The speed of time
Advertisement
One of the consequences of special relativity is that any moving clock slows down in accordance with a precise mathematical formula. The faster the clock is moving the more it slows down. The clock slows down for a very good reason; time itself, for anything moving, slows down relative to a stationary observer. This idea was put forward in 1905 by Albert Einstein and has since been tested many times. In particular, the accuracy of atomic clocks has allowed us to verify the effects of time dilation at even very modest speeds. This page gives examples of how time is slowed down at a variety of speeds, but concentrates on speeds that we can encounter as a matter of routine. The effects of time dilation only start to become apparent at speeds close to the speed of light (300,000 km per second, or 186,300 miles per second). For more detailed information on time dilation see this page, and you can find a time dilation calculator here.
A table of time
At the speeds we travel at in everyday life time dilation is so small that it's not detectable to all but atomic clocks. Even if you were to spend your whole life travelling in a fast modern jet your time dilation as measured by a stationary observer would still not even register on a digital watch, as we will see later in this page. While the equation dealing with time dilation isn't especially difficult it's perhaps more instructive to see some of the results listed as a table, such as in the one below. Here we see the percentage of the speed of light (c) against its dilated percentage, so that, for example, travelling at 90% of the speed of light time will slow down to 43.6% of its usual rate:
From the table it’s easy to see that time is only altered a little until we get to speeds above about 50% of that of light, and then the effect becomes ever more dramatic. Beyond 99% of the speed of light time slows down very rapidly indeed, as shown here:
So as we get ever closer to the speed of light time slows down until it almost stops. However, note that it’s not possible to reach 100% of the speed of light (as explained on this page) and so time never quite comes to a complete standstill. You can experiment with different percentages of by light using the Time Dilation Calculator here, all the way up to 99.99999999999999% of the speed of light.
Example 1: A flight over the Atlantic
Now let’s look at a some examples of dilation at relatively low speeds. For most of us the biggest and longest time dilation we ever experience is on a long flight, such as over the Atlantic Ocean. Such a flight takes about six hours and the average speed is about 550 mph (880 km/h). Now, 550 mph is undoubtedly very fast in human terms but the only change we have to make to a watch at the end of such a flight is to adjust it to the local time, and we don't need to alter it to take time dilation into account. Does this mean there is no time dilation during the flight? No! It's just that the time dilation, even at the speed of a jet, is so small that our everyday watches can't measure it. Remember, all moving clocks run slowly, even slowly moving ones.
Six hours at 550 mph (880 km/h). Time dilation?
So how much time do we "save" on such a flight? To answer this we must first convert hours into seconds and miles per hour into metres per second. When we do this we get:
6 hours = 21,600 seconds 550 mph = 244 metres per second
The next thing we need to do is to "plug" these numbers into the time dilation equation and carry out the calculation. In doing so I will use a form of the equation that can be written more easily using a word processor (the results are exactly the same as using the more usual format of the equation and can be checked here):
dilated time = initial time x (1 – V^2/c^2)^(1/2) = 21600 x [1 – (244 ms^-1)^2 / (3 x 10^8 ms^-1)^2]^(1/2) = 21600 x 0.999 999 999 999 7 = 21599.999 999 99 s
So how much time has been saved? In other words, by how much has time been dilated? To answer this we simply subtract the dilated time from the time as observed by a stationary observer:
21600s  – 21599.999 999 99 s = 0.000 000 1 seconds
This is a tiny amount of time! Can this prediction be checked? Amazingly, yes it can, using atomic clocks. Atomic clocks work by monitoring the natural vibrational frequency of atoms and are very accurate. In 1971 two scientists, J. Hafele and R Keating, borrowed four atomic clocks from the U.S. Naval Observatory, put them on commercial airliners and flew them around the world. When compared with similar atomic clocks back in the U.S. they found that the clocks slowed down by the tiny, but very real amount predicted by Einstein’s time dilation equations. This experiment has since been carried out many times using ever more accurate atomic clocks. Each time the results have been in accordance with what Einstein said they would be. Time dilation, even at low speeds, is real.
Example 2: The lifetime of a pilot
In the previous example we looked at a single flight over the Atlantic Ocean and found that the time difference was very small. What happens, however, if we spend a large part of our lives making such journeys. Surely the time dilations all add up and we can live longer! This is only half true and depends on an individual's "frame of reference". In the first place anyone making such journeys would still feel time passing normally, so they may live longer according to an external observer but would still experience time passing the way everyone else does. In the second place, as we shall see, the amount of time "saved" is still very tiny.
Do aircrew live longer?
