and positioned symmetrically to the objects.  Assume, the spherical blue and green objects are uniformly moving on parallel trajectories in opposite directions with the same by magnitude velocity u={±v,±w}, so their ideally elastic collision occurs at a point on the x-axis when both spheres are tangent to this axis.
This assures the mirror-like trajectories before and after the collision.

Summary of x,y-components (v,w) of velocity vectors u for both objects before and after the collision can be presented in the following table

 Object Before (x,y) After (x,y) Green (−v,−w) (−v,w) Blue (v,w) (v,−w)

Since in the β-frame both objects have the same mass and speed magnitude and move in the opposite directions before and after the collision, the total momentum of both should be zero before and after the collision. The Law of Conservation of Momentum is held.

Obviously, the Law of Conservation of Momentum as a vector must be held separately for its X- and Y-components.

Let's now switch to another reference frame and apply the Law of Conservation of total Momentum of both objects.

Consider an inertial reference frame α{X,Y,Z,T}.
Assume that β-frame {x,y,z,t} is uniformly moving along the X-axis of this α-frame with some constant speed s maintaining parallelism and directionality of all axes, and at some initial time both frames coincide.

Recall the laws of relativistic addition of velocities presented earlier in the lecture Adding Y-, Z-Velocities within Einstein View part of this course
uαx =
 s + uβx 1 + s·uβx /c²
uαy =
 uβy·√1−(s/c)² 1+s·uβx /c²

Let's calculate the X- and Y-components of both green and blue objects in α-frame before and after the collision.

1. The horizontal x-component of the green object's velocity in β-frame is, as we stated above, uβx=−v before and after the collision.
Therefore, in α-frame before and after the collision its X-component is
uαx =
 s + (−v) 1 + s·(−v)/c²

2. The value of vertical y-component of velocity of a green object in β-frame is, as we stated above, −w before the collision and w after the collision.
Therefore, before the collision in α-frame, substituting uβy=−w and uβx=−v, the vertical component of the green object is
uαy =
 (−w)·√1−s²/c² 1 + s·(−v)/c²

3. After the collision in α-frame, substituting uβy=w and uβx=−v, the vertical component of the green object is
uαy =
 w·√1−s²/c² 1 + s·(−v)/c²

4. Let's examine the blue object now.
In the original β-frame its horizontal speed along x-axis is v before and after the collision, while its vertical speed along y-axis is w before and −w after the collision.

Using the same formulas of adding velocities, the horizontal speed of the blue object in α-frame before and after the collision is
uαx =
 s + v 1 + s·v/c²

5. The value of vertical component of velocity of a blue object in β-frame is, as we stated above, w before the collision and −w after the collision.

Therefore, before the collision in α-frame, substituting uβy=w, the vertical component of the blue object is
uαy =
 w·√1−s²/c² 1 + s·v/c²

6. After the collision in α-frame, substituting uβy=−w the vertical component of the blue object is
uαy =
 −w·√1−s²/c² 1 + s·v/c²

Summary of velocities in α-frame:
Object Speed Value
Green X before
 s−v 1 − s·v/c²
Blue X before
 s+v 1 + s·v/c²
Green X after
 s−v 1 − s·v/c²
Blue X after
 s+v 1 + s·v/c²
Green Y before
 −w·√1−s²/c² 1 − s·v/c²
Blue Y before
 w·√1−s²/c² 1 + s·v/c²
Green Y after
 w·√1−s²/c² 1 − s·v/c²
Blue Y after
 −w·√1−s²/c² 1 + s·v/c²

Not surprisingly, the lengths² of velocity vectors, being the same in the β-frame for both objects lβ² = v² + w², is different in the α-frame.
Green object's velocity vector length² in α-frame (lα²) is
[(s−v)²+w²(1-s²/c²)] /(1−s·v/c²)²

Blue object's velocity vector length² in α-frame (lα²) is
[(s+v)²+w²(1-s²/c²)] /(1+s·v/c²)²

Newtonian momentum of an object of mass m moving in two-dimensional Euclidean inertial reference frame with some velocity vector u={v,w} is a vector u={m·v,m·w}.
The rules of addition of velocities when observing the movement from another inertial reference frame are simple vector addition derived from Galilean transformation of coordinates.

