The x,y Minkowski diagram
The Minkowski diagram is a 4-diamentional diagram made up of six 2-D planes. The x,y Minkowski diagram is one of these planes. Normally when we show a single plane of the Minkowski diagam it is the x,t plane, which compares time axis with the x axis where the motion occurs. The x,y plane compares two spatial axes. Figure 1 shows the observer's x,y coordinates at time T = 0 and at T = 1. This is a square grid plane in blue lines. Superimposed over these is the moving object's coordinates in red lines. The two separate diagrams represent the coordinate systems at two different instants in time. The object is moving past the observer with a relative speed of 0.8c along the x-axis. Because of the object's relative movement, its coordinate system is contracted to (1-(v/c)2)1/2 = 0.6 its original length in the x direction. On the left we see the origin of the object's coordinate system overlays the origin of the observer's coordinate system. On the right we see that the object's coordinate system has moved to the right by 0.8 space units. One space unit is the distance that light would travel in one time unit. These drawings could also represent the x,z plane by changing y to z.
The Minsowski x,y ellipses
To study events with the x,y diagram we must take into account time. Since time does not appear on the x,y Minkowski diagram we must use several drawing or a composite drawing to illustrate events occurring at different times. Consider what happens when a light is emitted from a source, then travels through space the same distance in all directions and is then reflected back to the source. This allows us to compare time with the x,y plane, To visualize this examine number 1 of fig. 2. The circle, that is not moving relative to the observer, and has a radius of one SU (space unit). The time it would take for a light signal to leave a point in the center of the circle, travel to any point on the circumference (blue arrows), and be reflected back to the center point (red arrows), would always take 2 time units.
Now consider that same circle moving past the observer with a velocity of 0.8c. Because of the Lorentz transformations all the lengths in the circle in the direction of motion (the x direction) are contracted to 0.6 their original lengths. However, there is no linear contraction in lengths perpendicular to the direction of motion (the y direction). This causes the circle to become an ellipse. This ellipse is called the space ellipse, since this is the amount space occupied at one instant in time. This ellipse is shown in 2. of figure 2, at t=0 the instant a light is emitted from its center.
In 3. of fig. 2 we see that because of the contraction, the distance from the center of the space ellipse and the left side of the ellipse (point b) is (1-(v/c)2)1/2 = 0.6 space units. The left side of the ellipse is moving to the right with a speed of v = 0.8c, while the light emitted from the center of the ellipse with speed c. Thus the time for the light to reach the left side of the ellipse is t = (1-(v/c)2)1/2/((c+v)/c) = 0.6/1.8 = 1/3 TU (time unit). During this time the ellipse has moved t(v/c) = (1/3)(0.8) = 0.266... space units to the right. Also during this time the light has moved ct = 1(1/3) = 1/3 space units away from its point of origin in all directions. Since 1/3 + 0.266.. = 0.6 space units, the light signal has arrived at point b on the circumference of the space ellipse and starts to be reflected back toward the center.
In 4.of fig. 2 we see the time for the light to reach the top of the ellipse is t = 1/(1-(v/c)2)1/2 = 1.6667 time units and the space ellipse has moved t(v/c) = (1.66...)(0.8) = 1.33... space units to the right. During this time the light has moved ct = 1x1.66... = 1.66... space units away from its point of origin, in all directions. Here the light has reached point a on the circumference at the top (also the bottom) of the space ellipse and starts to be reflected back toward the center of the ellipse. The angle at which the light traveled to reach this point was Ψ = arcsin v/c.
In 5. of fig. 2 we see the time for the light to reach the right side of the ellipse is t = (1+(v/c)2)1/2 /(1-v/c) = 0.6/0.2 = 3 time units and the ellipse has moved t(v/c) = (3.0)(0.8) = 2.4 space units to the right. During this time the light has moved ct = 1x3.0 = 3.0 space units away from its point of origin in all directions. Here the light has reached point f on the circumference at the right side of the ellipse and starts to be reflected toward the center.
In 6. of fig. 2 we see the time for the light to travel from the center of the ellipse to the right side of the ellipse and back to the center is t = 3.0+0.333 time units. During this time the ellipse has moved t(v/c) = (3.33...)(0.8) = 2.66... space units to the right, while the light has moved ct = 1(3.33...) = 3.33... space units away from its point of origin in all directions. Here the all the light that had been reflected from the circumference of the ellipse has reached center point of the space ellipse at the same time.
