Here is a list of free math books.Yes free free math books.........

Professor Jim Herod's Multivariable Calculus

Calculus,by Gilbert Strang is made available through MIT's OpenCourseWare electronic publishing initiative.

Linear Methods of Applied Mathematics, by Evans Harrell and James Herod.

Yet another one produced at Georgia Tech is Linear Algebra, Infinite Dimensions, and Maple, by James Herod.

One more recent one by Professor Herod is Partial Differential Equations.

Complex Analysis, Complex Variables , by Robert Ash and W. P. Novinger. This is a substantial revision of the first edition of Professor Ash's complex variables text originally published in 1971.

Professor E.H. Connell of the University of Miami has made available on the web his book Elements of Abstract and Linear Algebra. An introductory algebraic topology book, Algebraic Topology I, by Professor Allen Hatcher, of Cornell University, is available, and Professor Hatcher promises the second volume, Algebraic Topology II, will be ready soon.

The Geometry and Topology of Three-Manifolds, by William Thurston. This is an electronic edition of the 1980 lecture notes distributed by Princeton University.

Professor Jim Hefferon of Saint Michaels's College has made available his undergraduate textbook Linear Algebra.

Another elementary linear algebra textbook is Elementary Linear Algebra, by Keith Matthews.

Introduction to Probability, by Charles Grinstead & J. Laurie Snell.

An Introduction to Probability and Random Processes, by Gian-Carlo Rota and Kenneth Baclawski. This is the 1979 manuscript of the work Professor Rota had been working on for some time. It is made available through the efforts of David Ellerman.

Professor Herbert Wilf (and the publisher, A. K. Peters) has made available his book generatingfunctionology.

Perhaps the greatest textbook of them all is Euclid's Elements.

Originally published by Springer-Verlag, the book A Course in Universal Algebra, by Stanley Burris, and H. P. Sankappanavar, is available online.

Professor Robert Ash has written and made available Abstract Algebra:The Basic Graduate Year.

Another one by Professor Ash is A Course In Algebraic Number Theory.

Professor Ash has also completed and made available A Course in Commutative Algebra.

The Calculus Bible is an elementary calculus textbook from Professor G. S. Gill of the Brigham Young University Mathematics Department.

Calculus Without Limits, by John C. Sparks.

Originally published by Prindle, Weber & Schmidt but currently out of print, Elementary Calculus: An Approach Using Infinitesimals, by Professor H. Jerome Keisler, is now freely available online.

Handbook of Applied Cryptography, by Alfred J. Menezes , Paul C. van Oorschot, and Scott A. Vanstone, is freely available thanks to the publisher, CRC Press.

Graph Theory, by Reinhard Diestel.

Available for self-study from The Trillia Group is Basic Concepts of Mathematics, by Elias Zakon.

Another one from The Trillia Group is An Introduction to the Theory of Numbers by Leo Moser.

Yet another from The Trillia Group is Mathematical Analysis I, by Elias Zakon.

Thanks to Malaspina Great Books, Mechanism of the Heavens (1831), by Mary Somerville, is available online. This second edition was prepared by Russell McNeil.

Lecture Notes on Optimization, by Pravin Varaiya. This is a re-issue of a book out of print since 1975. It is an introduction to mathematical programming, optimal control, and dynamic programming.

A Manual of Mathematical Ilustration, by Bill Casselman, shows, at several levels of sophistication, how to use PostScript for producing mathematical graphics.

Chebyshev and Fourier Spectral Methods (2nd. Edition), by John P. Boyd. This is the free online version of the Dover 2001 edition.

A Problem Course in Mathematical Logic, by Stefan Bilaniuk .

Concepts and Applications of Inferential Statistics, by Richard Lowry.

To be published soon by Cambridge University Press, A Computational Introduction to Number Theory and Algebra, by Victor Shoup will nevertheless remain freely available on-line.

Out of print for sometime, but freely available is Graph Theory with Applications, by J. A. Bondy and U. S. R. Murty.

Yet another one out of print, but now freely available is Convergence of Stochastic Processes, by David Pollard.

Designed for undergraduate physics students is Mathematical Tools for Physics, by James Nearing.

Elementary Number Theory, by William Stein.

