How to do an infinite number of things before breakfast
Itamar Pitowsky of the Hebrew University of
Jerusalem in Israel has shown that if the travelling twin can accelerate his
spaceship sufficiently strongly he can record a finite amount of the universe's
time on his own proper time clock while his twin brother, who is not
accelerating, records an infinite amount of proper time elapsing on his
clock.
How to do an infinite number of
things before breakfast
29 January 2005
From New Scientist Print Edition.
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John D. Barrow
John D. Barrow is a cosmologist at
the University of Cambridge and the author of
The Infinite
Book, published this week by
Jonathan Cape
Enlarge image
Supertask
YOU get up, shower, eat something, clean your teeth and leave the house. It doesn't sound like much, so why are you always running late? Is it because there's a physical limit to the number of things you can get done in a fixed amount of time?
Sadly, that's not going to wash as a reason for being late to work. In fact, the laws of physics suggest that your hurried morning routine is as nothing compared with what you could theoretically achieve before your day gets going. You can, in theory, do an infinite number of things before breakfast. And while research into this "supertasking" is unlikely to speed up your exit from the house, it is no idle speculation: it goes to the heart of modern physics. Build a machine that can do an infinite number of things in a finite time and you may have found a way to probe the fundamental structure of our universe. How's that for a reason to get out of bed?
Hermann Weyl, a contemporary of Einstein, was the first physicist to argue that the possibility of an "infinity machine" should be taken seriously. Although he didn't believe that such a thing could exist, neither did he manage to prove that it couldn't. In work published in 1949 he imagined a machine that would complete step one in ½ minute, step two in ¼ minute, step three in 1/8 of a minute, and so on for infinitely many steps. Although the idea has some bizarre consequences (See "The genie of the lamp") it is easy to show that such an infinite series of steps will be completed in a finite time - in this case, 1 minute.
Infinite disorder
Whether a machine such as Weyl's could really exist depends on a rather subtle argument to do with the second law of thermodynamics, which states that the entropy of a closed system - roughly a measure of its disorder - must always increase. Subdividing a timeline into an infinite number of steps is not the same as identifying an infinite number of physically distinct acts. Each of the acts processes information and so must generate entropy. Unless you can find some way for each step to generate sufficiently less entropy than its predecessor, Weyl's infinite number of steps will produce an entropy explosion.
Such practical considerations throw up enormous obstacles to every attempt to create an infinity machine, but let's press on with the principle: at first glance the infinity machine seems conceivable at least. So can you build one? In 1992 Jeff Xia of Northwestern University in Evanston, Illinois, showed that it might be possible. He imagined taking four particles of equal mass and arranging them in two pairs orbiting with equal but opposite spin in two parallel planes (see Diagram). He then introduced a fifth, much lighter particle that moves back and forth along the perpendicular through the mass centres of the two binary pairs. Xia showed that the system of particles will expand to infinite size in a finite time.
How does this happen? With each run back and forth between the two binary pairs, the oscillating particle joins one of the pairs in their gravitational dance, destabilising the configuration. Xia showed that the lighter particle gets ejected as a result, and the binary system recoils outwards to conserve momentum. The lighter particle then travels across to the other binary pair and the process repeats, progressively accelerating the binary pairs apart so strongly that the separation between the pairs and the distance travelled by the oscillating particle both become infinite in finite time, and the light particle undergoes an infinite number of oscillations in the process. These oscillations are performing an infinite number of physically distinct tasks in a finite time: a supertask. Although the setup is unrealistic in that the mass particles are infinitesimally small and the initial conditions are rather improbable, it nevertheless conserves energy and is an exact solution of Newton's laws of motion and gravitation.
Unfortunately (or perhaps fortunately), Einstein's theory of relativity forbids this behaviour. Einstein showed that no information can be transmitted faster than the speed of light and that gravitational forces cannot become arbitrarily strong - both of which would have to be violated in Xia's scenario (because of the zero size of the mass particles and the speeds at which they move). Nor can masses get arbitrarily close to each other, which means they can't recoil with arbitrarily high acceleration: when two objects of mass M get closer than a distance 4GM/c2, where G is Newton's gravitation constant and c is the speed of light, then a "horizon" of no return forms around them - a black hole. Their fate is then sealed.
