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file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/A Brief History “But it's turtles all the way down!” .. theories outlined above assume we are rational beings who are free to observe the universe as we want and to. Check out this -: A brief history of time. sidi-its.info sidi-its.info The theory of everything. Stephen Hawking was the Lucasian Professor of Mathematics at the University of Cambridge for thirty years and the recipient of numerous awards and honors.


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The Theory of Everything - Stephen Hawking, Physics, Astronomy, quality Science ebooks available for free download in Digital Format. Read "The Theory Of Everything" by Stephen Hawking available from Free. Because every raindrop is a HOPE ebook by Mansi Sharma, Sankalp Kohli. The Universe in a Nutshell ALSO A BLACK HOLES BY STEPHEN BRIEF AND HISTORY BABY HAWKING OF T I M E U N I V E R S E S AND O T H E R ESSAYS.

Stephen Hawking. T h e right of Stephen Hawking to be identified as the author of this work has been asserted in accordance with sections 77 and 78 of the Copyright Designs and Patents Act 1 9 8 8. A catalogue record for this book is available from the British Library. ISBN 0 5 9 3 0 4 8 1 5 6 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publishers. How this can he reconciled with quantum theory. Could an advanced civilization go back and change the past?

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I am aware I can opt out at any time. Overview Specs. From Rustam Soft: How can you tell if an Android app is safe? Please Wait. Add Your Review. You are logged in as. Instead, we should adopt the principle of p-brane democracy: All the p-branes could be found as solutions of the equations of supergravity theories in 10 or 11 dimensions.

While 10 or 11 dimensions doesn't sound much like the spacetime we experience, the idea was that the other 6 or 7 dimensions are curled up so small that we don't notice them; we are only aware of the remaining 4 large and nearly flat dimensions.

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Special cases are I must say that personally, I have been reluctant to believe in extra dimensions. But as I am a positivist, the question "Do extra mem- dimensions really exist? Often, some or all of the p-dimensions are curled up like a torus. We hold these truths to be All self-evident: The membranes can be seen better if they string curled up curled up into a torus are curled up.

The dualities suggest that the different string theories are just different expressions of the same underlying theory, which has been named M-theory. But what has convinced many people, including myself, that one should take models with extra dimensions seriously is that there is a web of unexpected relationships, called dualities, between the models.

These dualities show that the models are all essentially equivalent; that is, they are just different aspects of the same underlying theory, which has been given the name M-theory. These dualities show that the five superstring theories all describe the same physics and that they are also physically equivalent to supergravity Fig. One cannot say that superstrings are more fundamental than supergravity, or vice versa.

Rather, they are different expressions of the same underlying theory, each useful for calculations in different kinds of situations. Because string theo- Heterotic-0 Heterotic-E ries don't have any infinities, they are good for calculating what happens when a few high energy particles collide and scatter off M-theory unites the five string theories within a single theoretical each other.

However, they are not of much use for describing how the energy of a very large number of particles curves the universe or framework, but many of its prop- forms a bound state, like a black hole. For these situations, one erties have yet to be understood. It is this picture that I shall mainly use in what follows. The model has rules that determine the history in imaginary time in terms of the history in real time, and vice versa.

Imaginary time sounds like something from science fiction, but it is a well-defined mathematical concept: You can't have an imaginary number credit card bill. One can think of ordinary real numbers such as 1 , 2 , - 3. Imaginary numbers can then be represented as corresponding to positions on a vertical line: Thus imaginary numbers can be thought of as a new kind of number at right angles to ordinary real numbers. Because they are a mathematical construct, they don't need a physical realization; one can't have an imaginary number of oranges or an imaginary credit card bill Fig.

One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real.

All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons.

So what is real and what is imaginary? Is the distinction just in our minds? But the real time direction was distin- from the space directions because it guished from t h e three spatial directions; the world line or history increases only along the history of an observer unlike the space directions, of an observer always increased in t h e real time direction that is, which can increase or decrease along time always m o v e d from past to future , but it could increase or that history.