For this example we will look at an airline pilot. For simplicity let's say that our pilot spends his or her whole career on the Atlantic route, flying (on average) 25 hours a week for 40 years at an average speed of 550 mph (880 km/h). This is undoubtedly a lot of "high" speed travelling but how much time will our pilot "save" due to time dilation? As in the previous example we must first convert the figures we have into more suitable units:
25 hours x 40 years = 52,000 hours 550 mph = 244 metres per second
Note that we didn't have to convert the hours into seconds. Using hours will work just as well and our initial answer will also be in hours. Plugging these numbers into the time dilation equation gives us an answer of:
dilated time = initial time x (1 – V^2/c^2)^(1/2)
= 52000 hours x [1 – (244 ms^-1)^2 / (3 x 10^8 ms^-1)^2]^(1/2) = 52000 hours x 0.999 999 999 999 7 = 51599.999 999 984 4 hours
Subtracting this value from the value measured by an external observer gives us a total time "saved" of:
52000 hours – 51599.999 999 984 4 hours = 0.000 000 015 6 hours
This is perhaps better expressed as seconds, in which case we find that in a lifetime of flying our airline pilot saves a total of 0.000056 seconds as compared to an external observer. Again, this is measurable by atomic clocks, but not in the least bit noticeable to the pilot or anyone else. There must be a better way! Let's try going even faster and for much longer...
Example 3: The Voyager Project
In 1977 two remarkable spacecraft were launched by NASA, they were called Voyager 1 and Voyager 2. During the following decade these two small probes visited the largest planets in our Solar System and provided spectacular and breathtaking information about our local neighbourhood. Although expected to function and send back data until at least 2020 they have both long since left our planetary system and are heading out into deep space where, it is tentatively hoped, they may one day many millions of years from now be found by other inhabitants of our galaxy. Just in case they're found "gramophone" records have been attached to each spacecraft. These records contain information about us. Each contains greetings in many languages, such as English, Hindi and even Latin. They can also play music from many cultures, including everything from Bach to Australian Aborigine songs. A map has also been supplied. It tells anyone that finds it where we are and "when" we are. We have been beaming signals into space since at least the invention of television, but this is our first intentional solid message. If there really is someone out there we might just be able to say hello.
The Voyager Golden Record -- A greeting from Planet Earth
One of the most notable things about the two Voyager spacecraft is their speed. Voyager 1 is travelling at around 35,500 mph (57,500 km/h) and this is the craft we will concentrate on. The Earth is about 24,000 miles in circumference and so it would take much less than an hour for Voyager 1 to circumnavigate the whole planet. The fastest speed that a human being has so far travelled is during the journey to the Moon and back. To get to the Moon took about three days. For Voyager 1 it would take only seven hours. Apart from the Sun the nearest star to us is Alpha Proxima. This faint star is about 4.37 light years away from us. That is, travelling at 186,300 miles per second it would take 4.37 years to get there (as measured by Earth clocks!). At the speed Voyager 1 is travelling it would reach our nearest stellar neighbour in about 80,000 years. What would be the time dilation experienced? In other words, travelling at the speed it is how much younger will Voyager 1 be than an Earth-bound observer 80,000 years from now? In this case we will use the actual speed as opposed to the percentage of the speed of light that we have used before. This makes no difference to the calculations as long as we make sure that the units are the same, i.e. either both as an absolute speed or both as a percentage. As always, we have to convert the units to something we can easily work with:
Voyager 1 = 35,000 mph
= 57,000 km/h = 16,000 metres per second     (about 10 miles a second)
We know the (stationary) time is 80,000 years so now we have all the information we need to solve the time dilation equation:
dilated time = initial time x (1 – V^2/c^2)^(1/2) = 80000 years x [1 – (16000 ms^-1)^2 / (3 x 10^8 ms^-1)^2]^(1/2) = 80000 years x (0.9999999971556)^1/2 = 80000 years x 0.9999999985778 = 79999.99988622 years
We now subtract the measured time according to an external observer:
80000 years – 79999.99988622 years = 0.00011378 years
If we convert this to a more convenient time scale we find that it is just under one hour. After 80,000 years travelling through space at 35,500 mph Voyager 1 will be only about one hour "younger" than the Earth! In comparison to the speed of light Voyager 1, for all its outstanding contributions to the understanding of our Solar System, is very, very slow. So slow in fact that it's possible that our distant descendants could intercept the craft before any alien civilisation gets the chance. Indeed, Voyager 1 could turn out to be biggest find in the archaeology of the distant future. Personally, I hope it's left to go on its way.