Since X-component of each object's velocity before and after a collision in α-frame are the same, the X-component of the total momentum of both objects, calculated according to classical Newtonian definition as a product of an object's mass by its velocity vector, would obviously be the same before and after the collision.

The situation is completely different along Y-axis.
If we try to check the law of conservation of momentum calculated according to classical Newtonian definition as mass times speed along each axis, we obtain the following results.

Before a collision the total Newtonian momentum of both green and blue objects along Y-axis is
 −m·w·√1−s²/c² 1 − s·v/c²
+
 m·w·√1−s²/c² 1 + s·v/c²

After a collision the total Newtonian momentum of both green and blue objects along Y-axis is
 m·w·√1−s²/c² 1 − s·v/c²
+
 −m·w·√1−s²/c² 1 + s·v/c²

The only cases when they are equal is if s·v=0 or w=0, that is either s=0, which means β-frame is not moving relative to α-frame or v=0, that is no horizontal movement, or w=0, that is no vertical movement.

Classical Newtonian definition of a momentum is not working in relativistic mechanics in general cases..
To address this problem, the relativistic momentum is defined differently than the classical Newtonian one.

Relativistic momentum of an object is its Newtonian-like momentum (product of mass and velocity vector) multiplied by factor γ=1/1 − l²/c², where l is the magnitude (length) of a velocity vector.

Modified this way, the relativistic momentum is preserved in the collision we use as an example.
The step-by-step proof of the Law of Conservation of Relativistic Momentum can be viewed on page Proof of Conservation of Relativistic Momentum (there are a lot of formulas there, so open it in a new tab for clarity by right-click and choosing to open in a new tab).

Let's summarize our results.
Assume, in our reference frame two objects of mass m each are moving towards each other on a collision course. Their velocities have the same magnitude u and we assume ideal elastic collision between them.
We have proven the law of conservation of their total relativistic momentum in the course of their collision, where the relativistic momentum of each is calculated according to this formula
p = γ·m·|u| =
 m·|u| √1−|u|²/c²

The interpretation of the denominator in the expression for relativistic momentum is a matter of opinion.
Some physicists prefer considering the mass of an object as changing with the speed and call the expression
γ·m = m/√1−|u|²/c²
a relativistic mass.

Some others prefer to relate the factor γ as applicable to an entire relativistic momentum, leaving mass unchanged and calling it a rest mass, emphasizing this by using a subscripted version m0
p = γ·m0·|u| =
 m0·|u| √1−|u|²/c²

## Tuesday, September 12, 2023

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

Notes to a video lecture on UNIZOR.COM

along Y-, Z-axis

Let's complete our formulas for adding velocities with those related to Y- and Z-axis, while assuming that the movement of one reference frame relatively to another is along the X-axis.

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 some direction with X-, Y- and Z-components of its velocity vector uβx, uβy and uβz correspondingly.
At time t=0 this object is at the origin of β-frame.

Our task is to determine the components uαx, uαy and uαz of the velocity of this object in α-frame, using Lorentz Transformation.

In the previous lecture we discussed the analogous task but restricted only to an object moving in the β-frame along X-axis.
This lecture is about a more general movement of the object along any direction.
Obviously, the formula derived in the previous lecture for a movement along X-axis is fully applicable to the X-component of the general velocity vector, and out task is restricted to add Y- and Z-components of the velocity vector.

We will use exactly the same methodology for Y- and Z-components as we used for X-component.