A Modern Course on Curves and Surfaces, by Richard Palais.

A First Course in Linear Algebra, by Rob Beezer.

Group Theory, by Pedrag Civitanovic.

Shlomo Sternberg has written Theory of Functions of a Real Variable.

Lie Algebras Semi-Riemann Geometry and General Relativity

Advanced Calculus, by Lynn Loomis and Schlomo Sternberg

Originally published by Springer-Verlag and now out of print, Non-Uniform Randon Variate Generation, by Luc Devroye is now, thanks to the author, freely available.

Difference Equations to Differential Equations, by Dan Sloughter.

The Calculus of Functions of Several Variables is another one by Professor Sloughter.

Notes on Differential Equation, by Bob Terrell.

Sets, Relations, Functions, by Ivo Düntsch and Günther Gediga.

Another one by Düntsch and Gediga is Rough Set Data Analysis.

Predicative Arithmetic, by Edward Nelson.

Toposes, Triples and Theories, by Michaele Barr and Charles Wells.

Information Theory, Inference, and Learning Algorithms, by David J. C. MacKay is published by Cambridge University Press, but is, nevertheless, freely available online.

Linear Partial Differential Equations and Fourier Theory , by Marcus Pivato.

Another one by Professor Pivato is Voting, Arbitration, and Fair Division: The Mathematics of Social Choice.

Introduction to Vectors and Tensors, Volume 1, Linear and Multilinear Algebra, and Introduction to Vectors and Tensors, Volume 2, Vector and Tensor Analysis by Ray M. Bowen and C.-C.Wang, are revised versions of books originally published by Plenum Press in 1976.

Another one by Professor Bowen and originally published by Plenum Press is Introduction to Continuum Mechanics for Engineers.

Numerical Methods and Analysis for Engineers, by Douglas Wilhelm Harder.

Analysis of Functions of a Single Variable, by Lawerence Baggett, was originally written to be used for a one semester senior course, but the author suggests that it is more appropriate for first year graduate students.

Convex Optimization, by Stephen Boyd, and Lieven Vandenberghe is freely available thanks to Cambridge University Press.

Mathematics Under the Microscope, by Alexandre Borovik, is, according to the author, an attempt "to start a dialogue between mathematicians and cognitive scientists."

Introduction to Statistical Signal Processing, by R. M. Gray and L. D. Davisson is, according to Professor Gray, a "...much revised version of the earlier text Random Processes: An Introduction for Engineers, Prentice-Hall, 1986, which is long out of print." The current book is published by Cambridge University Press.

Not simply an online textbook, but certainly in the same spirit is the Topology Webcourse project undertaken by Topology Atlas.

This is, I suppose, not a textbook, but nevertheless an interesting reference: The Matrix Cookbook, by Kaare Brandt Petersen, and Michael Syskind Pedersen.

Not really a textbook either, Constants, by Steven Finch, is, nevertheless, a nice collection of essays. The title pretty much describes the subject.

So you got your math books........

## Friday, June 1, 2007

### Free math books

## Sunday, May 20, 2007

## Thursday, May 10, 2007

## Friday, May 4, 2007

## Tuesday, May 1, 2007

## Monday, April 9, 2007

### A new way to test time travel

When i was searching for time travel at google i got an article at newscientist titeled

THE title of Heinrich Päs's latest paper might not

mean much to you. To

those who know their theoretical physics, however,

"Closed timelike

curves in asymmetrically warped brane universes"

contains a revelation.

It suggests that time machines might be far more

common than we ever

thought possible.

Forget trawling the universe in search of rotating

black holes or exotic

wormhole tunnels that could supposedly let us hop

from one instant to

another. According to Päs, a physicist at the

University of Hawaii at

Manoa, and his colleagues, the door to a time

machine could be anywhere

and everywhere in our universe. And unlike most

other scenarios for time

travel, we can test this one here on Earth. "I

think the ideas presented

are wonderful and exciting," says Bill Louis,

a physicist at Los Alamos

National Laboratory in New Mexico and co-spokesperson

for the MiniBoone

neutrino experiment at Fermilab, near Chicago. "The

question is are they

true or not."

