How does Einstein’s light clock explain time dilation?
Posted: Mon Sep 26, 2022 1:42 pm
Here's a thing I've been working on. Might submit it to the magazine:
The principle of relativity
Have you ever dropped something, a ball for instance? It’s a bit of a daft question, because who hasn’t? We’ve all done it and we all know that anything we drop will fall at our feet. We also know that if we jump straight up in the air, we land back where we started. It even happens if we are moving. If we drop a ball on a train, one that is travelling at 100mph say, the ball isn’t left behind to crash against the oncoming end of the carriage, and while your fellow passengers might think you a bit weird, if you jump in the air for no apparent reason, you can confidently expect to land on the same spot in your carriage, even though it’s further along the railway tracks from where you jumped. For extra fun, you can jump up and drop a ball at the same time and when your feet land on the same moving patch of lino they took off from, that’s where your ball will be. The reason being that everything in the carriage, the passengers, any balls they happen to be carrying, even the air is moving at the same speed as the train, and will continue to do so until something slows them down or makes them change direction. Think about riding a bike. If you stop peddling, you don’t immediately fall over, instead you keep freewheeling until the friction and air-resistance gradually bring you to a halt. Then you fall over.
Because the air is carried along in what is effectively a sealed bubble, that’s true even if you are riding your bike on a train. In fact whatever you do on the train, the result looks exactly the same as doing it on the platform. So when you drop a ball on a train, it drops to the ground in the same time it would take if it were dropped by a stranger standing on the platform. The difference is that whatever you drop, it carries on travelling at 100mph, so instead of falling straight down, the ball falls in an arc. Or at least that’s what it looks like to the stranger on the platform, watching as the train rattles past. To you on the train, your ball falls straight down, and instead it is the stranger’s ball that appears to fall in an arc. So who’s right? Well, we all know that the Earth is spinning around the North and South Poles. Wherever you are on Earth, it will take you 24 hours to spin around the poles. The circumference of Earth at the equator is 24,901 miles, so to go round the poles once in 24 hours means spinning at a little over 1000mph. The further from the equator, the more that speed drops and there are two latitudes where the Earth is spinning at 100mph. As it happens, they are so far North or South that they are in the Arctic Ocean or Antarctica respectively, but in principle there are two railways that could be built, one around each pole, on which the speed of the train is exactly matched by the speed of the Earth turning. In that case it’s tempting to think that, depending on which way round the train is going, either it’s moving at 200 mph, or not at all. Of course neither is true, because as the Earth spins on its axis, it orbits the sun in a galaxy that is cartwheeling through a universe that is expanding. With all the pirouettes Earth performs as it travels through space, how fast we, or any balls we drop, are really going is anyone’s guess. There isn’t some fixed universal grid we can mark our progress by; all we can measure is how fast we are moving relative to something else; some point or frame of reference - so 100mph is 100mph relative to a given point on the surface of the Earth. And so long as your speed remains constant, whatever you drop will land at your feet; it’s as if in your own little bubble it is the rest of the universe that is moving past beyond the window. This fact is known as the principle of relativity and in principle it doesn’t matter how fast you are going, everything in your bubble behaves and looks exactly the same as at any other speed.
The face in the mirror
The story goes that Albert Einstein was thinking about a possible exception. The speed of light was known to be 186 000 miles per second - fast enough to go round the Earth roughly seven and a half times in one second. It’s a staggering speed, but it is finite and unlike balls and people jumping up and down, it is unaffected by how fast the source is going. It is also the fastest that anything can move. Einstein wondered what would happen if we could travel at that speed. Suppose you were travelling at the speed of light; you might think this would be a good time to check yourself out in the mirror. So you hold your mirror out in the direction you are travelling. To your horror, your reflection has disappeared. Why? Well, if you are travelling at the speed of light, the light reflected from your face can’t reach the mirror to be reflected back. In other words, there would be an exception to the principle of relativity, a speed at which things look different. Except Einstein decided that isn’t what would happen. Instead, as you accelerated, time itself would progressively slow down until at the speed of light, time stops.
