Relativity Measurements

How can an observer measure the lengths or times occurring in another system that is passing him with a very high velocity? He can not hold up a yardstick and compare it with a yard-stick in the system that is flashing past him.

To measure lengths in the other system he can time how long it takes for two points in the other system to pass him. To measure relative velocity he could time how long it takes for one point in the other system to pass two points in the observer's system. To do this, however, he must be sure the clocks at the two different points are synchronized. To measure time in the other system he must be able to compare his clocks to the clocks in the other system as they pass each other.

To visualize how this could be done we will image two very large identical spacecraft. Each has a large clock in the front and a large clock in the rear. In both vehicles the clocks are 3000 meters apart. This distance between the clocks LO, (3 000) meters is about 1.86 miles The time it takes light to travel the 3000 meters between the two clocks is 10 microseconds (fig. 1).

For ease of understanding, instead of using large and small quantities, we can create our own time-space units. One space unit SU is 3000 meters long and one time unit TU last 10 microseconds. Therefore, light will travel the distance of one SU (LO) in the time span of one TU (LO/c).

Fig. 1   Two Identical Imaginary Spacecraft
Fig. 1 Two Identical Imaginary Spacecraft

To synchronize the clocks using light beams

First we will synchronize the two clocks when the spacecraft is at rest relative to our frame of reference. In figure 2A the rear clock is set at 0 and started. At the same instant a beam of light is started toward the front clock. It will take one time unit for the beam of light to reach the front clock. In figure 2B when the beam arrives at the front clock, the clock is instantly set at 1.0 TU and started. The rear clock has been running and arrives at 1.0 TU at the same instant as the front clock reads 1.0 TU. Both clocks are at one, running and synchronized. They are in phase with each other. The beam of light was reflected when it arrived at the front clock and is returning to the rear clock. In figure 2C the light beam returns to the rear clock and another time unit has passed, so that both clocks read 2 TU.

Fig. 2 Synchronizing the static clocks
Fig. 2 Synchronizing the static clocks

To synchronize the clocks using light beams while moving

Now we will do the same thing with the spacecraft moving relative to our frame of reference with a high constant velocity (here 0.6c). Because of its velocity the spacecraft will be shorter by L = LO(1+v2/c2)1/2 = 0.8 SU and its time will move slower by T = TO/(1+v2/c2)1/2 = 1.25. In figure 3A the rear clock is set at 0 TU and started. At the same instant a beam of light is started toward the front clock. The time t1 it would take for light to travel the length of the rocket is 0.8 TU. However the front clock is moving away from its original position. Therefore the light beam will have to travel a longer time t2, to reach the front clock. In figure 3B when the beam arrives at the front clock, the clock is instantly set at 1.0 TU and started. The rear clock has been running while the beam was traveling. To determine how long the beam traveled we note that

ct2 = ct1+vt2 or t2 = ct1/(c-v) where ct1 = L = LO(1+v2/c2)1/2 = 0.8 TU, v/c = 0.6

thus t2 = t1/(1-v/c) =0.8/(1.0-0.6) = 2 TU

Therefore the time it takes the beam to travel from the rear clock to the front clock is t2 = 2 TU (time units). However, since the clocks on this moving spacecraft are moving slower by 1.25 TU, then the time t'2 showing on the rear clock will be

t'2 = 2 TU/1.25 = 1.60 T'U.

The front clock is set 1.0 TU (fig. 3B) and the beam of light was reflected. When it arrived at the front clock in fig. 3C we see that the time t3 for the beam to return from the front clock to the rear clock is

t3 = ct1 /(c+v) = t1/(1+(v/c). Here again ct1 = L = 0.8, therefore t3 = 0.8/1.6 = 0.5 TU.

Since the clocks on the moving spacecraft move slower by 1.25 TU these clocks will indicate that only

t'3 = 0.5/1.25 = 0.4 T'U

has passed. When this added to the rear clock, makes it will read 2.0 T'U, and added to the front clock makes it read 1.4 T'U. If the beam is reflected back to the front clock, it will take 1.6 of its time units. So when the beam returns the front clock will read 3.0 T'U and the rear clock will read 3.6 T'U. Each time the beam returns to the front clock, this clock will be on an odd whole number, and will indicate that two time units have passed since the beam left this clock. Each time the beam returns to the rear clock, this clock will be on an even whole number, and will indicate two time units have passed since the beam left this clock. This is exactly like the synchronized clocks on the static spacecraft. Therefore these clocks are synchronized, but out of phase with each other. THE REAR CLOCK IS ALWAYS GOING TO BE (V/C)(Lo/C) = LOV/C2 AHEAD OF THE FRONT CLOCK. The readings on the clocks must be expressed in time units. Since TO is the time it would take light to travel the distance LO then LO = TOC thus LOV/C2 = TOV/C. Therefore, THE REAR CLOCK IS ALWAYS GOING TO BE TOV/C AHEAD OF THE FRONT CLOCK. This effect is called "the lack of simultaneity". It is not just an arbitrary means of synchronizing clocks, but a real time difference.