Louis is right to be cautious. Although nothing in

the laws of nature

appears to rule out time travel, physicists have

always been uneasy

about it because it makes a mockery of causality,

the idea that cause

always precedes effect. Violating causality would

play havoc with the

universe, for instance, allowing you to travel back

in time and prevent

your own birth.

Such paradoxes are what led Stephen Hawking to

propose his "chronology

protection conjecture". This basically says that

some principle of

physics, perhaps as yet undiscovered, will always

come to the rescue and

prevent time travel from happening. Yet no one

had been able to flesh

out the details until three years ago when several

groups of researchers

claimed that string theory, physicists' best stab

at a prospective

"theory of everything", was beginning to close the

door on time machines

(New Scientist, 20 September 2003, p 28).

All very well, except theoretical physicists are

notorious for never

taking no for an answer. Päs and his colleagues

Sandip Pakvasa of the

University of Hawaii and Thomas Weiler of Vanderbilt

University in

Nashville, Tennessee, have been re-examining

string theory. It views the

fundamental building blocks of the universe not

as point-like particles

but vibrating strings of energy; the faster the

vibration the greater

the mass of the particle.

Such vibrating strings can account for the myriad

interactions of all

the known subatomic particles, such as quarks and

electrons. But there

is a catch - it only works if the strings vibrate

in a space-time with

10 dimensions rather than the four with which we

are familiar.

Proponents of the idea maintain that the extra

dimensions are either so

fantastically small that we have not noticed them,

or so large and

warped in such a way that, again, they have remained

hidden from view.

This has led to the suggestion that our universe

may be like a

four-dimensional membrane or "brane" adrift in

a higher-dimensional

space-time. All of the particles and forces in

our universe would be

trapped in our brane like flies on fly-paper,

so we would have no

knowledge of any dimensions other than the four

we experience, even

though our brane might be floating in a 10-dimensional

space-time, or

"bulk". "If it is, then there is the possibility of

short cuts through

higher-dimensional space," says Päs. "It's such short

cuts that make

time travel possible."

?Time machines might be far more common than we

ever thought possible?

It is not too difficult to visualise such a short

cut. Suppose our

brane-universe is bent back on itself within a

large extra dimension,

making it the four-dimensional equivalent of a

pancake folded in two.

Then you could imagine leaving the brane at one

point, travelling a

short distance through the bulk and re-entering

the brane at a point far

away from your starting point.

There is a problem with this picture, however.

Although we can visualise

a universe where such a short cut is possible,

it cannot be our

universe. That is because the space-time of such

a severely folded brane

is not compatible with Einstein's special theory

of relativity, which

posits a "Euclidean geometry" where space is

perfectly flat. Since

numerous tests of special relativity have shown

that its predictions in

our locality are accurate to better than 1 part

in a million, it is very

unlikely that our universe is shaped like a folded pancake.

Instead Päs, Pakvasa and Weiler consider a space-time

where our universe

is a flat brane that is immersed in a bulk whose own

dimensions are

seriously warped. Because the brane is flat, special

relativity still

applies there. Yet in the bulk, Päs, Pakvasa and

Weiler have found that

the large dimensions can be distorted in such a

way that special

relativity does not apply within them. This means

that anything moving

through the fifth dimension can break one of the

founding principles of

special relativity: it can travel faster than the

speed of light as we

know it.

This has dramatic consequences for inhabitants stuck

on the brane. To

them, any entity that takes a short cut through the

bulk appears to

vanish and then pops up again at some point on the

brane far sooner than

it could have had it kept to the brane. For some

inhabitants of the

brane world, the entity appears to have travelled

faster than the speed

of light. Weirder still, to others it has also

travelled backwards in

time. That's because special relativity says that

from certain frames of

reference, faster-than-light travel is equivalent

to travelling

backwards in time. "Such off-brane short cuts can

appear as 'closed

timelike curves'," says Päs - again, that's code

for time machines.

Escape from the brane

The trouble with this idea is that it assumes there

is some way to

escape the confines of the brane and travel out into

the bulk. How could

we do this? Fortunately string theory provides a way

out. In the theory

almost all of the building blocks of matter are

represented by strings

whose ends are forever anchored to the brane.

This means they can never

escape into the fifth dimension and take a short

cut through space-time.