But that does not mean that relativity forbids all infinity machines. Indeed, the theory actually opens up interesting new possibilities for supertasks because it reveals that observers experience time in a way that depends on their relative motion and the acceleration they experience. Could it be that an observer moving relative to a computer could see it perform an infinite number of computations, even though someone stationary with respect to the computer witnesses only a finite number?
“Researchers have come up with universes where supertasks are possible”
The answer to this hinges on the intricacies of the famous "twin paradox". In this scenario, one of a pair of twins stays at home on Earth while the other ventures away on a space flight at almost the speed of light. After a while the spacecraft decelerates, turns around and returns home at similarly high speed. Relativity predicts that the traveller will return to find his twin much older than himself. That is because, according to relativity, clocks accelerating or moving at high speed with respect to you appear to run slow. So, can we send the twin on so extreme a trip that he returns to find his stay-at-home twin's laptop has carried out an infinite number of computations?
At first glance, yes - the theory of relativity permits universes where this is possible. But the idea of a supertask is no longer quite so clear-cut. In whose time do we measure the "finite" time for the infinite number of tasks? The machine's or the observer's?
It seems only natural to require that the infinite number of tasks must be accomplished in a finite time as measured by a clock moving with the supertasking machine. We can call this its "proper" time, and we can call the infinite accomplishment a proper supertask. A pseudo-supertask, on the other hand, is when an observer sees a moving machine carry out an infinite sequence of actions in what appears (to the observer) to be a finite interval.
A pseudo-supertask such as the twin's computation certainly seems to be possible - in principle - without doing violence to the structure of space and time and the laws of relativity. Itamar Pitowsky of the Hebrew University of Jerusalem in Israel has shown that if the travelling twin can accelerate his spaceship sufficiently strongly he can record a finite amount of the universe's time on his own proper time clock while his twin brother, who is not accelerating, records an infinite amount of proper time elapsing on his clock. Pitowsky asks whether this state of affairs permits the existence of a "Platonist computer" - one that carries out an infinite number of operations along some trajectory through space-time and prints out an answer after a finite time as observed by someone else.
Alas, in this simple example the observer who measures the infinite history cannot have access to all the information that it contains - it just cannot reach him. That's because, in order to stay in touch with the computer and maintain the flow of information, the receiver also has to accelerate dramatically. Eventually the necessary g-forces become stupendous and tear him apart, no matter what he is made of.
Such practical issues are a common pitfall for simple attempts to use theoretical pseudo-supertasks to solve infinite problems: rendering them effective as proper supertasks would mean violating inescapable physical constraints. Suppose, for example, the twins grow up to become ambitious young mathematicians, determined to uncover the truth (or otherwise) of the famous unproven claim known as Goldbach's conjecture, which states that all even numbers greater than 2 are equal to the sum of two prime numbers. So far it has only been checked for even numbers no bigger than 16 digits long.
The travelling twin becomes fanatical in this quest and decides to sacrifice himself so that they can learn the truth. He takes a trip in his spaceship and steers it towards a black hole. The gravitational pull of the black hole will attract him inexorably, and accelerate him towards its centre. He knows that it will only take a finite amount of his proper time before he is torn to pieces by gravity, but the stay-at-home twin will see an infinite amount of his own proper time elapse before his brother is destroyed. Not only is this comforting in a fraternal sort of way, it would also (in theory) allow him to see the result of an infinite number of computer calculations being radioed to him from his brother's computer.
Even here we have a problem, though. The black hole prevents the information escaping the event horizon so, alas, this supertask is censored - unless, of course, there is a way for information to get out of the black hole ungarbled (New Scientist, 22 January, p 28).
However, a related scenario hints at what type of situation might permit a proper supertask to occur. Imagine the twins had both fallen through the black hole's event horizon. Although they would both end up being torn to pieces by the tidal forces of gravity near the centre of the black hole, one of them might have at least been able to send the required information to the other. If we now change the nature of the singularity inside the black hole, we can make this situation workable.