The object in β-frame moves according to this formulas for its coordinates
x(t) = uβx·t
y(t) = uβy·t
z(t) = uβz·t

Obviously,
dx(t)/dt = uβx.
dy(t)/dt = uβy.
dz(t)/dt = uβz.

According to Lorentz transformation,
T =
 t + v·x/c² √1−(v/c)²
X =
 x + v·t √1−(v/c)²
Y = y
Z = z

Our task is to find all the components of the velocity of an object in α-frame, that is to find
uαx=dX/dT
uαy=dY/dT
uαz=dZ/dT

All calculations we did in the previous lecture, when motion of an abject was restricted to X-direction, are good for getting uαx, but we will repeat them here.

Using an explicit form of functions x(t), y(t) and z(t), the above formulas of Lorentz Transformation are
T =
 t + v·uβx·t/c² √1−(v/c)²
X =
 uβx·t + v·t √1−(v/c)²
Y = uβy·t
Z = uβz·t

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

The final formula for X-component of the object's velocity uαx in α-frame is

uαx =
 v + uβx 1 + v·uβx /c²

Let's analyze the Y-component of the object's velocity.
Since
dY/dt = (dY/dT)·(dT/dt),
we can express the uαy=dY/dT as
uαy = dY/dT = (dY/dt)/(dT/dt)
and we can easily resolve our problem as follows
dY/dt = dy/dt = uβy
As before,
dT/dt = (1+v·uβx /c²)/1−(v/c)²
Therefore,

uαy =
 uβy·√1−(v/c)² (1+v·uβx /c²)

In particular, if an object moves along y-axis in β-frame (uβx=0), its Y-axis speed in α-frame is
uαy = uβy·√1−(v/c)²

The Z-component of the velocity is analogous to Y-component.
Since
dZ/dt = (dZ/dT)·(dT/dt),
we can express the uαz=dZ/dT as
uαz = dZ/dT = (dZ/dt)/(dT/dt)
and we can easily resolve our problem as follows
dZ/dt = dz/dt = uβz
As before,
dT/dt = (1+v·uβx /c²)/1−(v/c)²
Therefore,

uαz =
 uβz·√1−(v/c)² (1+v·uβx /c²)

In particular, if an object moves along z-axis in β-frame (uβx=0), its Z-axis speed in α-frame is
uαz = uβz·√1−(v/c)²

## Friday, September 8, 2023

### Conservation Laws and Noether Theorem: UNIZOR.COM - Relativity 4 All - C...

Notes to a video lecture on UNIZOR.COM

Conservation Laws
and Noether Theorem

In this course we are trying to support every formula or a statement with a relatively rigorous proof, based on earlier proven properties and formulas.

This particular lecture is about an issue, which, on one hand, is extremely important for the theory, but, on the other hand, involves a significant mathematical effort that goes beyond the scope of this course.

So, the following is a theoretical item (Noether's Theorem) presented without a proof.

The Noether's Theorem was published in 1918 by Emmy Noether, a brilliant German mathematician, who was regarded by such scientists as Albert Einstein as the most important woman in the history of mathematics.
Her main contribution to Physics was to validate the only experimentally confirmed laws of conservation by a theoretical proof based on much more fundamental properties of the Universe.

We wish every statement to be based on some earlier proven statement or a theorem. Analyzing these earlier statements, we find them to be based on even earlier ones etc.

Inevitably, we would come to a statement that we have to take as an axiom without a proof.
Obviously, we would like to accept as axioms such statements that correspond to our general feelings about the Universe, seem to us as natural and not contradicting our intuition.

The laws of conservation of energy, linear momentum or angular momentum were widely accepted by physicists for centuries based on experimental data.
However, there was always a doubt whether these laws were indeed the universal laws of nature or just our not always perfect results of experiments.
The Nouther's Theorem actually stated that these laws of conservation are intimately related to properties of our space and time.
In particular, they are consequences of uniformity of our space and time.