But there are two crucial exceptions: the hypothetical

carrier of the

gravitational force, called the graviton, and a fourth

type of neutrino

called a sterile neutrino (after the three ordinary

kinds of neutrino).

In string theory these are represented by closed loops

of string. Since

they have no ends attached to the brane, they are free

to leave and

travel into the bulk.

String theorists have pointed to this property of

gravitons to explain

why gravity is tremendously weaker than nature's

other fundamental

forces, such as electromagnetism. The idea is that

gravity is so weak

because a large proportion of gravitons leak away

into the extra

dimensions of the bulk. More intriguingly, however,

their ability to

take short cuts through the bulk also means gravitons

and sterile

neutrinos are potential time travellers. "If we can

manipulate them, we

can study time travel experimentally," says Päs.

None of this will be easy. No one has ever spotted a

graviton or a

sterile neutrino, and the odds of detecting them are

slim, to say the

least (New Scientist, 18 March, p 32). Trillions of o

rdinary neutrinos

pass through our bodies every second, yet we feel

nothing because they

so rarely interact with electrons and atoms.

Sterile neutrinos are even

less communicative because they are thought to

interact only via the

feeble gravitational force and the exchange of

the elusive Higgs boson -

an as yet undetected particle believed to endow

all particles with mass.

Päs and his colleagues point out that a quirk

of quantum mechanics could

save the day. According to the laws of quantum

physics, neutrinos can

flip from one kind to another. Experiments in

Japan and the US designed

to detect neutrinos on the rare occasions they

do interact with matter

have confirmed that neutrinos spewed out by the

sun and those from space

do indeed change type. This phenomenon should

affect sterile neutrinos

too, changing them into ordinary, detectable

neutrinos and back again.

What's more, the odds of this happening increase

whenever the density of

the material the neutrinos are travelling through

changes abruptly.

?A quirk of quantum mechanics could allow us to test

time travel right

here on Earth?

This has inspired Päs and his colleagues to propose

an experiment that

could test their ideas. They suggest sending a

beam of ordinary

neutrinos through the Earth, from a research station

at the South Pole

towards a detector located at the equator. When they

enter the ground,

some of the neutrinos will flip into sterile neutrinos.

Capable of

taking a short cut though the extra dimensions of the

bulk, these

sterile neutrinos will reach the other side of the

Earth first,

apparently having travelled faster than light. As

they pass out of the

ground into the air again, they will flip back into

ordinary neutrinos,

which can be detected. Because the Earth is rotating,

these

faster-than-light neutrinos can appear to have arrived

before they set off.

Such an experiment is beyond our current technological

capabilities but,

remarkably, Päs says it is a realistic proposition within

the next 50

years. Of course, it requires two things. The first is the

existence of

sterile neutrinos. While many physicists are keen on the

idea of sterile

neutrinos, they are barely beyond theoretical flights of

fancy. The

other is that we live in an asymmetrically warped space-time,

as Päs

prescribes. How plausible is this?

When Einstein's came up with his general theory of

relativity, he showed

us how space-time can be warped or flat, but his

equations tell us

nothing about the actual shape of our universe -

merely that different

shapes are possible. For instance, cosmologists have

no way of knowing

if space stretches out to infinity or curves back on

itself. This opens

the door to many different types of time machines,

some more plausible

than others.

One famous solution to Einstein's equations, formulated

by mathematician

Kurt Gödel, describes a universe that rotates rapidly.

Instead of

travelling in straight lines, light will appear to

travel in a spiral.

Gödel realised that this allows a traveller to outrun

light and return

to their starting point before they left. In other

words, Gödel's

rotating universe is a time machine. "But we know we

don't live in such

a universe," says Päs.

Another time machine exists inside rotating

black holes, where

space-time becomes so warped that space and time

change places. The

trouble is, as Päs points out, rotating black holes

are inaccessible to

us. Then there is the space-time surrounding an

infinitely long, rapidly

rotating cylinder, as proposed by physicist Frank

Tipler. Päs is quick

to dismiss it too. "It requires huge masses rotating

unphysically fast,"

he says.

Among the other leading contenders are wormholes,

microscopic tubes of

space-time that act as tunnels from one point to

another. But before you

climb into one, there is a problem: wormholes snap

shut in an instant

unless propped open by a supply of something called

exotic matter.