Imagine that, instead of a black hole, the twins are inside a "closed" universe that first expands and then contracts towards a "big crunch" at some time in the future. A space-time point of infinite density still looms, but in this scenario there is no barrier to a separate "outside". In 1986 Frank Tipler of Tulane University in New Orleans and I showed that, in this situation, an infinite amount of communicable information-processing can occur, and it is all accessible to anyone.
“Any attempt to transmit the output from a supertask will destroy the receiver”
This universe wobbles in shape infinitely often because of oscillating gravitational waves moving in different directions as space-time plummets towards the big crunch. Each wobble is a physically distinct operation that can process information, and the whole collapse to the crunch completes a supertask. And since the computer is the whole universe, we don't need it to send the information anywhere. The only hitch is information storage: if the universe is finite in space and time, an infinite number of bits of information requires some skilful handling.
Subsequent to our work, other researchers have now devised a systematic set of conditions that, if satisfied, make for universes that allow supertasks to be completed. Such universes have become known as Malament-Hogarth (MH) universes after the University of Chicago philosopher David Malament, and a former University of Cambridge research student, Mark Hogarth, who in 1992 investigated the conditions under which they were theoretically possible, and what level of computational complexity they admitted.
MH universes are intriguing mathematical possibilities, but their properties seem to suggest that if we want to supertask we'll need to give up certain cherished notions. The future in an MH universe, for example, is not uniquely and completely determined by the state of the universe at the present time: things can happen for no reason. Also worrying, but by no means disastrous, is the fact that time travel is permitted in some MH universes. And there are alarming prospects for observers: being able to perform a supertask means that any amount of radiation they emit, no matter how small, gets compressed to zero wavelength and amplified to infinite energy. Any attempt to transmit the output from a supertask will destroy the receiver.
This last problem in particular seems to rule out the possibility of an infinity machine that wouldn't destroy us in the process of reading its output. But the general idea of MH universes does open up a rather intriguing possibility. The collection of universes in which supertasks are possible includes the example of "anti de Sitter" space. This is something rather like our universe, but comes equipped with a force that pulls space-time into itself - the converse of Einstein's famous cosmological constant that seems to be stretching the space and time of our universe. We already know that anti de Sitter space plays a role in the string theories that may provide us with an ultimate "theory of everything".
So perhaps there's something profound about supertasking. Doing an infinite number of things before breakfast might just provide physicists with a way to explore the deep structure of the real world.
From issue 2484 of New Scientist magazine, 29 January 2005, page 28
Enlarge image
Supertask
YOU get up, shower, eat something, clean your teeth and leave the house. It doesn't sound like much, so why are you always running late? Is it because there's a physical limit to the number of things you can get done in a fixed amount of time?
Sadly, that's not going to wash as a reason for being late to work. In fact, the laws of physics suggest that your hurried morning routine is as nothing compared with what you could theoretically achieve before your day gets going. You can, in theory, do an infinite number of things before breakfast. And while research into this "supertasking" is unlikely to speed up your exit from the house, it is no idle speculation: it goes to the heart of modern physics. Build a machine that can do an infinite number of things in a finite time and you may have found a way to probe the fundamental structure of our universe. How's that for a reason to get out of bed?
Hermann Weyl, a contemporary of Einstein, was the first physicist to argue that the possibility of an "infinity machine" should be taken seriously. Although he didn't believe that such a thing could exist, neither did he manage to prove that it couldn't. In work published in 1949 he imagined a machine that would complete step one in ½ minute, step two in ¼ minute, step three in 1/8 of a minute, and so on for infinitely many steps. Although the idea has some bizarre consequences (See "The genie of the lamp") it is easy to show that such an infinite series of steps will be completed in a finite time - in this case, 1 minute.
Infinite disorder
Whether a machine such as Weyl's could really exist depends on a rather subtle argument to do with the second law of thermodynamics, which states that the entropy of a closed system - roughly a measure of its disorder - must always increase. Subdividing a timeline into an infinite number of steps is not the same as identifying an infinite number of physically distinct acts. Each of the acts processes information and so must generate entropy. Unless you can find some way for each step to generate sufficiently less entropy than its predecessor, Weyl's infinite number of steps will produce an entropy explosion.