Voyager -- Earth’s messenger
Amazon
Advertisement
Advertisement  
Advertisement  
The speed of time
One of the consequences of special relativity is that any moving clock slows down in accordance with a precise mathematical formula. The faster the clock is moving the more it slows down. The clock slows down for a very good reason; time itself, for anything moving, slows down relative to a stationary observer. This idea was put forward in 1905 by Albert Einstein and has since been tested many times. In particular, the accuracy of atomic clocks has allowed us to verify the effects of time dilation at even very modest speeds. This page gives examples of how time is slowed down at a variety of speeds, but concentrates on speeds that we can encounter as a matter of routine. The effects of time dilation only start to become apparent at speeds close to the speed of light (300,000 km per second, or 186,300 miles per second). For more detailed information on time dilation see this page, and you can find a time dilation calculator here.
A table of time
At the speeds we travel at in everyday life time dilation is so small that it's not detectable to all but atomic clocks. Even if you were to spend your whole life travelling in a fast modern jet your time dilation as measured by a stationary observer would still not even register on a digital watch, as we will see later in this page. While the equation dealing with time dilation isn't especially difficult it's perhaps more instructive to see some of the results listed as a table, such as in the one below. Here we see the percentage of the speed of light (c) against its dilated percentage, so that, for example, travelling at 90% of the speed of light time will slow down to 43.6% of its usual rate:
From the table it’s easy to see that time is only altered a little until we get to speeds above about 50% of that of light, and then the effect becomes ever more dramatic. Beyond 99% of the speed of light time slows down very rapidly indeed, as shown here:
So as we get ever closer to the speed of light time slows down until it almost stops. However, note that it’s not possible to reach 100% of the speed of light (as explained on this page) and so time never quite comes to a complete standstill. You can experiment with different percentages of by light using the Time Dilation Calculator here, all the way up to 99.99999999999999% of the speed of light.
Example 1: A flight over the Atlantic
Now let’s look at a some examples of dilation at relatively low speeds. For most of us the biggest and longest time dilation we ever experience is on a long flight, such as over the Atlantic Ocean. Such a flight takes about six hours and the average speed is about 550 mph (880 km/h). Now, 550 mph is undoubtedly very fast in human terms but the only change we have to make to a watch at the end of such a flight is to adjust it to the local time, and we don't need to alter it to take time dilation into account. Does this mean there is no time dilation during the flight? No! It's just that the time dilation, even at the speed of a jet, is so small that our everyday watches can't measure it. Remember, all  moving clocks run slowly, even slowly moving ones.
Six hours at 550 mph (880 km/h). Time dilation?
So how much time do we "save" on such a flight? To answer this we must first convert hours into seconds and miles per hour into metres per second. When we do this we get:
6 hours = 21,600 seconds 550 mph = 244 metres per second
The next thing we need to do is to "plug" these numbers into the time dilation equation and carry out the calculation. In doing so I will use a form of the equation that can be written more easily using a word processor (the results are exactly the same as using the more usual format of the equation and can be checked here):
dilated time = initial time x (1 – V^2/c^2)^(1/2) = 21600 x [1 – (244 ms^-1)^2 / (3 x 10^8 ms^-1)^2]^(1/2) = 21600 x 0.999 999 999 999 7 = 21599.999 999 99 s
So how much time has been saved? In other words, by how much has time been dilated? To answer this we simply subtract the dilated time from the time as observed by a stationary observer:
21600s  – 21599.999 999 99 s = 0.000 000 1 seconds
This is a tiny amount of time! Can this prediction be checked? Amazingly, yes it can, using atomic clocks. Atomic clocks work by monitoring the natural vibrational frequency of atoms and are very accurate. In 1971 two scientists, J. Hafele and R Keating, borrowed four atomic clocks from the U.S. Naval Observatory, put them on commercial airliners and flew them around the world. When compared with similar atomic clocks back in the U.S. they found that the clocks slowed down by the tiny, but very real amount predicted by Einstein’s time dilation equations. This experiment has since been carried out many times using ever more accurate atomic clocks. Each time the results have been in accordance with what Einstein said they would be. Time dilation, even at low speeds, is real.
In the previous example we looked at a single flight over the Atlantic Ocean and found that the time difference was very small. What happens, however, if we spend a large part of our lives making such journeys. Surely the time dilations all add up and we can live longer! This is only half true and depends on an individual's "frame of reference". In the first place anyone making such journeys would still feel time passing normally, so they may live longer according to an external observer but would still experience time passing the way everyone else does. In the second place, as we shall see, the amount of time "saved" is still very tiny.
Do aircrew live longer?