Emmy Nouther took as an axiom some quite fundamental and well accepted statements that our time is uniform and totally symmetrical, our space is uniform and symmetrical, one direction in space is no different than another.
Based just on this uniformity and symmetry, she has proven the validity of our laws of conservation.

More precisely,
(a) if we accept the uniformity of time, the law of conservation of energy follows as a mathematical consequence;
(b) if we accept the linear uniformity of space, the law of conservation of linear momentum follows as a mathematical consequence;
(c) if we accept the directional uniformity of space, the law of conservation of angular momentum follows as a mathematical consequence.

There is no doubts that it's significantly easier and much more natural to accept the uniformity of time as an axiom than the law of conservation of energy, and, similarly, the rest of the prepositions of the Nouther's Theorem.
That's why this theoretical result is as fundamentally important for Physics as axioms of Euclid for Geometry.

## Saturday, August 19, 2023

### Past and Future in Minkowski Space+Time: UNIZOR.COM - Relativity 4 All -...

Notes to a video lecture on UNIZOR.COM

Past and Future
in Minkowski Space+Time

We continue analyzing the movements restricted to a two-dimensional XY-plane for visual representation of the three-dimensional Minkowski XYT-coordinates in space+time.

Assume, a point at present time t=0 is located at the origin of XY-plane.
As it moves along any trajectory (X(T),Y(T)) on a plane, the world point that represents this movement will be at position (X(T),Y(T),T) in the three-dimensional Minkowski space+time.

The distance from the world point (X,Y,T) to the T-axis equals to
D(T) = √X²(T)+Y²(T)

As we know, the speed of movement cannot exceed a speed of light in vacuum c.
Therefore, during time T the point on the XY-plane cannot deviate in its movement from the origin by a distance greater than c·T and the world point (X,Y,T) cannot deviate from the T-axis by the same distance c·T.

From this follows that for any time T the following inequality is held
d(T) ≤ c·T or
X²(T) + Y²(T) ≤ (c·T)²

The equation
X² + Y² = (c·T)²
represents a conical surface in three-dimensional XYT-space.
Therefore, the inequality
X²(T) + Y²(T) ≤ (c·T)²
represents the inside of this cone.

Below is a graphical representation of this with indication that if our present time is T=0, the inside of an upper cone represents all possible points reachable in the future, while the inside of a lower cone represents all possible points in the past where the moving point could have been located in the past. If a point in the past (for negative time T) was located outside the lower cone, there is no way it could reach the origin of coordinates at current time T=0, because its speed would have to exceed the speed of light c.

Analogously, any point in the future (for positive time T) outside of an upper cone could not be reached from the origin during the time T, because its speed would have to be greater than the speed of light c.

Therefore, we can state that all world lines that represent all the past and future possible movements of an object located at the origin of coordinates at time T=0 must lie inside the cone described by an equation
X² + Y² = (c·T)²
This cone is called the light cone because its shape (actually, the angle at the top) explicitly depends on the speed of light c.

Of course, any position on a world line can be interpreted as the beginning of a motion with all prior positions along a trajectory considered as the past and all subsequent as the future.
In this case we can require that at any position the world line should be contained within a light cone constructed at it, which can be schematically illustrated by this picture. Extending the definition of the light cone from three-dimensional Minkowski XYT-frame (that we used for better graphical representation) to a practical four-dimensional Minkowski XYZT-frame (that, unfortunately, we cannot visually represent), we can define a similar light cone as
X² + Y² + Z² = (c·T)²

Now let's express the same thing in term of metrics, assuming the distance between two world points in four-dimensional Minkowski XYZT-frame, presented in the previous lecture Space+Time, is expressed as
D²(A,B) =
= c²·(T
B−TA)² − (XB−XA)² −
− (Y
B−YA)² − (ZB−ZA

The condition for two world points to be inside of a light cone (that is, for them to represent the state of some object at two different moments of time, thus belonging to its world line) can now be expressed as the square of a distance between them in Minkowski space+time, as defined in the above formula, to be non-negative, which makes the distance itself to be a positive real number.