The imaginary time direc- decrease in any of t h e three spatial directions. In o t h e r words, one tion of quantum theory, on the other hand, is like another space direction, so can increase or decrease. On the o t h e r hand, because imaginary time is at right angles to real time, it behaves like a fourth spatial direction.

As one moves north, the circles of latitude at constant distances from the South Pole become bigger corresponding to the universe expanding with imaginary time. The universe would reach maximum size at the equator and then contract again with increasing imaginary time to a single point at the North Pole. Even though the universe would have zero size at the poles, these points would not be singularities, just as the North and South Poles on the Earth's surface are perfectly regular points.

This suggests that the origin of the universe in imaginary time can be a regular point in spacetime. Because all the lines of longitude meet at the North and South Poles, time is standing still at the poles; an increase of imaginary time leaves one on the same spot, just as going west on the North Pole of the Earth still leaves one on the North Pole.

Imaginary time as degrees of longitude which meet at the North and South Poles 61 T H E U N I V E R S E Information falling into black hole T h e area formula for the e n t r o p y — o r number of internal s t a t e s — o f a black hole suggests that information about what falls into a black hole may be stored like that on a record, and played back as the black hole evaporates.

It is in this imaginary sense that time has a shape. To see some of the possibilities, consider an imaginary time spacetime that is a sphere, like the surface of the Earth. Suppose that imaginary time was degrees of latitude Fig. T h e n the history of the universe in imaginary time would begin at the South Pole. It would make no sense to ask, " W h a t h a p p e n e d before the beginning? T h e South Pole is a perfectly regular point of the Earth's surface, and the same laws hold there as at other points.

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T h i s suggests that the b e g i n n i n g of the universe in imaginary time can be a regular point of spacetime, and that the same laws can hold at the beginning as in the rest of the universe. T h e quantum origin and evolution of the universe will be discussed in the next chapter. A n o t h e r possible b e h a v i o r is illustrated by taking imaginary time to be degrees of longitude on the Earth. All the lines of longitude meet at the N o r t h and S o u t h Poles Fig. T h i s is very similar to the way that ordinary time appears to stand still on the horizon of a b l a c k h o l e.

We have c o m e to r e c o g n i z e that this standing still of real and imaginary time either b o t h stand still or neither does means that the s p a c e t i m e has a temperature, as I discovered for black holes.

N o t o n l y does a b l a c k h o l e have a t e m perature, it also behaves as if it has a quantity called entropy. T h e entropy is a measure of t h e n u m b e r of internal states ways it c o u l d be configured on the inside that the black h o l e c o u l d have w i t h o u t looking any different to an outside observer, w h o can o n l y observe its mass, rotation, and c h a r g e.

It equals t h e area of the horizon of the black h o l e: Information a b o u t the quantum states in a region of spacetime may be s o m e h o w c o d e d on t h e boundary of the region, which has t w o dimensions less.

T h i s is like t h e way that a hologram carries a t h r e e - d i m e n s i o n a l image on a two-dimensional surface. T h i s is essential if we are to be able to predict the radiation that c o m e s out of black holes. If we can't do that, we won't be able to predict the future as fully as we t h o u g h t.

It seems we may live on a 3 - b r a n e — a four-dimensional three space plus o n e time surface that is the b o u n d a r y of a five-dimensional region, with the remaining dimensions curled up very small. T h e state of the world on a brane e n c o d e s what is h a p p e n i n g in the five-dimensional region. Is the universe actually infinite or just very large?

And is it everlasting or just long-lived? Isn't it presumptuous of us even to make the attempt? Despite this cautionary tale, I believe we can and should try to understand the universe.

The Universe in a Nutshell

We have already made remarkable progress Above: Etruscan vase painting, 6th century B. We don't yet have a c o m p l e t e picture, but this may not be far off. Hubble space telescope lens and mirrors being upgraded by a space shuttle mission. Australia can be seen below. Galaxies can have various shapes and sizes; they can be either elliptical or spiral, like our own Milky Way.