Unlike the familiar stuff found on Earth, which always

feels the pull of

gravity, this exotic matter has repulsive gravity,

halting the

wormhole's collapse. "We don't know whether such matter

exists and if it

is stable," says Päs.

Päs confesses that the scenario his team has examined

also requires

exotic matter to warp the fifth dimension, but he

still maintains that

it is more plausible than the other scenarios.

What sets their

space-time apart from wormholes is that the hypothetical

exotic matter

is hidden away in the higher-dimensional bulk instead

of roaming around

the brane. If it exists, this might explain why we have

never seen it.

Understandably, the idea is not without its critics.

Sydney Deser of

Brandeis University in Waltham, Massachusetts, is

convinced, as was

Einstein, that time machines are not possible and

does not like

"unphysical" exotic matter. He believes that concealing

it in the bulk,

as Päs's team suggests, is little better than a scenario

in which it is

out in the open. "It's only a matter of degree," he says.

Päs, however, points out that the kind of space-times

he and his

colleagues have considered can do away with a number

of problems that

have plagued general relativity. For instance,

faster-than-light

connections between far-flung parts of the cosmos

would have allowed

heat to flow back and forth across the early universe.

This would have

evened out any temperature variations, explaining the

uniformity that

cosmologists observe. This could provide an alternative

to the theory of

inflation, in which cosmologists believe space-time was

stretched

unimaginably fast after the big bang, and which would

also account for

the evenness in temperature. While the majority of

cosmologists believe

in inflation, no one has explained the detailed

physics behind it.

Others do not find asymmetrically warped space-times

so plausible. "I

certainly think the idea is interesting, but I have

some worries," says

Tony Padilla of the University of Barcelona in Spain.

"For a start, I

think it is premature to claim that these space-times

are 'natural'. One

needs to examine their stability first, and in this

case I would expect

the solution to be unstable, although I could be wrong."

Padilla concedes that it is possible that we may one

day find a stable

brane universe with the properties described by

Päs's team. "I'm just

not convinced we are there yet," he says.

John Cramer of the University of Washington in

Seattle agrees that the

work outlines some interesting ideas. "The scheme,

however, requires

asymmetrically warped brane universes - and our

universe may not be one

of these," he cautions. "Nevertheless, it's a

fascinating proposal."

Of course, if time travel is possible in the way

Päs envisages, it may

be accessible only to special particles like sterile

neutrinos and

gravitons, and therefore won't cause much havoc in

the everyday

universe. Päs takes a pragmatic view of all this.

As long as the

possibility of time machines remains, he believes it

is worth exploring

experimentally. "Even if time travel is not possible,

by manipulating

particles like sterile neutrinos we can explore the

physics that

intervenes to prevent it," he says.

?Anything travelling through the fifth dimension

can move faster than

the speed of light as we know it?

The first answers might come soon courtesy of the

MiniBoone neutrino

experiment at Fermilab. It could confirm the existence

of sterile

neutrinos and short cuts in extra dimensions as early as

this year. Then

again, if time travel really is true, maybe the answer

has already been

published.

From issue 2552 of New Scientist magazine, 22 May 2006,

page 34

At last, a way to test time travel

But now that is not available.I suddenly found it at

http://bama.ua.edu/~physics/news/pas.txt

`Here is the article:`

THE title of Heinrich Päs's latest paper might not

mean much to you. To

those who know their theoretical physics, however,

"Closed timelike

curves in asymmetrically warped brane universes"

contains a revelation.

It suggests that time machines might be far more

common than we ever

thought possible.

Forget trawling the universe in search of rotating

black holes or exotic

wormhole tunnels that could supposedly let us hop

from one instant to

another. According to Päs, a physicist at the

University of Hawaii at

Manoa, and his colleagues, the door to a time

machine could be anywhere

and everywhere in our universe. And unlike most

other scenarios for time

travel, we can test this one here on Earth. "I

think the ideas presented

are wonderful and exciting," says Bill Louis,

a physicist at Los Alamos

National Laboratory in New Mexico and co-spokesperson

for the MiniBoone

neutrino experiment at Fermilab, near Chicago. "The

question is are they

true or not."