Special Theory of relativity
In mathematics, time takes on a life of its own - it becomes this abstract ‘thing’ we can add or subtract as easily as we might pour or store water in a jug. That flexibility is matched by our experience. We all have our perception of time and whether we feel time passing quickly or slowly depends on variables that are unique to each of us and every occasion. But however you feel time passing, the sun will set at exactly the time it was always going to, and that is because what we call time is determined by where the Sun is in the sky. A day is simply the Earth spinning until the Sun appears in the same place in the sky. That unit of time is not much good for tasks such as boiling an egg, so a day is divided into 24 hours, which in turn are divided into 60 minutes of 60 seconds. And it is those seconds that would slow down and stop in your bubble as you accelerated to the speed of light, or rather it is the things we count to mark seconds that would stop.
One of the ideas Einstein used to show what happens to ‘time’ is a device he made up called a light clock. 1. What this clock counts is a pulse of light that bounces up and down.
2. And to Einstein, who is moving at the same speed as the clock, that’s exactly what appears to happen.
3. But to someone watching the train go by, the light clock moves.
4. So instead of seeing the light travel as Einstein sees it, the stranger on the platform sees the light travel at an angle. Since the light pulse has travelled further, it must take longer.
However, the principle of relativity dictates that if the stranger has a clock, Einstein sees hers ticking as she sees his. In other words, both would see the other’s clock tick slower than their own and in the conditions of special relativity, that is true. The thing is, special relativity describes what you will see in one special circumstance. That circumstance being that two frames of reference, two bubbles, are moving at a constant velocity relative to each other. Velocity is speed in a straight line, so two bubbles moving at a constant relative velocity will move further and further apart once they have passed each other, unless and until at least one bubble changes direction; only then would it be possible to bring the clocks together and compare them to find out which one has ticked less.
Suppose instead of going in a straight line, Einstein was on a train that was going in a circle, on one of those hypothetical tracks near the poles for example. If then the stranger were standing at the pole, it would be obvious who was moving relative to who, because the stranger would see Einstein’s light beam take a diagonal route. By contrast Einstein could look out of his window and watch the stranger’s light pulse bounce straight up and down. In that case, he would see the stranger’s clock tick faster than his own.
This is not just an anomaly of light clocks; in 1971 scientists Joseph Hafele and Richard Keating, put atomic clocks on commercial airliners. With another clock in Washington DC as their standard, they flew around the world both westwards and eastwards. When they compared the clocks that went around the world with the one that stayed in Washington they found that by flying westwards clocks gain time, whereas flying eastwards, they lose it. That might seem weird, but it’s only from a point of view on the ground in Washington. In this idealised version, we can watch from a point above the North Pole. 1. The planes leave Washington at sunset.
2. By midnight both planes are the same distance from Washington, but at opposite sides of the North Pole.
3. At dawn the planes pass each other on the other side of the world to Washington. But while the plane flying east has gone round the North Pole, the plane flying west doesn’t appear to have moved.
4. Midday in Washington. The plane flying east is half way round the pole again and still the plane flying west is where it was at the beginning.
The planes land again at sunset in Washington. Seen from above the North Pole, the plane heading west has flown around the world without going anywhere, Washington DC has been around the world once, and the plane flying east has been round twice. If the planes were carrying light clocks, we would see the pulses of light stretched out in Washington and even more in the plane flying east, so from this point of view, the clock on the westward plane ticks at the same rate as any we might have with us. The clock in Washington ticks slower, but not as slowly as the eastward bound clock.
The actual differences in times were measured in billionths of a second, far too little for anyone to notice, but the principle of relativity states that even if we were travelling at speeds fast enough that the difference in time would be noticeable, we wouldn’t actually notice any difference in our own bubble. Nor would we, because every molecular and chemical reaction that is life and even consciousness depends on the interaction of particles in our bodies and brains. However much the path the pulse of light in a light clock is stretched, the movement of particles in our bodies and brains is stretched by exactly the same amount. So however much a clock we are travelling with slows, we slow down by the same amount. And if it were possible to travel at the speed of light, every particle of your being would be travelling at the speed of light, and since nothing can travel faster than light, there is no chance for any interaction between particles in your carriage, including those in your body and brain; nothing would happen in your bubble. The world would keep spinning, but for you time would stop.