The dust in the spiral arms blocks our view of the universe in FIG. We find that the galaxies are distributed the outer region of the spiral Milky Way galaxy.

T h e stellar dust in the spiral arms blocks our view within the roughly uniformly throughout space, with some local concentra- plane of the galaxy but we have a tions and voids. The density of galaxies appears to drop off at very clear view on either side of that plane. As far as we can tell, the universe goes on in space forever see page 7 2 , Fig. Although the universe seems to be much the same at each position in space, it is definitely changing in time.

This was not realized until the early years of the twentieth century. Up to then, it was thought the universe was essentially constant in time. It might have existed for an infinite time, but that seemed to lead to absurd conclusions.

If stars had been radiating for an infinite time, they would have heated up the universe to their temperature. T h e observation that we have all made, that the sky at night is dark, is very important. It implies that the universe c a n n o t have existed forever in the state we see today.

S o m e t h i n g must have happ e n e d in the past to make the stars light up a finite time ago, which means that t h e light from very distant stars has not had time to reach us yet. T h i s would explain why the sky at night isn't glowing in every d i r e c t i o n. However, discrepancies with this idea b e g a n to appear with the observations by V e s t o S l i p h e r and Edwin H u b b l e in t h e s e c o n d decade o f the twentieth century. In T h e Doppler effect is also true of light order for them to appear so small and faint, the distances had to be so great that light from them would have taken millions or even billions of years to reach us.

This indicated that the beginning of the universe couldn't have been just a few thousand years ago. But the second thing Hubble discovered was even more remarkable.

Astronomers had learned that by analyzing the light from other galaxies, it was possible to measure whether they are moving toward us or away from us Fig. To their great surprise, they had found that nearly all galaxies are moving away. Moreover, waves. If a galaxy were to remain at a constant distance from Earth, characteristic lines in the spectrum would appear in a normal or standard position.

However, if the galaxy is moving away from us, the waves will appear elongated or stretched and the characteristic lines will be shifted toward the red right. If the galaxy is moving toward us then the waves will appear to be compressed, and the lines will be blue-shifted left. It was Hubble who recognized the dramatic implications of this discovery: The universe is expanding Fig.

The discovery of the expansion of the universe was one of the great intellectual revolutions of the twentieth century. It came as a total surprise, and it completely changed the discussion of the origin of the universe. If the galaxies are moving apart, they must have been closer together in the past. From the present rate of expansion, we can estimate that they must have been very close together indeed ten to fifteen billion years ago.

As described in the last chapter, Roger Penrose and I were able to show that Einstein's general theory of relativity implied that the universe and time itself must have had a beginning in a tremendous explosion.

We are used to the idea that events are caused by earlier events, w h i c h in turn are caused by still earlier events. W h a t caused it? T h i s was not a question that m a n y scientists w a n t e d to address. T h e y tried to avoid it, either by claiming, like t h e Russians, that t h e universe didn't have a b e g i n n i n g or by maintaining that t h e origin of the universe did not lie within the realm of s c i e n c e but b e l o n g e d to metaphysics or religion.

In my opinion, this is n o t a position a n y true scientist should take. We must try to understand the beginning of the universe on the basis of science.

It may he a task beyond our powers, hut we should at least make the attempt. W h i l e the t h e o r e m s that Penrose and I proved s h o w e d that the universe must have had a beginning, t h e y didn't give much information about the nature of that b e g i n n i n g.

T h e y indicated that the universe began in a big bang, a point where the w h o l e universe, and everything in it, was scrunched up into a single point of infinite density. At this point, Einstein's general t h e o r y of relativity would have broken down, so it c a n n o t be used to predict in what m a n n e r t h e universe began. O n e is left with the origin of the universe apparently being b e y o n d the scope of s c i e n c e.