Louis is right to be cautious. Although nothing in

the laws of nature

appears to rule out time travel, physicists have

always been uneasy

about it because it makes a mockery of causality,

the idea that cause

always precedes effect. Violating causality would

play havoc with the

universe, for instance, allowing you to travel back

in time and prevent

your own birth.

Such paradoxes are what led Stephen Hawking to

propose his "chronology

protection conjecture". This basically says that

some principle of

physics, perhaps as yet undiscovered, will always

come to the rescue and

prevent time travel from happening. Yet no one

had been able to flesh

out the details until three years ago when several

groups of researchers

claimed that string theory, physicists' best stab

at a prospective

"theory of everything", was beginning to close the

door on time machines

(New Scientist, 20 September 2003, p 28).

All very well, except theoretical physicists are

notorious for never

taking no for an answer. Päs and his colleagues

Sandip Pakvasa of the

University of Hawaii and Thomas Weiler of Vanderbilt

University in

Nashville, Tennessee, have been re-examining

string theory. It views the

fundamental building blocks of the universe not

as point-like particles

but vibrating strings of energy; the faster the

vibration the greater

the mass of the particle.

Such vibrating strings can account for the myriad

interactions of all

the known subatomic particles, such as quarks and

electrons. But there

is a catch - it only works if the strings vibrate

in a space-time with

10 dimensions rather than the four with which we

are familiar.

Proponents of the idea maintain that the extra

dimensions are either so

fantastically small that we have not noticed them,

or so large and

warped in such a way that, again, they have remained

hidden from view.

This has led to the suggestion that our universe

may be like a

four-dimensional membrane or "brane" adrift in

a higher-dimensional

space-time. All of the particles and forces in

our universe would be

trapped in our brane like flies on fly-paper,

so we would have no

knowledge of any dimensions other than the four

we experience, even

though our brane might be floating in a 10-dimensional

space-time, or

"bulk". "If it is, then there is the possibility of

short cuts through

higher-dimensional space," says Päs. "It's such short

cuts that make

time travel possible."

?Time machines might be far more common than we

ever thought possible?

It is not too difficult to visualise such a short

cut. Suppose our

brane-universe is bent back on itself within a

large extra dimension,

making it the four-dimensional equivalent of a

pancake folded in two.

Then you could imagine leaving the brane at one

point, travelling a

short distance through the bulk and re-entering

the brane at a point far

away from your starting point.

There is a problem with this picture, however.

Although we can visualise

a universe where such a short cut is possible,

it cannot be our

universe. That is because the space-time of such

a severely folded brane

is not compatible with Einstein's special theory

of relativity, which

posits a "Euclidean geometry" where space is

perfectly flat. Since

numerous tests of special relativity have shown

that its predictions in

our locality are accurate to better than 1 part

in a million, it is very

unlikely that our universe is shaped like a folded pancake.

Instead Päs, Pakvasa and Weiler consider a space-time

where our universe

is a flat brane that is immersed in a bulk whose own

dimensions are

seriously warped. Because the brane is flat, special

relativity still

applies there. Yet in the bulk, Päs, Pakvasa and

Weiler have found that

the large dimensions can be distorted in such a

way that special

relativity does not apply within them. This means

that anything moving

through the fifth dimension can break one of the

founding principles of

special relativity: it can travel faster than the

speed of light as we

know it.

This has dramatic consequences for inhabitants stuck

on the brane. To

them, any entity that takes a short cut through the

bulk appears to

vanish and then pops up again at some point on the

brane far sooner than

it could have had it kept to the brane. For some

inhabitants of the

brane world, the entity appears to have travelled

faster than the speed

of light. Weirder still, to others it has also

travelled backwards in

time. That's because special relativity says that

from certain frames of

reference, faster-than-light travel is equivalent

to travelling

backwards in time. "Such off-brane short cuts can

appear as 'closed

timelike curves'," says Päs - again, that's code

for time machines.

Escape from the brane

The trouble with this idea is that it assumes there

is some way to

escape the confines of the brane and travel out into

the bulk. How could

we do this? Fortunately string theory provides a way

out. In the theory

almost all of the building blocks of matter are

represented by strings

whose ends are forever anchored to the brane.

