Unizor is a site where students can learn the high school math (and in the future some other subjects) in a thorough and rigorous way. It allows parents to enroll their children in educational programs and to control the learning process.
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·(TB−TA)² − (XB−XA)² −
− (YB−YA)² − (ZB−ZA)²
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.
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)²
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
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.
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
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 +
+ (YB−YA)² + (ZB−ZA)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 dβ²(A,B) = (xB−xA)² +
+ (yB−yA)² + (zB−zA)² =
= (XB−v·T−XA+v·T)² +
+ (YB−YA)² + (ZB−ZA)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)
yA = YA
zA = ZA
and xB = γ·(XB − v·TB)
yB = YB
zB = 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 +
+ (YB−YA)² + (ZB−ZA)²
dβ²(A,B) = (xB−xA)² +
+ (yB−yA)² + (zB−zA)²
As seen from the coordinate transformation above, xB−xA =
= γ·[(XB−v·TB) − (XA−v·TA)] =
= γ·(XB−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 +
+ (YB−YA)² + (ZB−ZA)² +
+ f(TB−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 +
+ (YB−YA)² + (ZB−ZA)² +
+ k·(TB−TA)²
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 =
(xB−xA)² + k(tB−tA)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. (XB−XA)² + k(TB−TA)² =
γ²·[(XB−v·TB) − (XA−v·TA)]² +
+ k·γ²·[(TB−v·XB/c²) −
− (TA−v·XA/c²)]²
Recall the metric in α-frame is Dα²(A,B) =
(XB−XA)² + k·(TB−TA)²
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) =
= (XB−XA)² − c²·(TB−TA)²
is equal to relativistic distance in α-frame Dβ²(A,B) =
= (xB−xA)² − c²·(tB−tA)²
and, therefore, is relativistic invariant.
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.
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
Using a traditional symbol γ = 1/√1−(v²/c²)
(which is greater than 1),
the length, as perceived by α-observer, is LαR = 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. .