T h i s was not a conclusion that scientists should be happy with. As Chapters 1 and 2 point out, the reason general relativity b r o k e down near the big bang is that it did not incorporate the uncertainty principle, the random element of quantum theory that Einstein had o b j e c t e d to on the grounds that G o d does not play dice.

However, all the evidence is that G o d is quite a gambler. You might think that operating a casino is a very c h a n c y business, because you risk losing m o n e y each time dice are thrown or the wheel is spun. But over a large number of bets, the gains and losses average out to a result that can be predicted, even though the result of any particular bet c a n n o t be predicted Fig.

T h e casino operators make sure the odds average out in their favor. T h a t is w h y casino operators are so rich. T h e o n l y c h a n c e you have of winning against them is to stake all your m o n e y on a few rolls of the dice or spins of the wheel.

It is the same with the universe. W h e n the universe is big, as it is today, there are a very large number of rolls of the dice, and the results FIG.

T h a t is why classical laws If a gambler bets on red for a large work for large systems. But when the universe is very small, as it was near in time to the big bang, there are only a small number of rolls of the dice, and the uncertainty principle is very important.

Because the universe keeps on rolling t h e dice to see what happens next, it doesn't have just a single history, as o n e m i g h t have t h o u g h t. Instead, t h e universe must have every possible history, e a c h with its own probability.

If t h e frontier of t h e universe was just at a normal point of space and time, we c o u l d go past it and claim t h e territory b e y o n d as part of the universe.

On the o t h e r hand, if the b o u n d a r y of the 80 number of rolls of the dice, one can fairly accurately predict his return because the results of the single rolls average out.

On the other hand, it is impossible to predict the outcome of any particular bet. However, a colleague named Jim Hartle and I realized there was a third possibility. M a y b e the universe has no boundary in space and time. At first sight, this seems to be in direct contradiction with the theorems that Penrose and I proved, which showed that the universe must have had a beginning, a boundary in time.

However, as explained in C h a p t e r 2, there is another kind of time, called imaginary time, that is at right angles to the ordinary real time that we feel going by.

In particular, the universe need have no beginning or end in imaginary time. Imaginary time behaves just like a n o t h e r direction in space. T h u s , the histories of the universe in imaginary time can be thought of as curved surfaces, like a ball, a plane, or a saddle shape, but with four dimensions instead of two see Fig. If the histories of the universe went off to infinity like a saddle or a plane, o n e would have the p r o b l e m of specifying w h a t the boundary c o n d i t i o n s were at infinity.

But o n e can avoid having to specify boundary c o n d i t i o n s at all if the histories of the universe in imaginary time are closed surfaces, like the surface of the Earth. T h e surface of the Earth doesn't have any boundaries or e d g e s. T h e r e are no reliable reports of p e o p l e falling off.

T h e universe would be entirely s e l f - c o n t a i n e d ; it wouldn't need wind the up anything clockwork outside and set to it going. Instead, e v e r y t h i n g in the universe would be d e t e r m i n e d by t h e laws of science and by rolls of the dice within the universe.

T h i s may sound presumptuous, but it is what I and m a n y o t h e r scientists believe. Even if the boundary condition of the universe is that it has no boundary, it won't have just a single history. It will have multiple histories, as suggested by Feynman. T h e r e will be a history in imaginary time corresponding to every possible closed surface, and each history in imaginary time will determine a history in real time.

T h u s we have a superabundance of possibilities for the universe. W h a t picks out the particular universe that we live in from the set of all possible universes? O n e point we can notice is that many of the possible histories of the universe won't go through the sequence of forming galaxies and stars that was essential to our own development. W h i l e it may be that intelligent beings can evolve without galaxies and stars, this seems unlikely.

The surface of the Earth doesn't have T h u s , the very fact that we exist as beings w h o can any boundaries or edges. Reports of ask the question " W h y is the universe the way it is?

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It implies it is o n e of the minority of histories that have galaxies and stars. On the far right are those open universes b that will continue expanding forever Those critical universes that are balanced between falling back on themselves and continuing to expand like cl or the double might inflation of c2 harbor intelligent life. Our own universe d is poised The double inflation could T h e inflation of our own universe to continue expanding for now.