The principle of relativity
Have you ever dropped something, a ball for instance? It’s a bit of a daft question, because who hasn’t? We’ve all done it and we all know that anything we drop will fall at our feet. We also know that if we jump straight up in the air, we land back where we started. It even happens if we are moving. If we drop a ball on a train, one that is travelling at 100mph say, the ball isn’t left behind to crash against the oncoming end of the carriage, and while your fellow passengers might think you a bit weird, if you jump in the air for no apparent reason, you can confidently expect to land on the same spot in your carriage, even though it’s further along the railway tracks from where you jumped. For extra fun, you can jump up and drop a ball at the same time and when your feet land on the same moving patch of lino they took off from, that’s where your ball will be. The reason being that everything in the carriage, the passengers, any balls they happen to be carrying, even the air is moving at the same speed as the train, and will continue to do so until something slows them down or makes them change direction. Think about riding a bike. If you stop peddling, you don’t immediately fall over, instead you keep freewheeling until the friction and air-resistance gradually bring you to a halt. Then you fall over.
Because the air is carried along in what is effectively a sealed bubble, that’s true even if you are riding your bike on a train. In fact whatever you do on the train, the result looks exactly the same as doing it on the platform. So when you drop a ball on a train, it drops to the ground in the same time it would take if it were dropped by a stranger standing on the platform. The difference is that whatever you drop, it carries on travelling at 100mph, so instead of falling straight down, the ball falls in an arc. Or at least that’s what it looks like to the stranger on the platform, watching as the train rattles past. To you on the train, your ball falls straight down, and instead it is the stranger’s ball that appears to fall in an arc. So who’s right? Well, we all know that the Earth is spinning around the North and South Poles. Wherever you are on Earth, it will take you 24 hours to spin around the poles. The circumference of Earth at the equator is 24,901 miles, so to go round the poles once in 24 hours means spinning at a little over 1000mph. The further from the equator, the more that speed drops and there are two latitudes where the Earth is spinning at 100mph. As it happens, they are so far North or South that they are in the Arctic Ocean or Antarctica respectively, but in principle there are two railways that could be built, one around each pole, on which the speed of the train is exactly matched by the speed of the Earth turning. In that case it’s tempting to think that, depending on which way round the train is going, either it’s moving at 200 mph, or not at all. Of course neither is true, because as the Earth spins on its axis, it orbits the sun in a galaxy that is cartwheeling through a universe that is expanding. With all the pirouettes Earth performs as it travels through space, how fast we, or any balls we drop, are really going is anyone’s guess. There isn’t some fixed universal grid we can mark our progress by; all we can measure is how fast we are moving relative to something else; some point or frame of reference - so 100mph is 100mph relative to a given point on the surface of the Earth. And so long as your speed remains constant, whatever you drop will land at your feet; it’s as if in your own little bubble it is the rest of the universe that is moving past beyond the window. This fact is known as the principle of relativity and in principle it doesn’t matter how fast you are going, everything in your bubble behaves and looks exactly the same as at any other speed.
The face in the mirror
The story goes that Albert Einstein was thinking about a possible exception. The speed of light was known to be 186 000 miles per second - fast enough to go round the Earth roughly seven and a half times in one second. It’s a staggering speed, but it is finite and unlike balls and people jumping up and down, it is unaffected by how fast the source is going. It is also the fastest that anything can move. Einstein wondered what would happen if we could travel at that speed. Suppose you were travelling at the speed of light; you might think this would be a good time to check yourself out in the mirror. So you hold your mirror out in the direction you are travelling. To your horror, your reflection has disappeared. Why? Well, if you are travelling at the speed of light, the light reflected from your face can’t reach the mirror to be reflected back. In other words, there would be an exception to the principle of relativity, a speed at which things look different. Except Einstein decided that isn’t what would happen. Instead, as you accelerated, time itself would progressively slow down until at the speed of light, time stops.