  Fig. 3   Synchronizing clocks on a vehicle with a relative velocity of 0.6c
Fig. 3 Synchronizing clocks on a vehicle with a relative velocity of 0.6c

Now let us use the clock readings, as these two spacecraft pass each other to measure relative velocity, spatial contraction and time dilation.

Clock readings at key points

Here we see a spacecraft that is static to the observer is being passed by an identical spacecraft with a velocity of 0.6c. Just as the two front clocks pass each other they are both at 0.0 TU

  Fig. 4  Clock readings as a moving vehicle passes a static vehicle with a speed of 0.6c
Fig. 4 Clock readings as a moving vehicle passes a static vehicle with a speed of 0.6c

To measure the relative velocity between the spacecrafts

One way to measure relative velocity between passing vehicles is to note the time a point on the passing vehicle (here its front clock) passes, the observer's front clock. Then to note the time that same point passes the observers rear clock. Both observers know that their front and rear clocks are 3000 meters or one SU apart. By examining the readings at key points (see fig. 4A), we see that the front clock on the vehicle static to the observer reads 0.00 TU when the moving front clock passes it. When the moving front clock passes the static rear clock (see fig. 4C), the static rear clock reads 1.6667. Using L = vt = 1 and t = 1.6667, the observer in the static vehicle determines the relative velocity (v) between the two vehicles as;

v = L/t = 1/1.6667 =0.6c

Likewise the front clock on the moving vehicle reads 0.00 TU when it passes the front clock on the static vehicle. In fig. 4B the rear clock on the moving vehicle reads 1.6667 TU when it passes the front clock on the static vehicle. Therefore the observer on the moving vehicle will calculate the relative velocity as;

v = L/t = 1/1.6667 = 0.6c

Even though the rear clock readings occur in different locations for the static and moving vehicles, they both calculate the same relative velocity between them.

To measure lengths on passing vehicles

To measure the length between the clocks on the passing vehicle, the observer on either vehicle will note his time when the front clock of the other vehicle passes him. Then he notes the time when the rear clock passes him. The observer on the static vehicle gets first time reading T1 (fig. 4A) as 0.0 TU and his second time reading T2 (fig. 4B) as 1.3333. Hence the total time T =T2-T1 = 1.3333. The static observer will calculate the length between the two moving clocks as

L = vT = 0.6x1.3333 = 0.8 SU.

The observer on the moving vehicle gets his first time reading (fig 4A) as 0.0 TU and the second time reading (fig. C) as 1.3333. These are the same time readings as the observer on the static vehicle had. Therefore both the moving and static observers measure lengths on the other vehicle as being 0.8 shorter than the same lengths on their own vehicle.

To measure the time rate on a passing vehicle

Both observers want to compare the rate of time passage in the passing system with their own. First the each observer must note his own clock and the passing clock readings as the front clock of the other vehicle passes him. Then when the rear clock passes him, he again needs to note his own clock reading and that of the passing clock. To calculate any difference in time rate, the observer must take the time difference (rear clock reading-front clock reading = DT) of the passing system and divide it by his own time difference.

time rate ratio = TO/T = time in observer's system / time in the other system

The observer on the moving vehicle will calculate the rate that time T passes on the static vehicle compared to the rate that time TO passes on his own vehicle as;

TO/T = 1.6667/1.3333 = 1.25 (from fig. 4A and fig. 4B).

The observer on the moving vehicle will calculate the rate that time passes on the static vehicle is 1.25 times slower than in his own system.

The observer on the static vehicle sees his own time as TO and time on the moving vehicle as T and will calculate;

TO/T = 1.6667/1.3333 = 1.25 (from fig. 4A and fig. 4C).

Here we see that the observer on the static vehicle will also calculates the rate that time passes on the other vehicle is 1.25 times slower than in his own system. Both observers see the same time dilation in the other system.

The same results could have been obtained by using the relativity equation for time dilation

T = TO/(1 - v2/c2)1/2 = 1.0/(1 - (0.6)2)1/2 = 1.25.

Clock readings when passing vehicles are both moving relative to the observer

Consider the same two vehicles both moving relative to the observer's frame of reference. In fig 5 vehicle R is moving past observer with a velocity of 0.6c toward the right. Its actual length L has contracted to

(1-0.62 ) 1/2 = 0.8 SU and his is moving slower by 1/0.8 = 1.25 TU. Vehicle L is moving past observer with a velocity of 0.8c toward the left. Its actual length L has contracted to (1-0.82) 1/2 = 0.6 SU and its time is moving slower by 1/0.6 = 1.6667 TU.

At the instant when the front clocks of the two vehicles pass each other, in fig 5A, they both read 0.0 TU. Both rear clocks read the same as their front clock plus their velocity ratio. That is R's rear clock reads 0.6 and L's rear clock reads 0.8.

In fig. 5B, since L's actual length is 0.6 TU, then the real time T it will take for R's front clock to pass from L's front clock to L's rear clock can be calculated from;

vLT + vRT = 0.6 SU therefore T = 0.6/(vL +vR ) = 0.6/(0.6 + 0.8) = 0.42857143TU

where vL is the velocity of vehicle L and vR is the velocity vehicle R.