The light was emitted from a point at the center of the moving space ellipse at an instant in time. The locus of points where the light reaches the circumference of the moving ellipse is itself an ellipse, stationary to the observer. This is called the time ellipse, where each point on its circumference occurs at a different time. The light was emitted at one of the foci of this ellipse. Then reflected from the circumference to the other focus of this ellipse at the instant when the center of the moving ellipse arrives at that point. This is illustrated in 6 of fig. 2. The path that light travels from one focus, to the circumference is a focal radius of the ellipse, The path that light travels from from the circumference to the other focus is also a focal radius of an ellipse. The ellipse is defined as the locus of points for which the sum of their distances (the focal radii) from two fixed points, the foci, is constant. This relationship can also be seen in the article on the Michelson-Morley experiment. The diagram can easily be expanded to 3 dimensions by considering the ellipses as ellipsoids. See fig. 3.
All of the distances in the x,y ellipses can be expressed as trigonometry functions
Fig. 4 illustrates the trigonometry functions for the angle ψ in a circle.
In fig. 5 the space ellipse has a relative speed of 0.8c to the observer. In the x,t Minkowski diagram the angle between the two time axes (t & t') is θ = arctan v/c. In the x,y diagram the angle between the y-axis and the path light must travel to reach a point at a distance of R, and perpendicular to the direction of motion is angle Ψ = arcsin v/c. All the distances in these diagrams can be expressed as trigonometry functions and they match the values of distances in the x,t Minkowski diagrams. Thus showing that these diagrams represent the Minkowski diagram for the x,y axes more effectively than the rectangular coordinates shown at the beginning of this article.
The dilated time T = TOγ , where γ = 1/(1-(v/c)2)1/2. The distances light must travel from a point to another point one unit away and perpendicular to their common motion and return to the first point (A1 + A2) is 2cT or 2sec Ψ. The distances light must travel from a point to another point one unit away along a line of their common motion and return to the first point (B1 +B2) is (cT + vT) + (cT - vT) or (sec Ψ + tan Ψ) + (sec Ψ - tan Ψ ). Both paths are the same distance, 2 sec Ψ = sec Ψ + tan Ψ + sec Ψ - tan Ψ .
Trigonometry functions used in the x,y elliptical Diagram
Angle θ is used for the hyperbolic functions in the x,t Minkowski diagram. θ = the angle between the observer's time axis t and the object's time axis t'. θ = arctan v/c
Angle Ψ is used for the trigonometry functions in the x,y Minkowski diagram. Ψ = the angle between the y-axis and the path light must travel to reach a point that is perpendicular to the direction of motion. Angle Ψ = arcsin tan θ = arcsin v/c
cos Ψ = spatial contraction = x = 1/γ = 1/cosh θ = sech θ [for v/c = 0.8, cos Ψ = 0.6 SU]
sin Ψ = relative velocity v/c = distance object will move in one observer's TU = (1-x 2) ½ = sinh θ /cosh θ = tanh θ [for v/c = 0.8, sin Ψ = 0.8 SU]
tan Ψ = distance object will move in one of object's time units = (1-x 2) ½/x = γv/c = tanh θ x cosh θ = sinh θ [for v/c = 0.8, tan y = 1.333SU]
cot Ψ = time it takes object to move one observer's space units = 1/tan Ψ = c/vγ = coth θ /cosh θ = csch θ [for v/c = 0.8, cot Ψ = 0.75TU]
csc Ψ = time for object length to pass observer = 1/sin Ψ = c/v = csc θ x coth θ [for v/c = 0.8, csc Ψ = 1.25 TU]
sec Ψ = distance light will move in one of object's time units = time dilation = γ = 1/cos Ψ = cosh θ [for v/c = 0.8, sec Ψ = 1.666 SU]
The circular and hyperbolic functions can be related by the Gudermannian function when the length of the sine of the circle is equal to the length of the tangent (tanh) of the hyperbola.
The hyperbolic angle theta is related to the circular angle psi by the relativity factor
The angle Ψ (psi) is said to be the gudermannian of angle θ (theta) when sin Ψ equals tan θ or tan Ψ = sinh θ. For the angle θ the hyperbolic function tanh θ is equal to the circular function tan θ. In fig. 7 the angle Ψ = arcsin v/c and the angle θ = arctan v/c.