M a n y scientists dislike the anthropic principle because it seems rather vague and does not appear to have much predictive power. But the anthropic principle can be given a precise formulation, and it seems to be essential when dealing with the origin of the universe. M - t h e o ry, described in C h a p t e r 2, allows a very large number of possible histories for the universe. M o s t of these histories are not suitable for the development of intelligent life; either they are empty, last for t o o short a time, are too highly curved, or w r o n g in some o t h e r way.

Yet according to Richard Feynman's idea of multiple histories, these uninhabited histories can have quite a high probability see page 8 4. In fact, it doesn't really matter h o w many histories there may be that don't contain intelligent beings. We are interested o n l y in the subset of histories in w h i c h intelligent life develops. T h i s intelligent life need not be anything like humans.

Little green aliens would do as well. In fact, t h e y might do rather better. T h e human race does not have a very g o o d record of intelligent behavior.

As an example of the power of the a n t h r o p i c principle, consider the number of directions in space. It is a matter of c o m m o n experience that we live in three-dimensional space. But w h y is space three-dimensional?

W h y isn't it two, or four, or some o t h e r number of dimensions, as in science fiction? In M-theory, space has nine or ten dimensions, but it is thought that six or seven of the directions are curled up very small, leaving three dimensions that are large and nearly flat Fig. W h y don't we live in a history in w h i c h eight of the dimensions are curled up small, leaving o n l y two dimensions that we n o t i c e?

If it had a gut that w e n t right through it, it would divide the animal in two, and the p o o r creature would fall apart. On the o t h e r hand, if there were four or m o r e nearly flat directions, the gravitational force b e t w e e n two bodies would increase m o r e rapidly as t h e y a p p r o a c h e d each other.

T h i s would mean that FIG. T h u s , although the idea of multiple histories would allow any n u m b e r of nearly flat directions, time that expands in an inflationary o n l y histories with three flat directions will contain intelligent manner.

O n l y in such histories will the question be asked, " W h y does space have three dimensions?

It determines a history of the universe in the real time that we experience, in which the universe is the same at every point of space and is expanding in time.

In these respects, it is like the universe we live in. But the rate of expansion is very rapid, and it keeps on getting faster. Such accelerating expansion is called inflation, because it is like the way prices go up and up at an ever-increasing rate. W h i l e t h e universe is inflating, matter could n o t fall 9! T h u s a l t h o u g h histories of t h e universe INFLATION in imaginary time that are perfectly round spheres are allowed by the notion of multiple histories, t h e y are not of m u c h interest.

However, histories in imaginary time that are slightly flattened at the south pole of the spheres are m u c h m o r e relevant Fig. In this case, the corresponding history in real time will expand in an accelerated, inflationary manner at first. But then the expansion 3. After July the phase of hyperinflation began. All confidence in money vanished and the price index rose faster and faster for will begin to slow down, and galaxies can form. In order for intelli- fifteen months, outpacing the printing gent life to be able to develop, the flattening at the S o u t h Pole must presses, which be very slight.

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T h i s will mean that the universe will expand initially could not produce money as fast as it was depreciating. By late , 3 0 0 paper mills were by an enormous amount.

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T h e record level of m o n e t a r y inflation working at top speed and printing occurred in G e r m a n y between the world wars, when prices rose bil- companies had 2, presses running lions of t i m e s — b u t the amount of inflation that must have occurred day and night turning out currency. Instead, the histories in imaginary time will be a w h o l e family of slightly deformed spheres, each of w h i c h corresponds to a history in real time in which the universe inflates for a long time but not indefinitely.

We can then Although slightly irregular histories ask w h i c h of these allowable histories is the most probable. It turns b and c are each less probable, out that t h e most p r o b a b l e histories are not c o m p l e t e l y smooth but there are such a large number of have tiny ups and downs Fig. T h e ripples on the most prob- them that the likely histories of the universe will have small departures from smoothness. T h e departures from smoothness are of the order of o n e part in a hundred thousand.