Such practical considerations throw up enormous obstacles to every attempt to create an infinity machine, but let's press on with the principle: at first glance the infinity machine seems conceivable at least. So can you build one? In 1992 Jeff Xia of Northwestern University in Evanston, Illinois, showed that it might be possible. He imagined taking four particles of equal mass and arranging them in two pairs orbiting with equal but opposite spin in two parallel planes (see Diagram). He then introduced a fifth, much lighter particle that moves back and forth along the perpendicular through the mass centres of the two binary pairs. Xia showed that the system of particles will expand to infinite size in a finite time.
How does this happen? With each run back and forth between the two binary pairs, the oscillating particle joins one of the pairs in their gravitational dance, destabilising the configuration. Xia showed that the lighter particle gets ejected as a result, and the binary system recoils outwards to conserve momentum. The lighter particle then travels across to the other binary pair and the process repeats, progressively accelerating the binary pairs apart so strongly that the separation between the pairs and the distance travelled by the oscillating particle both become infinite in finite time, and the light particle undergoes an infinite number of oscillations in the process. These oscillations are performing an infinite number of physically distinct tasks in a finite time: a supertask. Although the setup is unrealistic in that the mass particles are infinitesimally small and the initial conditions are rather improbable, it nevertheless conserves energy and is an exact solution of Newton's laws of motion and gravitation.
Unfortunately (or perhaps fortunately), Einstein's theory of relativity forbids this behaviour. Einstein showed that no information can be transmitted faster than the speed of light and that gravitational forces cannot become arbitrarily strong - both of which would have to be violated in Xia's scenario (because of the zero size of the mass particles and the speeds at which they move). Nor can masses get arbitrarily close to each other, which means they can't recoil with arbitrarily high acceleration: when two objects of mass M get closer than a distance 4GM/c2, where G is Newton's gravitation constant and c is the speed of light, then a "horizon" of no return forms around them - a black hole. Their fate is then sealed.
But that does not mean that relativity forbids all infinity machines. Indeed, the theory actually opens up interesting new possibilities for supertasks because it reveals that observers experience time in a way that depends on their relative motion and the acceleration they experience. Could it be that an observer moving relative to a computer could see it perform an infinite number of computations, even though someone stationary with respect to the computer witnesses only a finite number?
“Researchers have come up with universes where supertasks are possible”
The answer to this hinges on the intricacies of the famous "twin paradox". In this scenario, one of a pair of twins stays at home on Earth while the other ventures away on a space flight at almost the speed of light. After a while the spacecraft decelerates, turns around and returns home at similarly high speed. Relativity predicts that the traveller will return to find his twin much older than himself. That is because, according to relativity, clocks accelerating or moving at high speed with respect to you appear to run slow. So, can we send the twin on so extreme a trip that he returns to find his stay-at-home twin's laptop has carried out an infinite number of computations?
At first glance, yes - the theory of relativity permits universes where this is possible. But the idea of a supertask is no longer quite so clear-cut. In whose time do we measure the "finite" time for the infinite number of tasks? The machine's or the observer's?
It seems only natural to require that the infinite number of tasks must be accomplished in a finite time as measured by a clock moving with the supertasking machine. We can call this its "proper" time, and we can call the infinite accomplishment a proper supertask. A pseudo-supertask, on the other hand, is when an observer sees a moving machine carry out an infinite sequence of actions in what appears (to the observer) to be a finite interval.
A pseudo-supertask such as the twin's computation certainly seems to be possible - in principle - without doing violence to the structure of space and time and the laws of relativity. Itamar Pitowsky of the Hebrew University of Jerusalem in Israel has shown that if the travelling twin can accelerate his spaceship sufficiently strongly he can record a finite amount of the universe's time on his own proper time clock while his twin brother, who is not accelerating, records an infinite amount of proper time elapsing on his clock. Pitowsky asks whether this state of affairs permits the existence of a "Platonist computer" - one that carries out an infinite number of operations along some trajectory through space-time and prints out an answer after a finite time as observed by someone else.
Alas, in this simple example the observer who measures the infinite history cannot have access to all the information that it contains - it just cannot reach him. That's because, in order to stay in touch with the computer and maintain the flow of information, the receiver also has to accelerate dramatically. Eventually the necessary g-forces become stupendous and tear him apart, no matter what he is made of.
