- Education and Science
What could cause the effects of Relativity
Let us begin with a brief review of some of the effects of relativity illustrated in figure 1. On the left we see two astronauts P and Q. Both are in identical spacecrafts. Here Q sees P passing him to the right at a speed of 0.8 times the speed of light. He sees P's spacecraft contracted. He sees that P's time is moving slower than his own time. He sees that P's clocks are not synchronized.
On the right P sees Q passing him to the left at a constant speed of 0.8c. He sees the same spatial contraction and time dilation in Q's spacecraft as Q saw in his.
We calculate this using the Lorentz transformations, but what could physically cause space to contract and time to dilate? Is there anything that can be used as an example to physically illustrate these effects?
All of the information an observer receives from an object is transmitted to him by electromagnetic waves. The standing electromagnetic wave can be used as an analogy to show how these effects might occur. In fig 2 we see an astronaut that is stationary relative to the observer. He is transmitting an electromagnetic wave into a box. The red line represents the varying electric fields and the blue line represents the varying magnetic fields
In fig 3, to make it easier to visualize, we will only examine the varying electric fields. The astronaut transmits wave (A), to a mirror at one end of the box. The wave (B), is the wave reflected back to the astronaut. These two waves add together to produce a resulting wave (C). This resulting wave is a standing wave. Even though all three waves occupy the same space, in the illustration they are drawn as separate lines for ease in visualization.
On each of the waves, the curve above the black center line is positive. The curve below the black center line is negative. The light blue circles are zero field points, neither positive nor negative. These are points where the waves A and B continually cancel each other. These points are nodes, where the electric field remains zero. These nodes remain stationary relative to the box. The distance between every other node is one wavelength.
In fig. 4 we see one Complete cycle of the Electric Fields is one time period. We can see that the waves in the box at T = 0 and T = 1 are identical.
In figure 5 shows the paths of the zero field points, trace out a rectangular coordinate system through time and space. Lines 1, 2 & 3 are the loci of the nodes. Lines 4, 5 & 6 are the loci of the points where the two waves cancel each other completely.
In figure 6 the system containing the standing electromagnetic wave is moving past the observer with a constant speed of 0.6c. Because of this we see the length of the box reduced to 0.8 its original length. The movement causes the waves to be affected by the Doppler Effect. The observer would see the wavelength of wave A, the wave transmitted by the astronaut, as compressed to one half the length of the original wavelength.
In figure 7 the observer would see the wavelength, of wave B, the reflected wave, as stretched out to twice the length of the original wavelength.
In figure 8 we see the astronaut and waves at three different time increments. The time increments are indicated in the observer's one quarter time units. The waves are drawn extended across the page and independent of the box. This is to see how the resulting ZFP's line up with the moving box. An asterisk is placed on each wave to indicate its velocity c. The Doppler affected waves combine to form a new resulting wave. This new resulting wave contains more zero field points in the box than the static standing wave we saw early. These points occur in two sets. The 1st set of points moves through space, with the same speed as the box. They are indicated by the light blue circles. These are the nodes and they are now closer together by 0.8 than when the box was not moving relative to the observer. This is the same amount of contraction as in the box. Thus the nodes remain stationary to the box. These are in the same location in the box as when the box was not moving relative to the observer. The 2nd set of ZFP, indicted by orange inverted Ts; move faster than the speed of light. These are the ZFP's of a group wave traveling on the standing electromagnetic wave. More about the 2nd set of ZFPs later.
In figure 9 we see it now takes more of the observer's time for the cycles of the waves, in the box, to repeat. We can see that the waves in the box at T = 0 and T = 1.25 are identical. This illustrates the time dilation.
Relative velocity 0.6c
Figure 9 compares the observer's standing wave with the astronaut's standing wave. Here we see the resulting spatial contraction and time dilation caused by the Doppler Effect.
In figure 10a if solid matter, atoms and molecules, were held together by standing electromagnetic waves, then it is easy to see why matter is affected by relativity. This also applies the space between objects moving at the same speed relative to the observer. This offers a possible answer to the earlier question. The Doppler Effect might cause space to contract and time to dilate.
The Minkowski diagram is a two system diagram, that compares the coordinate system of the moving object with the coordinate system of the observer. In figure 11 on the left is the x, t Minkowski diagram for a relative speed of 0.6c. This diagram was drawn by plotting the object's coordinates on a Cartesian plane using the x & t inverse Lorentz transformations. On the right we see where all the zero field points of the resulting wave of electric fields are plotted over a period of time. The red lines are the loci of the nodes. The blue lines are the loci of the points where the two waves cancel each other completely. These lines also draw the Minkowski diagram.