Nevertheless, although t h e y are e x t r e m e l y small, we have managed to observe them as small variations in the microwaves that c o m e to us from different directions in space. T h e C o s m i c Background Explorer satellite was launched in 1 9 8 9 and made a map of the sky in microwaves.

T h e different colors indicate different temperatures, but the whole range from red to blue is only about a ten-thousandth of a degree. So in principle, at least, the instrument, time. C O B E map is the blueprint for all the structures in the universe. T h e r e seem to be various possibilities, d e p e n d ing on the amount of matter in the universe. If there is m o r e than a certain critical amount, the gravitational attraction b e t w e e n the galaxies will slow them down and will eventually stop them from flying apart.

S o , again, things will c o m e to an end, but in a the big crunch in which all matter will be sucked back into a vast cataclysmic gravity well. Either way, the universe will last a g o o d few billion years m o r e Fig.

As well as matter, the universe may contain what is called "vac- FIG, 3. T h i s means that it has a gravitational effect on the expansion flicker their fuel. But, remarkably e n o u g h , the effect of vacuum energy is the opposite of that of matter. M a t t e r causes the expansion to slow down and can eventually stop and reverse it. On the other hand, vacuum e n e r g y causes the expansion to accelerate, as in inflation.

However, it may not have been a mistake at all. As described in C h a p t e r 2, we now realize that quantum t h e o r y implies that spacetime is filled with quantum fluctuations. In a supersymmetric theory, the infinite positive and negative energies of these g r o u n d state fluctuations c a n c e l out b e t w e e n particles of different spin. But we wouldn't e x p e c t the positive and negative energies to c a n c e l so c o m p l e t e l y that there wasn't a small, finite a m o u n t of vacuum energy left over, because the universe is not in a supersymmetric state.

M a y b e this is another example of the FIG. A history with a larger vacuum energy would not have formed galaxies, so would not contain beings w h o could ask the question: We can show the well estimated.

T h e dotted line shows t h e boundary of the region in w h i c h intelligent life could d e v e l o p Fig. Fortunately, all three regions have a c o m m o n intersection. If the matter density and vacuum energy lie in this intersection, it means that t h e expansion of the universe has begun to speed up again, after a l o n g period of slowing down.

It seems that inflation may be a law of nature. In this c h a p t e r we have seen h o w the b e h a v i o r of the vast universe can be understood in terms of its history in imaginary time, which is a tiny, slightly flattened sphere. It is like Hamlet's nutshell, yet this nut e n c o d e s e v e r y t h i n g that happens in real time.

So H a m l e t was quite right. We could be b o u n d e d in a nutshell and still count ourselves kings of infinite space. The of the complicated planets in apparent motion the can sky be explained by Newton's laws and has no influence on personal fortunes. T h a t is w h y astrology is so popular. A s t r o l o g y claims that events on Earth are related to the m o t i o n s of the planets across the sky.

T h i s is a scientifically testable h y p o t h e s i s , or would be if astrologers stuck their necks out and made definite p r e d i c t i o n s that c o u l d be tested. S t a t e m e n t s such as "Personal relations may b e c o m e intense" or "You will have a financially rewarding opportunity" can never be proved wrong.

W h e n C o p e r n i c u s and G a l i l e o discovered that the planets orbit the Sun rather than the Earth, and N e w t o n discovered the laws that govern their m o tio n, astrology b e c a m e e x t r e m e l y implausible.

W h y should the positions of o t h e r planets against the b a c k g r o u n d sky as seen from Earth have any correlations with the m a c r o m o l e c u l e s on a minor planet that call themselves intelligent life Fig.

Yet this is what astrology would have us believe. T h e success of Newton's laws and o t h e r physical theories led to the idea of scientific determinism, which was first expressed at the beginning of the nineteenth century by the French scientist the Marquis de Laplace.

