Special Relativity Speedometer
Colin Apke
University of Colorado
Boulder, Colorado
80310
CSGC
2/6/2016
Introduction:
Velocity can be measured with many different methods, including GPS,
accelerometers, and more, but for objects with a large enough velocity, the principles of
special relativity can also be used. Einstein predicted, and was later proven to be correct,
that time is only constant per a frame of reference. If an object, such as a rocket, is
orbiting the Earth at .1c (.1 times the speed of light), then its interpretation of time passed
during its flight would differ from the time recorded on the surface of the Earth. Though
we may not posses a rocket that can fly at close to .1c, the same conclusion can be made
of any rocket launching from the Earth, but instead of using its velocity to find its time
dilation, we could use its compression of time to figure out its velocity. The relativity
speedometer would use two computer clocks, running over a GHz cycle, and compare
their points in time with each other, and this comparison should show a minute difference
between the two. For even better accuracy, if a link could be made between the rocket
and a base station, then the two clocks could constantly check themselves and give
updated information about the velocity of the rocket. This comparison between the two
clocks, if both are able to operate at very high frequencies, should show a shift in the
clock cycles, where the rocket-bound clock is very gradually falling behind the Earth
clock as its velocity increases. This experiment would not require any external equipment
besides a second clock to compare with the rocket bound clock, eliminating the need to
make contact with a satellite which can be tricky if the rocket is moving too fast, and also
does not require multiple dimensional analysis as would be required of the accelerometer
approach. The principles of special relativity do not require a direction, but only a
velocity, meaning that the rocket could be going in any direction, but still give the same
CSGC
1
2/6/2016
result of its velocity in that direction. For interplanetary and even interstellar travel, this
could prove to be a handy way to keep track of a space ships velocity.
The Theory:
The speed of light is constant in all frames of reference. No matter what speeds a
frame obtains, the speed of light will be the same speed it would be to someone in a
different frame of reference. This means that a fast moving object can not discern the fact
that it is in a different frame of reference by observations of its frame of reference.
However, if a comparison is made between two frames, then the difference between the
two frames can be measured. Using the Earth as one frame of reference and a rocket as
the other, we should be able to detect a difference between the two as the rocket’s
velocity changes when compared to that of the Earth’s.
Since the speed of light must remain constant in the rocket’s frame of reference as
well as the Earth’s frame of reference, other properties of the rocket compared to the
Earth’s must become dilated as the rocket approaches the speed of light. This spawns
from the concept of simultaneity, which states that events of simultaneity are frame
dependent. This means that an even that appears simultaneous in one frame occurs at two
different times to another frame of reference. This distortion of time has become a very
important concept of special relativity, since it means that time is also frame dependent.
Since time dilates between two moving frames, the difference between the times recorded
in one frame can be compared to that of another frame to discern the speed of one frame
to another.
CSGC
2
2/6/2016
The Application:
Through a lot of complicated calculations, we find that the difference of time
between two frames of reference is the change in time measured on the inertial frame
divided by the square root of one minus the velocity of the moving frame squared divided
by the speed of light squared, which looks like: t Earth 
t Rocket
v
1  Rocket
c2
2
. Looking at this, it
can be noted that for speeds much smaller than the speed of light, the dilation will be
close to impossible to measure. This is why people driving in their cars to work do not
notice a difference in time from the moment the step into the car till the moment they step
out of it, but there is still a difference. The problem is measuring the dilation between the
two requires a device that can measure a difference of a couple nanoseconds.
Before we can figure out what kind of time dilation we want to measure, we must
figure out the smallest velocity to be measured. Using the equation above, we can make a
table of time dilations for an object moving at a set speed for 1 second, which is shown
below.