In figure 12 the diagram on the left, the slope of each line relative to the time axis t represents a speed, similar to the speedometer in a car. The diagram on the right compares two coordinate systems with a relative velocity of 0.6c. The t' axis has a slope of v/c = 0.6 representing the speed of the object as 0.6c. The DeBroglie wave, the wave associated with particles of matter, has a velocity c2/v. On the diagram this velocity would be represented by (c2/v)/c = c/v. The x' axis has a slope of c/v. The slope of the x' axis represents the velocity to the observer of the group wave.
There are two wavelengths associated with every elemental particle of matter (electron, proton, etc.). One of these is the DeBroglie wavelength and the other is the Compton wavelength.
The Compton wavelength λO for a particle at rest is Planck's constant times the speed of light divided by the rest energy of the particle hc/mOc2 = h/(mOc). This is derived from the Compton effect.
If we consider the Compton wavelength as the wavelength of an electromagnetic wave emitted by a subatomic particles at rest, then the wave would have a frequency of f O= c/λO = mOc2/h.
The velocity for an electromagnetic wave is c = 2.9979x108 m/s.
Its period T, the time it takes one cycle to occur is the wavelength divided by c, T = λ/c = h/(mOc2).
The Compton wavelength for an electron with a relative velocity of 0.6c is 2.4263 x 10-12 m with a frequency of 1.2356x1020 Hz.
When a standing wave produced by objects emitting a Compton electromagnetic wave is moving relative to the observer, all of its wave attributes will be affected by relativity. This can be seen in figure 13, the first set of ZFP's. Here we see the paths of the first set of zero field points in a standing electromagnetic wave. The slope of these paths is the velocity of the of object v/c, here 0.6c. All of the attributes of the standing electromagnetic wave affected by relativity can be found between these lines. The horizontal distance between of every other these lines is the contracted wavelength (0.8 λO). The vertical distance from the x-axis to the point 0',1' represents the period of the wave (1.25TO). The frequency is the number of cycles per observer time unit. There no way to show this on the diagram. However, in a standing EM wave, both the frequency and the wavelength are the reciprocal of the time period, thus the wavelength can be used to show the frequency (0.8).
The DeBroglie wave is a wave associated with matter, because sometimes a particle behaves as a wave.
The DeBroglie wave is an odd wave. If the object producing the wave is not moving relative to the observer, then the DeBroglie wave has an infinite velocity and an infinite wavelength.
The wavelength λ of a particle of matter is Planck's constant (h = 6.626x10-34 J/Hz) divided by its momentum (λ= h/p = h/mv = h/γmOv). Note the relativity factor γ = 1/[1-(v/c)2]1/2. When v = 0.6c then γ = 1.25.
The frequency f of the DeBroglie wave is the energy of the particle of matter divided by Planck's constant
(f = E/h = mc2/h = mOc2γ/h). The velocity (vDB) of the wave is vDB = fλ = c2/v.
The period (the time for one cycle to occur) is the wavelength divided by c T = λ/c = h/γmOvc.
An electron has a rest mass mO = 0.911x10-30kg. For an electron with a relative velocity of 0.6c (1.799x108m/s), its DeBroglie wave has a: wavelength of 3.2351x10-12 m and a frequency of 1.5445 x 1020 Hz.
If we were to divide the attributes of the DeBroglie wave by the same attributes of the Compton wave, we have a set of ratios. These ratios can be used to compare the attributes of the second set of zero field points with the attributes of the original electromagnetic wave from the non-moving source. Table 3-1, below shows the ratios of the DeBroglie attributes divided by the Compton wave attributes.
Table 3-1 Ratios of DeBroglie attributes over Compton attributes
Wavelength ratioλ/λO = (h/γmOv)/( h/mOc) = (c/v)/γ for v = 0.6c,λ/λO = 1.6666/1.25 = 1.3333 λO
Frequency ratio f/fO = (mOc2γ/h)/( mOc2/h) = γ for v = 0.6c, f = 1/T = γ = 1.25 fO
Wave velocity ratio (c2/v)/c = c/v for v = 0.6c, c/v = 1/0.6 = 1.6667 c
Period ratio TD/TO = (h/γmOvc)/(h/mOc2) = γ(v/c) for v = 0.6c, TD/TO = γ(v/c) = 1.25(0.6) = 0.75 TO
In figure 14 we see the paths the second set of zero field points in the same standing electromagnetic wave. The slope of these paths is c/v, the velocity of the DeBroglie wave. Consider that the wavelength of the original electromagnetic wave were the Compton wavelength for a particle of matter. Then, all of the attributes of the DeBroglie wave for the same particle of matter with a velocity of v can be found between these lines. Like the horizontal distance between of every other of these lines (the solid lines) is the DeBroglie wavelength (1.333...λO). The vertical distance from the x-axis to the point 1',0' represents the period of the DeBroglie wave (0.75TO). The horizontal distance from the T-axis to the point 1',0' represents the frequency of this wave (1.25TO). This presents the possibility that the DeBroglie wave or DeBroglie like wave is a traveling group wave on a standing electromagnetic wave.