Laplace suggested that if we knew the positions and velocities of all the particles in the universe at one time, the laws of physics should allow us to predict what the state of the universe would be at any o t h e r time in the past or in the future Fig. In o t h e r words, if scientific determinism holds, we should in principle be able to predict the future and wouldn't need astrology.

Of course, in practice even s o m e t h i n g as simple as Newton's theory of gravity produces e q u a t i o n s that we can't solve exactly for more than t w o particles. Furthermore, the equations often have a property known as c h a o s , so that a small c h a n g e in position or velocity at o n e time can lead to c o m p l e t e l y different behavior at later times.

As t h o s e w h o have seen Jurassic Park know, a tiny disturbance in o n e place can cause a m a j o r c h a n g e in another. T h e trouble is the sequence of events is not repeatable. T h e next time FIG. That is why weather forecasts are so unreliable. Thus, although in principle the laws of quantum electrodynamics should allow us to calculate everything in chemistry and biology, we have not had much success in predicting human behavior from mathematical equations.

Nevertheless, despite these practical difficulties most scientists have comforted themselves with the idea that, again in principle, the future is predictable.

At first sight, determinism would also seem to be threatened by the uncertainty principle, which says that we cannot measure accurately both the position and the velocity of a particle at the same time. The more accurately we measure the position, the less accurately we can determine the velocity, and vice versa. The Laplace version of scientific determinism held that if we knew the positions and velocities of particles at one time, we could determine their positions and velocities at any time in the past or future.

But how could we even get started if the uncertainty principle prevented us from knowing accurately both the positions and the velocities at one time? However good our computer is, if we put lousy data in, we will get lousy predictions out.

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A wave function is a number at each point of space that gives the probability that the particle is to be found at that position. The rate at which the wave function changes from point to point tells how probable different particle velocities are. Some wave functions are sharply peaked at a particular point in space.

In these cases, there is only a small amount of uncertainty in the position of the particle. That means the probability distribution for the velocity is spread over a wide range. In other words, the uncertainty in the velocity is large. Consider, on the other hand, a continuous train of waves. Now there is a large uncertainty in position but a small uncertainty in velocity.

So the description of a particle by a wave function does not have a well-defined position or velocity. It satisfies the uncertainty principle. We now realize that the wave function is all that can be well defined. We cannot even suppose that the particle has a position and velocity that are known to God but are hidden from us.

Such "hidden-variable" theories predict results that are not in agreement with observation. Even God is bound by the uncertainty principle and cannot know the position and velocity; He can only know the wave function.

The rate at which the wave function changes with time is given by what is called the Schrodinger equation Fig. Therefore, there is still determinism in quantum theory, but it is on a reduced scale. Instead measures of time, but we can use of being able to predict both the positions and the velocities, we can the Schrodinger equation in any of predict only the wave function.

This can allow us to predict either the these times to predict what the wave function will be in the future. Thus in quantum theory the ability to make exact predictions is just half what it was in the classical Laplace worldview. Nevertheless, within this restricted sense it is still possible to claim that there is determinism. However, the use of the Schrodinger equation to evolve the wave function forward in time that is, to predict what it will be at future times implicitly assumes that time runs on smoothly everywhere, forever.

This was certainly true in Newtonian physics. Time was assumed to be absolute, meaning that each event in the history of the universe was labeled by a number called time, and that a series of time labels ran smoothly from the infinite past to the infinite future. This is what might be called the commonsense view of time, and it is the view of time that most people and even most physicists have at the back of their minds.

However, in 1 9 0 5 , as we have seen, the concept of absolute time was overthrown by the special theory of relativity, in which time was no longer an independent quantity on its own but was just one direction in a four-dimensional continuum called spacetime.

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However, the spacetime of special rela- where time stood still. At these points, time would not increase in any direction. Therefore, one could not use the tivity is flat. This means that in special relativity, the time measured Schrodinger equation to predict what by any freely moving observer increases smoothly in spacetime the wave function will be in the future.