Such practical issues are a common pitfall for simple attempts to use theoretical pseudo-supertasks to solve infinite problems: rendering them effective as proper supertasks would mean violating inescapable physical constraints. Suppose, for example, the twins grow up to become ambitious young mathematicians, determined to uncover the truth (or otherwise) of the famous unproven claim known as Goldbach's conjecture, which states that all even numbers greater than 2 are equal to the sum of two prime numbers. So far it has only been checked for even numbers no bigger than 16 digits long.
The travelling twin becomes fanatical in this quest and decides to sacrifice himself so that they can learn the truth. He takes a trip in his spaceship and steers it towards a black hole. The gravitational pull of the black hole will attract him inexorably, and accelerate him towards its centre. He knows that it will only take a finite amount of his proper time before he is torn to pieces by gravity, but the stay-at-home twin will see an infinite amount of his own proper time elapse before his brother is destroyed. Not only is this comforting in a fraternal sort of way, it would also (in theory) allow him to see the result of an infinite number of computer calculations being radioed to him from his brother's computer.
Even here we have a problem, though. The black hole prevents the information escaping the event horizon so, alas, this supertask is censored - unless, of course, there is a way for information to get out of the black hole ungarbled (New Scientist, 22 January, p 28).
However, a related scenario hints at what type of situation might permit a proper supertask to occur. Imagine the twins had both fallen through the black hole's event horizon. Although they would both end up being torn to pieces by the tidal forces of gravity near the centre of the black hole, one of them might have at least been able to send the required information to the other. If we now change the nature of the singularity inside the black hole, we can make this situation workable.
Imagine that, instead of a black hole, the twins are inside a "closed" universe that first expands and then contracts towards a "big crunch" at some time in the future. A space-time point of infinite density still looms, but in this scenario there is no barrier to a separate "outside". In 1986 Frank Tipler of Tulane University in New Orleans and I showed that, in this situation, an infinite amount of communicable information-processing can occur, and it is all accessible to anyone.
“Any attempt to transmit the output from a supertask will destroy the receiver”
This universe wobbles in shape infinitely often because of oscillating gravitational waves moving in different directions as space-time plummets towards the big crunch. Each wobble is a physically distinct operation that can process information, and the whole collapse to the crunch completes a supertask. And since the computer is the whole universe, we don't need it to send the information anywhere. The only hitch is information storage: if the universe is finite in space and time, an infinite number of bits of information requires some skilful handling.
Subsequent to our work, other researchers have now devised a systematic set of conditions that, if satisfied, make for universes that allow supertasks to be completed. Such universes have become known as Malament-Hogarth (MH) universes after the University of Chicago philosopher David Malament, and a former University of Cambridge research student, Mark Hogarth, who in 1992 investigated the conditions under which they were theoretically possible, and what level of computational complexity they admitted.
MH universes are intriguing mathematical possibilities, but their properties seem to suggest that if we want to supertask we'll need to give up certain cherished notions. The future in an MH universe, for example, is not uniquely and completely determined by the state of the universe at the present time: things can happen for no reason. Also worrying, but by no means disastrous, is the fact that time travel is permitted in some MH universes. And there are alarming prospects for observers: being able to perform a supertask means that any amount of radiation they emit, no matter how small, gets compressed to zero wavelength and amplified to infinite energy. Any attempt to transmit the output from a supertask will destroy the receiver.
This last problem in particular seems to rule out the possibility of an infinity machine that wouldn't destroy us in the process of reading its output. But the general idea of MH universes does open up a rather intriguing possibility. The collection of universes in which supertasks are possible includes the example of "anti de Sitter" space. This is something rather like our universe, but comes equipped with a force that pulls space-time into itself - the converse of Einstein's famous cosmological constant that seems to be stretching the space and time of our universe. We already know that anti de Sitter space plays a role in the string theories that may provide us with an ultimate "theory of everything".
So perhaps there's something profound about supertasking. Doing an infinite number of things before breakfast might just provide physicists with a way to explore the deep structure of the real world.
From issue 2484 of New Scientist magazine, 29 January 2005, page 28
Posted: Thu - May 26, 2005 at 08:25 PM
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