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
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
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
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

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.


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.