Special Theory of relativity
In mathematics, time takes on a life of its own - it becomes this abstract ‘thing’ we can add or subtract as easily as we might pour or store water in a jug. That flexibility is matched by our experience. We all have our perception of time and whether we feel time passing quickly or slowly depends on variables that are unique to each of us and every occasion. But however you feel time passing, the sun will set at exactly the time it was always going to, and that is because what we call time is determined by where the Sun is in the sky. A day is simply the Earth spinning until the Sun appears in the same place in the sky. That unit of time is not much good for tasks such as boiling an egg, so a day is divided into 24 hours, which in turn are divided into 60 minutes of 60 seconds. And it is those seconds that would slow down and stop in your bubble as you accelerated to the speed of light, or rather it is the things we count to mark seconds that would stop.
One of the ideas Einstein used to show what happens to ‘time’ is a device he made up called a light clock. 1. What this clock counts is a pulse of light that bounces up and down.
2. And to Einstein, who is moving at the same speed as the clock, that’s exactly what appears to happen.
3. But to someone watching the train go by, the light clock moves.
4. So instead of seeing the light travel as Einstein sees it, the stranger on the platform sees the light travel at an angle. Since the light pulse has travelled further, it must take longer.
However, the principle of relativity dictates that if the stranger has a clock, Einstein sees hers ticking as she sees his. In other words, both would see the other’s clock tick slower than their own and in the conditions of special relativity, that is true. The thing is, special relativity describes what you will see in one special circumstance. That circumstance being that two frames of reference, two bubbles, are moving at a constant velocity relative to each other. Velocity is speed in a straight line, so two bubbles moving at a constant relative velocity will move further and further apart once they have passed each other, unless and until at least one bubble changes direction; only then would it be possible to bring the clocks together and compare them to find out which one has ticked less.
Suppose instead of going in a straight line, Einstein was on a train that was going in a circle, on one of those hypothetical tracks near the poles for example. If then the stranger were standing at the pole, it would be obvious who was moving relative to who, because the stranger would see Einstein’s light beam take a diagonal route. By contrast Einstein could look out of his window and watch the stranger’s light pulse bounce straight up and down. In that case, he would see the stranger’s clock tick faster than his own.
This is not just an anomaly of light clocks; in 1971 scientists Joseph Hafele and Richard Keating, put atomic clocks on commercial airliners. With another clock in Washington DC as their standard, they flew around the world both westwards and eastwards. When they compared the clocks that went around the world with the one that stayed in Washington they found that by flying westwards clocks gain time, whereas flying eastwards, they lose it. That might seem weird, but it’s only from a point of view on the ground in Washington. In this idealised version, we can watch from a point above the North Pole. 1. The planes leave Washington at sunset.
2. By midnight both planes are the same distance from Washington, but at opposite sides of the North Pole.
3. At dawn the planes pass each other on the other side of the world to Washington. But while the plane flying east has gone round the North Pole, the plane flying west doesn’t appear to have moved.
4. Midday in Washington. The plane flying east is half way round the pole again and still the plane flying west is where it was at the beginning.
The planes land again at sunset in Washington. Seen from above the North Pole, the plane heading west has flown around the world without going anywhere, Washington DC has been around the world once, and the plane flying east has been round twice. If the planes were carrying light clocks, we would see the pulses of light stretched out in Washington and even more in the plane flying east, so from this point of view, the clock on the westward plane ticks at the same rate as any we might have with us. The clock in Washington ticks slower, but not as slowly as the eastward bound clock.
The actual differences in times were measured in billionths of a second, far too little for anyone to notice, but the principle of relativity states that even if we were travelling at speeds fast enough that the difference in time would be noticeable, we wouldn’t actually notice any difference in our own bubble. Nor would we, because every molecular and chemical reaction that is life and even consciousness depends on the interaction of particles in our bodies and brains. However much the path the pulse of light in a light clock is stretched, the movement of particles in our bodies and brains is stretched by exactly the same amount. So however much a clock we are travelling with slows, we slow down by the same amount. And if it were possible to travel at the speed of light, every particle of your being would be travelling at the speed of light, and since nothing can travel faster than light, there is no chance for any interaction between particles in your carriage, including those in your body and brain; nothing would happen in your bubble. The world would keep spinning, but for you time would stop.