If the square of a distance between two points in the Minkowski space+time is negative and, consequently, the distance itself is imaginary complex number, these two points cannot belong to a world line that represents a motion of one point-object, because, to reach one point from another, this object would have to move with a speed faster than a speed of light in vacuum, which is impossible.

## Tuesday, August 15, 2023

### Minkowski Space+Time: UNIZOR.COM - Relativity 4 All - Minkowski View

Notes to a video lecture on UNIZOR.COM

Minkowski Space+Time

Imagine a point moving in a circle on a two-dimensional XY-coordinate plane.
As it moves, we can register its position on this plane and obtain its trajectory. Does this trajectory fully represent graphically the motion of a point?
The answer is NO, because this trajectory does not reflect the time when our point visited this or that location on a circle.
The point might uniformly rotate along this circle, it might stop every once in awhile or even start rotating in the opposite direction. Trajectory will be the same - a circle.

To better represent the motion, let's add a third dimension - the time axis T.
Now, if the point occurs at coordinates (x,y) on a plane at time t, we will represent it as a point with coordinates (x,y,t) in our three-dimensional XYT-coordinate space.
This point with added time coordinate is called the world point.

Assume, a point uniformly rotates along a circle with a constant angular speed. Then its three-dimensional XYT-trajectory will look like a helix. The above picture graphically fully represents the motion - uniform rotation of a point in a circle on a plane.
The line our world point is moving along is called the world line that fully represents the motion.

Granted, we live in a three-dimensional world and our movements are in three-dimensional XYZ-space.
So, to fully represent the motion, we have to add the fourth dimension - time.
The problem is, it's practically impossible to graphically present a four-dimensional space on a paper or a screen, or a whiteboard.

So, to preserve the clarity of the graphical representation of a motion, we will continue to use examples of motion on a plane, adding the time as the third dimension.
It should be assumed, however, that when we use XYT-coordinates, we imply the analogous usage of XYZT-coordinates.

Examples of World Lines

1. Being at rest at a fixed point on XY-plane

Being at rest can be described by equations
x(t) = x0
y(t) = y0
where
x0 is the X-coordinate of a location
y0 is the Y-coordinate of a location
Therefore, as time t increases, the world line, starting at the initial point on XY-plane, goes vertically retaining X- and Y-coordinates.
The projection of this world line onto XY-plane is a point with XY-coordinates (x0,y0). 2. Uniform movement along a straight line on an XY-plane

Since a point is uniformly moving on a plane along a straight line, it's X- and Y-coordinates are proportional to time
x(t) = x0 + a·t
y(t) = y0 + b·t
where
x0=x(0) is the starting X-coordinate at time t=0
y0=y(0) is the starting Y-coordinate at time t=0
a and b are real constants.

Therefore, if (x,y,t) is any world point that represent a position of a point on a plane (x,y) at time t, the following equations must be true:
x = x0 + a·t
y = y0 + b·t
t = t0 + c·t (where t0=0, c=1)
which is a classical representation of a straight line in 3D passing through a point (x0,y0,0) with X-, Y- and T-coordinates growth ratio, correspondingly, a, b and 1.

The projection of this world line onto XY-plane is a trajectory of a point on a plane - a straight line expressed by an equation
(x−x0)/a = (y−y0)/b 3. Straight line movement with constant acceleration on an XY-plane

Let's assume for simplicity that a point on an XY-plane starts movement at time t=0 at the origin of coordinates and moves along an X-axis with constant acceleration a.

Then, according to familiar rules of Newtonian mechanics, we can write
x(t) = a·t²/2
y(t) = 0

The world line of this motion lies in the plane y=0 and is a parabola. Distance in Space-Time

For better illustration, instead of considering our three-dimensional space and one-dimensional time, we continue restricting our analysis to two-dimensional space + one-dimensional time coordinates to be able to graphically represent the resulting three-dimensional space+time continuum.