Example 2: The lifetime of a pilot
For this example we will look at an airline pilot. For simplicity let's say that our pilot spends his or her whole career on the Atlantic route, flying (on average) 25 hours a week for 40 years at an average speed of 550 mph (880 km/h). This is undoubtedly a lot of "high" speed travelling but how much time will our pilot "save" due to time dilation? As in the previous example we must first convert the figures we have into more suitable units:
25 hours x 40 years = 52,000 hours 550 mph = 244 metres per second
Note that we didn't have to convert the hours into seconds. Using hours will work just as well and our initial answer will also be in hours. Plugging these numbers into the time dilation equation gives us an answer of:
dilated time = initial time x (1 – V^2/c^2)^(1/2)
= 52000 hours x [1 – (244 ms^-1)^2 / (3 x 10^8 ms^-1)^2]^(1/2) = 52000 hours x 0.999 999 999 999 7 = 51599.999 999 984 4 hours
Subtracting this value from the value measured by an external observer gives us a total time "saved" of:
52000 hours – 51599.999 999 984 4 hours = 0.000 000 015 6 hours
This is perhaps better expressed as seconds, in which case we find that in a lifetime of flying our airline pilot saves a total of 0.000056 seconds as compared to an external observer. Again, this is measurable by atomic clocks, but not in the least bit noticeable to the pilot or anyone else. There must be a better way! Let's try going even faster and for much longer...
Example 3: The Voyager Project
In 1977 two remarkable spacecraft were launched by NASA, they were called Voyager 1 and Voyager 2. During the following decade these two small probes visited the largest planets in our Solar System and provided spectacular and breathtaking information about our local neighbourhood. Although expected to function and send back data until at least 2020 they have both long since left our planetary system and are heading out into deep space where, it is tentatively hoped, they may one day many millions of years from now be found by other inhabitants of our galaxy. Just in case they're found "gramophone" records have been attached to each spacecraft. These records contain information about us. Each contains greetings in many languages, such as English, Hindi and even Latin. They can also play music from many cultures, including everything from Bach to Australian Aborigine songs. A map has also been supplied. It tells anyone that finds it where we are and "when" we are. We have been beaming signals into space since at least the invention of television, but this is our first intentional solid message. If there really is someone out there we might just be able to say hello.
The Voyager Golden Record -- A greeting from Planet Earth
One of the most notable things about the two Voyager spacecraft is their speed. Voyager 1 is travelling at around 35,500 mph (57,500 km/h) and this is the craft we will concentrate on. The Earth is about 24,000 miles in circumference and so it would take much less than an hour for Voyager 1 to circumnavigate the whole planet. The fastest speed that a human being has so far travelled is during the journey to the Moon and back. To get to the Moon took about three days. For Voyager 1 it would take only seven hours. Apart from the Sun the nearest star to us is Alpha Proxima. This faint star is about 4.37 light years away from us. That is, travelling at 186,300 miles per second it would take 4.37 years to get there (as measured by Earth clocks!). At the speed Voyager 1 is travelling it would reach our nearest stellar neighbour in about 80,000 years. What would be the time dilation experienced? In other words, travelling at the speed it is how much younger will Voyager 1 be than an Earth-bound observer 80,000 years from now? In this case we will use the actual speed as opposed to the percentage of the speed of light that we have used before. This makes no difference to the calculations as long as we make sure that the units are the same, i.e. either both as an absolute speed or both as a percentage. As always, we have to convert the units to something we can easily work with:
Voyager 1 = 35,000 mph
= 57,000 km/h = 16,000 metres per second     (about 10 miles a second)
We know the (stationary) time is 80,000 years so now we have all the information we need to solve the time dilation equation:
dilated time = initial time x (1 – V^2/c^2)^(1/2)
= 80000 years x [1 – (16000 ms^-1)^2 / (3 x 10^8 ms^-1)^2]^(1/2) = 80000 years x (0.9999999971556)^1/2 = 80000 years x 0.9999999985778 = 79999.99988622 years
We now subtract the measured time according to an external observer:
80000 years – 79999.99988622 years = 0.00011378 years
If we convert this to a more convenient time scale we find that it is just under one hour. After 80,000 years travelling through space at 35,500 mph Voyager 1 will be only about one hour "younger" than the Earth! In comparison to the speed of light Voyager 1, for all its outstanding contributions to the understanding of our Solar System, is very, very slow. So slow in fact that it's possible that our distant descendants could intercept the craft before any alien civilisation gets the chance. Indeed, Voyager 1 could turn out to be biggest find in the archaeology of the distant future. Personally, I hope it's left to go on its way.
Voyager -- Earth’s messenger
Amazon