Velocity
Time Dilation after 1 second
Time Dilation after 10 seconds
1 km/hr
4.29263138759000000184 x 10-19 s
4.29263138759000000184 x 10-18 s
10 km/hr
4.29263138925380018424 x 10-17 s
4.29263138925380018426 x 10-16 s
1 km/min
1.54534730019810141084 x 10-15 s
1.54534730019810238810 x 10-14 s
10 km/min
1.54534730007482825942 x 10-13 s
1.54534730007482825983 x 10-12 s
1 km/s
5.56325028031451679168 x 10-12 s
5.56325028031451679168 x 10-11 s
10 km/s
5.56325028491055521731 x 10-10 s
5.56325028491055521731 x 10-9 s
CSGC
3
2/6/2016
100 km/s
5.56325074451444041973 x 10-8 s
5.56325074451444041973 x 10-7 s
1000 km/s
5.56329670532907077078 x 10-6 s
5.56329670532907077078 x 10-5 s
104 km/s
5.56789705204583117681 x 10-4 s
5.56789705204583117681 x 10-3 s
105 km/s
6.07520004442040002425 x 10-2 s
.607520004442040002425 s
For comparison, 10 km/hr is about the speed you leave your drive way at, 10 km/min is
just slow of the cruising speed of 747 airplanes, and 1 km/s is a tad slow of the speed that
GPS satellites orbit the Earth at. Since we want to measure the velocity of a rocket, which
can vary between 1 km/min and 10km/sec, we would want a system capable of
measuring the difference of a trillionth of a second.
To tell the difference of a trillionth of a second, it would be desirable to use a
timer with a clock speed of in the terahertz cycle, though something in the range of
gigahertz would work as well, since the above examples are only for 1 to 10 seconds,
while a rocket, preferably a sounding rocket, will travel for many minutes, if not more.
This should allow a delay that can be detected between a clock on the rocket and a clock
on Earth.
The Clock:
Creating a reliable oscillator that runs in the gigahertz range is complicated and
involves complex construction and equipment that is unavailable and expensive. A
simpler solution is to use a computer clock, which usually have clock cycles in the
gigahertz range and are becoming faster each year. If a computer, running a very specific
set of instructions that are known to take a very specific amount of time, then those two
computer programs can be compared to measure the dilation of time. The problem now
CSGC
4
2/6/2016
becomes discerning the difference between an error in the clock cycle and the actual
dilation of time.
The problem with any oscillator for this project is that all oscillators have a set
error range, which means that they will have so many errors for every second. This
comes from imperfections in the circuitry, which causes the period in an oscillator to vary
by an amount that can lengthen or shorten a given clock cycle, which when returned to
Earth, would be indistinguishable with the actual dilation of time caused by special
relativity. The only way to beat this problem would be statistically. That is, to place a
large number of oscillators on the rocket, and then compare them all with a large number
of oscillators on the ground, and the average of those answers should give you the actual
affect of relativity, and not of clock errors.
Now space becomes an issue since a computer is a large device, which also draws
a large amount of power, which requires even more space, and that limits the number of
computers we can launch. Luckily, with the advance of technology, the computer is
becoming increasingly smaller and requiring less power, which means less space, but for
now there exist microchips that operate in the gigahertz range which can be used then the
difference between them can be collected to determine error, which should eliminate
some error, but not all error. Unfortunately, this may not be ready to use on smaller
rockets that fly at lesser speeds for smaller time intervals, since the error would mask the
time dilation. The larger the rocket and the farther it flies, the greater the accuracy
obtained from this method.
The Computers:
CSGC
5
2/6/2016
As stated above, using two computers would be ideal since they have high clock
cycles, and could produce a measurable dilation between the rocket’s frame of reference
and the Earth’s. If a set instruction, which was known to take a set amount of time, could
be sent through an endless loop, when the two clocks are then compared against each
other, then the dilation should become apparent. There could be many ways to check this
dilation, most involving a computer for the greatest accuracy, though some could even be
detectable using the human ear. For example, the endless loop could be outputting a sin
wave, and when the two are compared before the launch, the sin wave output would
sound very normal, but after the flight, if dilation occurred, then the two sin waves would
be off and we should hear interference. Measuring that interference would lead to a
velocity of the rocket, but for more accurate results, its better to have a computer do the
measurements.