In addition, to use the same units of measurement (length units, like meters) along all three XYT-dimensions, instead of just time t on the T-axis, we will use a scale factor - speed of light in vacuum c, so the measurement along the T-axis will be c·t in length units (like meters).
Since speed of light in vacuum c is a universal constant in all inertial reference frames, this does not change the character of the curves we construct.

One of the most important characteristic of any space is the method of calculating the distance between two points.
Since we talk about real physical space+time continuum, the definition of a distance between two points A and B should give the same result if measured in any inertial system of coordinates introduced in our space+time.
In other words, the distance between two points in space+time must be an invariant relative to coordinate transformation from one inertial system to another.

In the previous parts of this course, Galilean View and Einstein View, we have determined that the Principle of Relativity and constancy of the speed of light in vacuum in any inertial frame of reference dictate a new approach to transformation of coordinates from one inertial reference frame to another.
In particular, the proper transformation of coordinates is so-called Lorentz transformation that takes into account time dilation and changes in length of moving objects.

The old Euclidean definition of a distance d(A,B) between two points A and B is not an invariant relative to Lorentz transformation (see the lectures Non-Invariance in the Galilean View part and Lorentz Transformation in the Einstein View part of this course).

At the same time, in the lecture Relativity Metric of Einstein View part of this course we have introduce a new expression that is invariant relative to Lorentz transformation:
d²(A,B) = (XB−XA)² + (YB−YA)² + (ZB−ZA)² − c²·(TB−TA

In 1907 Hermann Minkowski, German physicist and mathematician, proposed to represent the world we live in as a four-dimensional space+time continuum (three space coordinates + one-dimensional time) and use the expression above as a definition of a distance between two world points in this Minkowski space+time.

Thus defined, the distance between two world points is an invariant relative to Lorentz transformation.

Note that many other definitions of invariant metric in Minkowski space+time can be derived by applying any function to an expression for d²(A,B) above.

For example, to have the value of the Minkowski distance between two points along a world line as a positive value (see the next lecture Past and Future), many physicists use the negative of this expression as a measure of a distance between two world points

D²(A,B) = c²·(TB−TA)² − (XB−XA)² − (YB−YA)² − (ZB−ZA

## Monday, August 7, 2023

### Is Time Absolute: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Is Time Absolute?

Since we discuss the Relativity in this course, let's discuss what exactly the term relativity means.

It is easy to see that a concept of a position is relative.
Consider a traveler on a cruise ship around the world who every day takes tea in the tea room at 5 o'clock.
From his perspective he takes his tea at the same place - in the tea room. But from a viewpoint of a person who lives in London the places are different because a cruise ship is moving along its course.

Another easy concept is a relativity of a movement. If an object A is attached to the ground, it's not absolutely at rest, as some observers might say, because the Earth is rotating around its own axis and around Sun.

If object A is positioned somewhere in space and object B is passing it, an observer at A might say that he is at rest and B is moving, while an observer at B can say that he is at rest and object A is moving.

Isaac Newton has built his theory of Mechanics with these principles in mind.
At the same time, to put his theory on a solid mathematical base he needed such entities as trajectory, speed, acceleration etc. - all those where we need to use time as a major component in calculations.

The time was understood then as something absolute, independent of position or motion, and the only progress in its understanding was to be able to measure it as precisely as possible.
Everybody intuitively understood that...
...except Einstein.

First of all, Einstein accepted as postulates two main principles developed by physicists up to 19th century:

Principle of Relativity which states that all physical processes, laws and equations must be identical in all inertial reference frames uniformly moving relative to each other
and
Principle of constancy of the speed of light in all inertial reference frames independently of their relative movements.
In other interpretation, the speed of light is independent of the speed of its source. It depends only on the properties of medium through which light propagates, like vacuum, air, glass etc.