We have seen how a moving standing electromagnetic wave can be used to physically illustrate what might cause space to contract and time to dilate. Also any standing electromagnetic wave that is moving relative to an observer with a speed of v, will produce a group wave on the standing wave that travels at speed of c2/v relative to the observer. When the object is a beam of electrons, the static wavelength is the Compton wavelength and the wavelength of the group wave, is the DeBroglie wavelength. Since plotting the pathes of zero field points draws the Minkowski diagram, all of these wave attributes can be found on this diagram.
The computer program ZFPpatrn uses ZFP’s of standing wave to draw Minkowski Diagram
This program is written in GW Basic. The program plots the zero field points of a standing electromagnetic wave in time and space. When you start the program, it will ask you to input the velocity (V/C) that you want. This can be any number from 0 to almost 1.0. When this program is run the zero field points draw the x,t Minkowski diagram for the velocity (V/C) you input at start of program. A scale of 1/10 th unit is drawn on the one time unit line and the one space unit line. This is to see that the t'-axis crosses the one time unit line at tv/c space units since an object with a velocity of v/c will move v/c space units in one time unit. Since the x'-axis is the inverse function of the t'-axis, the x'-axis will cross the one space line at v/c time units.
100 C=I: PI=3.1415927#: REM ZDPpatrn
110 REM Plots the position and time at which the sum of the leading and trailing
waves are at 0 amplitude
120 SCREEN 1,0:CLS:KEY OFF:COLOR 0,l
130 INPUT "V/C";VR:CLS
150 AJ=1.25:REM Adjustment variance in x,y pixels
160 LINE(0*AJ,15)-(0*AJ,199),2:REM Static time axis
170 LINE(0,199)-(300,198),2:REM Static spatial axis
180 LINE(100*AJ,199)-(100*AJ,0),2:REM line at distance of one wavelength
182 FOR SU=0 TO 200*AJ STEP 10*AJ
184 LINE (SU,99)-(SU,94),2:REM for horizontal scale
185 LINE (SU,199)-(SU,194),2:REM for horizontal scale
186 NEXT SU
190 LINE(320,99)-(0,99),2:REM line indicating one time period
192 FOR SU=-1 TO 199 STEP 10:REM for vertical scale
194 LINE (100*AJ,SU)-(105*AJ,SU),2:REM for vertical scale
195 LINE (1*AJ,SU)-(5*AJ,SU),2:REM for vertical scale
196 NEXT SU
200 B=(1-VR^2)^.5:TT=200/B:REM Relativity Factor and Period of Waves
210 Wl=200*((1-VR)/B):REM Wavelength of Wave moving to the right effected by the leading Doppler Effect
220 W2=200*((1+VR)/B):REM Wavelength of Wave moving to the left effected by the trailing Doppler Effect
240 FOR T=0 TO 200 STEP 1:REM Position on the Time Axis
250 FOR X=0 TO 320 STEP .5:REM Position on the Spatial Axis
260 Yl=10*COS(2*PI*((X-T)/Wl)):REM Equation for amplitude of wave moving right
270 Y2=10*-COS(2*PI*((X+T)/W2)):REM Equation for amplitude of wave moving left
280 Y=YI+Y2:REM The sum of the amplitudes of both waves
290 PQ=199-T:REM places 0 time at bottom of graph
draws points of 0 displacement
300 IF Y=0 THEN PSET(X*AJ,PQ):REM If current point on resultant wave is at 0 then a point is placed on the time-space graph
320 IF X>0 AND SGN(Y)<>SGN(RS) THEN PSET(X*AJ,PQ):REM If the current point on
the resultant wave has just passed through 0 the a point is drawn, on graph
330 RS=Y:REM Used to compare sign of present point to sign of next point
340 NEXT X
350 NEXT T:END
Since the plotting of these zero field points draws the Minkowski diagram illustrates that the effects of relativity on a standing electromagnetic wave are caused by the Doppler effect.
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