Since R's time TR is slower by 1/0.8 its clocks will show that

TR = 0.8 x 0.42857143 = 0.34285714 time units

have passed during the trip from L's front clock to L's rear clock. In fig. 5B we round the time to three decimal places so that R's front clock reads 0.343 and R's rear clock reads the same plus 0.6 to equal 0.943. Since L's time TL is slower by 1/0.6, therefore its clocks will show that

TL = vL x T = 0.6 x 0.42857143 = 0.25714286 time units

have passed since passing R's front clock. In fig. 5B L's front clock reads 0.257 and L's rear clock reads the same plus 0.8 which equals 1.057.

In 5C we see R's actual length is 0.8 SU, then the real time T, it will take for L's front clock to pass from R's front clock to R's rear clock can be calculated from;

vLT + vRT = 0.8 SU therefore T = 0.8/(vL + vR ) = 0.8/(0.6 + 0.8) = 0.57142857TU

Since R's time TR is slower by 1/0.8 its clocks will show that

TR = vR x T = 0.8 x 0.57142857 = 0.4571428 time units

have passed since passing L's front clock. In fig. 5C note that R's front clock reads 0.457 and R's rear clock reads the same plus 0.6 to equal 1.057. L's time is slower by 1/0.6 therefore its clocks will show that

TL = 0.6 x 0.57142857 = 0.34285714 time units

have passed since passing R's front clock. In fig. 5B L's front clock reads 0.343 and L's rear clock reads the same plus 0.8 to equal 1.143.

Fig. 5    Clock readings as two moving vehicles pass each other
Fig. 5 Clock readings as two moving vehicles pass each other

To measure relative velocity when both vehicles are moving

Just as before each observer will time how long it will take a point on the passing vehicle to move from his front clock to his rear clock. This time is T. Since both observers know that their front and back clocks are one SU apart, to find the relative velocity between them we divide one by the time T.

In fig. 5 we see that when the front clock on the other vehicle is passing either observer's rear clock, that rear clock reads 1.057 TU. Therefore both observers calculate their relative velocity

v = L/t = 1 SU/1.057 TU = 0.9459459c.

The same results could have been obtained by using the relativity velocity addition equation

v = (vL + vR )/(1 + (vL x vR /c2)) = (0.6+0.8)/(1+0.6x0.8) = 0.946c.

where vL is the velocity of vehicle L and vR is the velocity vehicle R.

To measure lengths and times in other system when both vehicles are moving

To measure length, both observers will time how long it will take to pass from the other craft's front clock to its rear clock. In fig. 5 we see that both observers time this event as taking 0.343 TU. Hence both observers will calculate the distance between the other vehicle's clocks as

L= vT= 0.946 SU/TU x 0.343 TU = 0.324 SU.

The same results could have been obtained by using the relativity equation for spatial contraction

L = LO(1 - v2/c2)1/2 = 1.0(1 - (0.946)2)1/2 = 0.324 SU.

To measure time on the passing vehicle, both observers will compare the time on the passing front clock to the time on the passing rear clock. This will tell them how long this event took in the other system. Then the observers will divide this time by their own time to obtain the time ratio between the two systems. In fig 5 both observers see the time difference in the passing clocks as 1.057 TU. Both observers know that the same event only took 0.343 TU in their own system. Therefore both observers calculate the time in the other system as 1.057/0.343 = 3.082 times slower than in their own vehicle.

T/TO = 1.0571429 TU/0.3428571 TU = 3.0833.

The same results could have been obtained by using the relativity equation for time dilation

Time dilation g = TO/(1 - v2/c2)1/2 = 1.0/(1 - (0.9459459)2)1/2 = 3.0833.

Doppler Effect

Table 1 shows the equations for the relativistic Doppler effect on electromagnetic waves.

Table 1 Relativistic Doppler Effect on electromagnetic waves
Table 1 Relativistic Doppler Effect on electromagnetic waves

In fig. 6, vehicle A is in the observer's frame of reference and object B is approaching A with a velocity of 0.6c. Both are emitting electromagnetic signals that have a period of one time unit. Since B is moving through the observer's frame with a velocity of 0.6c, its time unit is 1.25 times longer than A's time unit, therefore its the period of its signal is 1.25 times longer than A's. When A measures the period of B's signal, he finds that it is 1/2 his own time unit. Solving the Doppler equation T = TO(c-v/c+v)1/2 for v, A can determine that the relative velocity between A and B is 0.6c. Likewise when B measures the period of A's signal, he finds its 1/2 his time unit. Using the same calculations B also determines that the relative velocity between them is 0.6c.

Fig. 6  Doppler Effect on waves between objects approaching with a velocity of 0.6c
Fig. 6 Doppler Effect on waves between objects approaching with a velocity of 0.6c

We have seen how vehicles passing each other at a high velocity, will measure spatial contraction and time dilation in the other vehicle. Also how the Minkowski diagram can be used to trace signals from both vehicles to illustrate the Doppler effect.

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