If you apply the linear contraction to the tan of the gudermannian angle Ψ, it becomes the tan of the angle θ.
(1+v2/c2)1/2 tan Ψ = tan θ
When v/c = 0.6c The arcsin 0.6 = the gudermannian angle Ψ = 36.87O and the tan Ψ = tan 36.87O = 0.75. When v/c = 0.6c, the linear contraction = (1+v2/c2)1/2 = 0.8. Thus tan θ = tan Ψ(1+v2/c2)1/2 = 0.75(0.8) = 0.6 and tan θ = v/c = 0.6. Thus angle θ = 30.964O.
When v/c = 0.8c The arcsin 0.8 = the gudermannian angle Ψ = 53.13O and the tan Ψ = tan 53.13O = 1.333... When v/c = 0.8c, the linear contraction = (1+v2/c2)1/2 = 0.6. Thus tan θ = tan Ψ(1+v2/c2)1/2 = 1.333(0.6) = 0.8 and tan θ = v/c = 0.8. Thus angle θ = 38.6598O.
Space & time ellipses in two space and one time dimensions
Here we will examine the path of light in three dimensions.
Fig. 8 illustrates a circle with a radius of one SU, in the observer's system. That is the circle is not moving relative to the observer. A light is emitted from center of a circle at t=0 TU. This light travels (the blue arrows) out from the center in all directions. Here the arrows representing the paths of the light are only shown at each quadrant of the circle. At t=1TU the light reaches every point of the circumference of the circle (the green circle). Here the light is reflected from circumference back to center of a circle (the red arrows). All the reflected light arrives back at the center of the circle at t = 2TU.
In fig. 9 illustrates the same circle that is that is moving to the right at 0.8c relative to the observer. Because of its movement, the circle is contracted to an ellipse. This is the space ellipse. A light is emitted from center of this ellipse at t = 0 TU. This light travels (the blue arrows) out from the center in all directions. Since the circle is now an ellipse and it is moving, to the observer, the light will not arrive at its circumference all at the same time as in fig 8. The light will arrive at each different point of the circumference at different times. These points where the light meets the space ellipse's circumference forms a new ellipse stretched out through both space and time. This is called the time ellipse since each different point occurs at a different time. The point in space-time where the light was emitted form the center of the space ellipse is the first focus (F1) of the time ellipse. All the light reflected from the circumference will arrive at the same time at the center of the space ellipse when it is at the second focus (F2) of the time ellipse. The blue arrows show the light leaving the center at F1 and moving to the left till it strikes the circumference. Then reflected back to the center as it arrives at F2 (the red arrows).
We see the center of the spatial ellipse traveling along the object's time axes. However, the loci of the points of the time ellipse is traced out on the object's spatial plane when it located at one time unit t' from the object's spatial axis.
It is easy to see why the object's time axis t' is rotated to the observer's time axis t, because of the object's movement through time and space. This illustration helps to show why the object's spatial axis must be rotated by the same but opposite angle to the observer's space axis.
At the bottom in black we see all the images projected on to one of the observer's spatial plane.
The object's x',y' plane intersects the light cone at an angle
Here we will examine the paths of light in three dimensions, two spatial axes and one time axis, as related to the cone of light. If a light were emitted at the origin of the observer's x,y plane, then as this x.y plane moves through time, the light will travel across this plane as an expanding circle. If we observe this circle of light as it moves through time, it traces out a cone of light though time and space.
Fig. 10 shows the observer's xy spatial plane at the instant that t = 0. This plane is drawn in black lines. The vertical plane in blue lines is the observer's x,t plane at y = 1. This is the plane that the x,t Minkowski diagram is drawn on. The reddish plane is the object's x'y' plane at the instant when the object's time t' = 1. This plane is inclined to the observer's x,y plane by the angle θ = arctan v/c. This angle is due to the relative velocity of 0.6c between the observer and the object. We see that the intersection of the object's x',y' plane with the light cone emanating from the observer's origin is an ellipse. The t-axis is the path of the observer through time. The point P is a point on the object and the t'-axis is its path through time. The point P started at the object's coordinate origin 0' and moves with a speed of 0.6c, for one of its time units, to its present position. The path of point P is shown passing the observer at one space unit distance on the y-axis. The point's position on the hyperbola (on the x,y plane at y=1) is determined by its relative velocity (v/c) to the observer. The time it takes the point to move from t' = 0 to P (along line K) at a velocity of v/c, is the same amount of time it takes light to move from t = 0 (the vertex of the cone) to the point P.