Another possibility would be to have the computers transferring a very large
picture, bit by bit, and if dilation occurs, then the Earth computer would have completed a
larger portion of the picture then the rocket bound computer. Since a picture is much
more complicated then a sin wave, the dilation should become easier to detect and
measure. There exist an infinite amount of instructions that the computers could use, each
may be better or worse then the ones stated here, but these would still achieve their
purpose.
If the purpose was to measure the difference between two oscillating computers,
then the maximum amount of dilation possible between the two clocks has to be the
difference between two instruction loops. As stated, the desired method for measuring the
difference between the two clocks would be to have a computer running a very specific
CSGC
6
2/6/2016
set of code inside an endless loop, then compare the two computers to see where they are
in each loop. But what happens if the difference between the loops is nonexistent, or in
other words, that the dilation in time has caused the rocket-bound computer to loose
exactly one loop from the Earth-bound computer. Then the difference is impossible to
tell. This means that the loop between the two computers must be large enough so that it
is impossible for the rocket flight to exceed a velocity that would cause such a situation to
arise.
The Results:
Now comparing the two computers to see where they are in the loop becomes
tricky. A few methods were described above but both hinge on one very important
concept: Each computer must start and stop and the exact same time. It is unimportant
that the rocket-bound clock detect launch since up till the rocket launches since no
dilation should occur, with the same holding true after landing. Neither can be set to
expire on a timer as well, since that would defeat the purpose of detecting a shift between
the two if you have them stop after the clock records a set amount of time, since each
clock will view time occurring at a different rate according to the original theory. A
physical force must tell both to start and stop at the same instant in the same reference
plane, which means that they should be turned on together before launch and turned off
together after launch.
Now assuming that there was a perfect launch and no altitude anomalies occurred
and the rocket returned in one piece and all of the above instructions were followed, then
the dilation can be measured. All the dilation will appear to be, though, is a difference
CSGC
7
2/6/2016
between the two clocks, and it lacks a very accurate description of the instantaneous
velocities achieved at different points in time, but rather the average velocity of the
rocket for the entire flight. It is important to remember that the recorded time on the
rocket should not be greater then the time recorded on the earth, since the equation
describes the time change to be longer on Earth then it is on the rocket. Since up to this
point the rocket and the Earth were never in communication with each other, it is
impossible to tell the instant dilation occurs, but rather that it has occurred at some point
on flight. With a little knowledge of the flight, some assumptions can be made about the
trajectory of the rocket.
Lets assume in this case that the rocket was launched out of a cannon with no
parachute at a 90 degree angle. This tells us that the moment the rocket leaves the
cannon, it will be traveling at its greatest velocity it will achieve during its flight, and it
will fly straight up into the air, turn around, and in a perfect world, fall right back into the
cannon. If this were true of the rocket holding one of the clocks, and we already know the
dilation of the clock, then we could build a theoretical velocity over time graph for the
rocket that would fit within the above guidelines. In fact, the graph of dilation vs. time
would resemble the integral of the velocity vs. time graph. Since we now the sum of the
dilation, we should be able to work out a semi-accurate graph for the velocity of the
rocket. Below is an example with no values used to illustrate the point. As the velocity of
the rocket reaches its minimum, the dilation goes to 0, but as the rocket’s velocity
increases, so does the time dilation.
CSGC
8
2/6/2016
Dilation vs. Tim e
Velocity
Dilation
Velocity vs. Tim e
Tim e
Tim e
A real rocket will not follow this path since the engine on the rocket will take it
from 0 to its maximum velocity, it will slow as it reaches the top of its flight, then
increase until the parachute deploys then fall with a semi-constant velocity. There are a
lot more variables to deal with in that scenario, but the dilation should appear as it would
in the above example. Yet there are still a major number of assumptions taking place and
for a very accurate description of flight, it would be better if the two clocks could talk to
each other.