Based on these principles, common sense logic and simple mathematics Einstein came to a conclusion that, if we continue considering time as absolute and identical in all reference frames, the second principle of constancy of the speed of light contradicts the Principle of Relativity.

This topic was discussed in the previous lecture Postulates of Theory of Relativity.
Here we will try to dig a little deeper into what time is.

Consider the following example.

A spaceship is flying with a speed of v=150,000,000 meters per second (m/s) away from the Earth.
Speed of light in the air inside a spaceship is c=300,000,000 m/s.

In the middle of a cabin of this spaceship we place a source of light capable to produce a short flash in all directions.
On two walls of this cabin, the front and the back, we install sensors that register the time the light reaches the wall. The distance from the center of a cabin to either wall is L=90 meters (m).

We do assume that all clocks are absolutely synchronous and show the same time on a spaceship.

According to the Principle of Relativity, an observer inside a spaceship should have no difference in timing of a ray of light reaching the front or the back of the spaceship walls, whether a spaceship is moving or not, and this time to the front wall Tfront should be the same as to the back Tback and equal to the length covered by a ray of light (90m) divided by a speed of light (300,000,000m/s), which is equal to 0.3 microsecond (μs). Both rays, issued towards the front and to the back, should reach the sensors simultaneously.
Tfront = Tback = 0.3μs

Observer on Earth sees it differently.
In his view the front of a spaceship moves away from the source of light, while the back of a spaceship moves toward it.

Therefore, by the time Tfront needed for light to reach the front wall, this front wall will cover the extra distance
Dfront = v·Tfront
During this same time Tfront the ray of light has covered the initial distance of L=90(m) to the wall plus extra distance Dfront that wall moved away from it because a spaceship moves, which brings us to an equation
c·Tfront = L + Dfront or
c·Tfront = L + v·Tfront or
Tfront = L/(c−v) =
= 90/150,000,000
(s) = 0.6μs

Analogously,
Dback = V·Tback
c·Tback = L − Dback or
c·Tback = L − v·Tfront or
Tfront = L/(c+v) =
= 90/450,000,000
(s) = 0.2μs

Times for light to reach the front and the back walls of a cabin are different
Tfront = 0.6μs
Tback = 0.2μs
The same two events, light hitting the front and the back walls of a cabin in a spaceship, that seemed simultaneous for an observer inside a spaceship, are not simultaneous from an Earth observer's viewpoint.

Let's make a small modification in our experiment and change the functionality of a sensor attached to the back wall of a cabin in a spaceship.
The time it's supposed to register will be registered with a delay of 0.1μs.
An observer on a spaceship will see that the timing for the light to reach the front wall is still 0.3μs, but the timing for the back wall will be greater by 0.1μs, that is
Tfront = 0.3μs
Tback = 0.4μs
The back wall sensor will show greater time than the front wall one.

Let's see it from the Earth observer's view point.
The front wall sensor functions as before,
Tfront = 0.6μs
The same delay in the back wall sensor will register time by 0.1μs greater than before, that is
Tback = 0.3μs

We have an interesting situation.
The same two processes happening on a spaceship (the light reaching the front and the back walls) take different time with one (the light reaching the back wall) taking longer (Tback=0.4μs) than another (Tfront=0.3μs) from the spaceship observer, but the opposite inequality (Tback=0.3μs is shorter than Tfront=0.6μs) between timing is observed from the Earth.
What is a longer process for a spaceship is taking shorter time for an observer on Earth.

Summary

Time is relative.
One observer can see two events as simultaneous, while another might see them as occurring at different times.
A process can take one time from one observer's viewpoint, but another observer might see it as taking longer or shorter time.
If one even precedes another from one perspective, from another perspective might be seen as occurring in reverse order.
TIME IS RELATIVE, as everything else in our Universe.

## 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