At this instant the point P on the ellipse is the same point on the hyperbola. The point P is on the object's t'-axis and is at the end of the ellipse's minor axis on the circumference of the ellipse. The observer's t-axis passes though one of the ellipse's focus points at A. The angle of the object's plane containing the ellipse, to the observers x,y plane is θ = arctan v/c. Most of this can be more easily seen in the 2-dimensional three-view shown in figures 11, 12 and 13.
Three orthographic views of the light cone
The front view in fig. 11 shows the light cone and the x,t Minkowski diagram. The heavy red line at t'=1, parallel to the X'-axis, represents the edge view of the elliptical intersection of the light cone and the inclined x',y' plane. This x',y' plane is tangent to the hyperbola at point P and on the line t'=1. The length of this line is the major axis of the ellipse. This length is 2Sσ = twice S times the scale ratio σ. S is ½ the minor axis of the ellipse. The ratio σ is the unit scale in the object's system over the observer's unit scale
σ = ((γ)2 +(γ(v/c)) 2) 1/2 = ((1+(v/c) 2)/(1-(v/c) 2) 1/2 = 1.457738 when relative speed is 0.6c.
The ellipse is also shown in dotted lines rotated 90O around its major axis for reference. The minor axis of the ellipse is 2 units long. The distance of focus from center
f = σv/c = 1.457738(0.6) = 0.8746428
The left focus F1 of the ellipse is on the observer's time-axis and the center of the ellipse is on the object's time-axis.
The side view fig. 12 shows why, the position, where the hyperbola crosses the time-axis indicates the distance S of the object's closest approach to the observer.
In fig. 13, the top view, the ellipse is projected as it would appear on the observer's x,y plane. The length of the ellipse's major axis is 2γ. The ellipse's minor axis is 2 units in both inertial systems. The distance from either focus to the point P, through the observer's space, is γ = 1/(1-x2) ½ = sec Ψ = 1.25 space units. To the object the ellipse appears as a circle. In the object's system the left side of the ellipse is at x' = -1 and the right side is at x' = 1. Thus the major axis is 2 units in length; the same as the minor axis. This is the time ellipse. The spatial ellipse is not shown in these drawing.
To prove that the locus of points of light intercept with the circumference of moving ellipse is also an ellipse
Consider the rays of light emitted at one instant from the center of the ellipse of a moving object. Since the ellipse is moving through space the locus of points where the light rays intercept the circumference of the ellipse will be stretched out through both time and space. Figure 14 shows this locus of points as another ellipse. To see if this locus of points is really an ellipse we will use the computer program ellipse.
When this program is run, it will print the angle A for each increment of 10 degrees. DB is the distance light would travel from the center (F1) of the moving ellipse, to its circumference. DC is the distance for the light to return to the center at F2. The sum (DB+DC) is distance the light must travel for each increment of arc. When the program is run, the sum of DB+DC is a constant and it is equal to 2 /(1+v2/c2))1/2. Thus, illustrating that the locus of points where the light rays intercept the circumference of the moving ellipse is itself an ellipse. Since the definition of an ellipse is the set of all points P in a plane such that the sum of the distances of P from two fixed points F' and F (the two foci of the ellipse) is constant.***
***Analytic Geometry by Gordon Fuller
In this article we have seen the two system coordinate diagram that compares the x,y diagram with the x',y' diagram. We have seen how movement in the x',y' coordinate system causes a circle to become an ellipse. This is called the spatial ellipse. When rays of light emitted at one instant from the center of this spatial ellipse, the locus of points where the light rays intercept the circumference of the ellipse is itself an ellipse, the time ellipse. We have seen how the trigonometry measurements of the x,y ellipses equals the hyperbolic measurements of the Minkowski x,t diagram for the same velocity. We have seen the locus of points where the light cone intersects the x',y' plane is an ellipse.