If the two clocks could talk to each other, then we could calculate an
instantaneous velocity since the two clocks could tell the dilation every second. This
would require a few more complications, since now the speed of light and the distance
the clocks are from each other must now be taken into account. First off, if the two clocks
are going to stay in communications, then one clock must transmit and the other must
receive and calculate the dilation. Since the rocket is the defining factor in the size of the
rocket bound clock, it’s probably better that the ground clock transmit since that requires
much more equipment. Also, the rocket clock must realize that it will take some time for
the ground clock’s signal to reach it. To solve that problem, before launch, the two should
CSGC
9
2/6/2016
synchronize themselves such that the ground clock will be sending a signal every x
seconds and the rocket clock should be receiving those signals every x seconds on the
ground. When the rocket launches, the signals should take either more or less time to
reach the rocket depending on the orientation of the rocket. The rocket then should be
able to tell how far away it is from the ground station and then compare where it was
when the ground station sent its signal then compare those two signals for their dilation.
For smaller projects or projects where it is impossible to stay in communication
with the rocket, it would be better to use the clocks as separate systems and not link them.
At worst, the results will show the average velocity of the rocket during its flight, but
with a lot of math, it should give a good representation of the rocket’s velocity for the
flight. For an experiment which requires more detail, then the communication link would
add a lot more detail as it could provide instantaneous dilation which would give
instantaneous velocities for the rocket. Each method simply depends on the need of the
experiment.
The Other Methods:
There are other methods which would also give the velocity of a rocket, as stated
in the introduction of this paper, and each method has its pros and cons. For example,
with the GPS system, that is the best way on Earth to get very reliable data on velocity
and a number of other useful data points, but making the link with the satellites can be
tricky, and what if the experiment is not on Earth? What if someone wanted to test the
velocity of a rocket on Mars, which does not have a GPS system? In that case,
accelerometers can be used to determine the velocity at different instances.
CSGC
10
2/6/2016
Accelerometers are small, and are getting cheaper every year, but only give the
acceleration at certain times on the rocket. What happens if the rocket is in free fall?
Then the accelerometers would register zero when in fact there is most likely a large
velocity.
A speedometer relying on special relativity has its own complications, such as
getting the accuracy out of the clocks as well as starting and stopping both clocks at the
exact same time, but they also have their strengths. For example, even in free fall where
the acceleration is zero, the dilation effect will still occur because of the velocity of the
rocket, and also does not require the complicated link process as does GPS, which is also
controlled by the military and getting permission for that kind of information is nearly
impossible. Then again, for slower velocities over shorter time intervals, the relativity
speedometer would not register dilation. For different experiments, each method has its
strengths and weaknesses, and for different situations, one could be better than the other.
Goals:
Obviously there are complications in the system that would require a lot of
perfecting before the system can be called reliable, but it may become a solution for
interstellar space travel. For those people who were wondering, if the two clocks are in
communication with each other, then won’t they know the velocity just from that? The
answer is sometimes yes. It would be possible to calculate the velocity from the change in
signal intervals, but what if the rocket is flying parallel to the signals, so that they arrived
at the same times. The rocket might assume that it had stopped moving when in fact it
still might have a large velocity. A good example of this is a satellite in a geosynchronous
CSGC
11
2/6/2016
orbit. If the signal is originating from the same spot on Earth, then the signal will always
reach the satellite at the same time. The relativity speedometer would ignore that fact and
line up markers to check for dilation, which should occur at those speeds. So there are
definite strengths to the system, but they still depend on the situation for which they are
applied.
Throughout this paper, I have used the words “if dilation occurs”, which implies
that there may be variables which I have not considered which would interfere with the
experiment, but none of it comes from the actual theory itself, since Einstein’s theory of
special relativity has been proven many times. The best example is in a particle
accelerator, where the half life’s of particles moving a very high velocities (approaching
the speed of light) show a delay before the particle decays away. The theory is solid, but
the application may have many problems which still need to be worked out, and the best
way to do that would be to actually perform the experiment. If possible, it would be nice
to have a second, well documented method for finding the velocity of a rocket as well as
using the relativity speedometer, so that when the flight is over, the two can be compared
to check the accuracy of this experiment. Until then, this experiment only stands on
theory and waits to be tested.
CSGC
12
2/6/2016