This means they can never

escape into the fifth dimension and take a short

cut through space-time.

But there are two crucial exceptions: the hypothetical

carrier of the

gravitational force, called the graviton, and a fourth

type of neutrino

called a sterile neutrino (after the three ordinary

kinds of neutrino).

In string theory these are represented by closed loops

of string. Since

they have no ends attached to the brane, they are free

to leave and

travel into the bulk.

String theorists have pointed to this property of

gravitons to explain

why gravity is tremendously weaker than nature's

other fundamental

forces, such as electromagnetism. The idea is that

gravity is so weak

because a large proportion of gravitons leak away

into the extra

dimensions of the bulk. More intriguingly, however,

their ability to

take short cuts through the bulk also means gravitons

and sterile

neutrinos are potential time travellers. "If we can

manipulate them, we

can study time travel experimentally," says Päs.

None of this will be easy. No one has ever spotted a

graviton or a

sterile neutrino, and the odds of detecting them are

slim, to say the

least (New Scientist, 18 March, p 32). Trillions of o

rdinary neutrinos

pass through our bodies every second, yet we feel

nothing because they

so rarely interact with electrons and atoms.

Sterile neutrinos are even

less communicative because they are thought to

interact only via the

feeble gravitational force and the exchange of

the elusive Higgs boson -

an as yet undetected particle believed to endow

all particles with mass.

Päs and his colleagues point out that a quirk

of quantum mechanics could

save the day. According to the laws of quantum

physics, neutrinos can

flip from one kind to another. Experiments in

Japan and the US designed

to detect neutrinos on the rare occasions they

do interact with matter

have confirmed that neutrinos spewed out by the

sun and those from space

do indeed change type. This phenomenon should

affect sterile neutrinos

too, changing them into ordinary, detectable

neutrinos and back again.

What's more, the odds of this happening increase

whenever the density of

the material the neutrinos are travelling through

changes abruptly.

?A quirk of quantum mechanics could allow us to test

time travel right

here on Earth?

This has inspired Päs and his colleagues to propose

an experiment that

could test their ideas. They suggest sending a

beam of ordinary

neutrinos through the Earth, from a research station

at the South Pole

towards a detector located at the equator. When they

enter the ground,

some of the neutrinos will flip into sterile neutrinos.

Capable of

taking a short cut though the extra dimensions of the

bulk, these

sterile neutrinos will reach the other side of the

Earth first,

apparently having travelled faster than light. As

they pass out of the

ground into the air again, they will flip back into

ordinary neutrinos,

which can be detected. Because the Earth is rotating,

these

faster-than-light neutrinos can appear to have arrived

before they set off.

Such an experiment is beyond our current technological

capabilities but,

remarkably, Päs says it is a realistic proposition within

the next 50

years. Of course, it requires two things. The first is the

existence of

sterile neutrinos. While many physicists are keen on the

idea of sterile

neutrinos, they are barely beyond theoretical flights of

fancy. The

other is that we live in an asymmetrically warped space-time,

as Päs

prescribes. How plausible is this?

When Einstein's came up with his general theory of

relativity, he showed

us how space-time can be warped or flat, but his

equations tell us

nothing about the actual shape of our universe -

merely that different

shapes are possible. For instance, cosmologists have

no way of knowing

if space stretches out to infinity or curves back on

itself. This opens

the door to many different types of time machines,

some more plausible

than others.

One famous solution to Einstein's equations, formulated

by mathematician

Kurt Gödel, describes a universe that rotates rapidly.

Instead of

travelling in straight lines, light will appear to

travel in a spiral.

Gödel realised that this allows a traveller to outrun

light and return

to their starting point before they left. In other

words, Gödel's

rotating universe is a time machine. "But we know we

don't live in such

a universe," says Päs.

Another time machine exists inside rotating

black holes, where

space-time becomes so warped that space and time

change places. The

trouble is, as Päs points out, rotating black holes

are inaccessible to

us. Then there is the space-time surrounding an

infinitely long, rapidly

rotating cylinder, as proposed by physicist Frank

Tipler. Päs is quick

to dismiss it too. "It requires huge masses rotating

unphysically fast,"

he says.

Among the other leading contenders are wormholes,

microscopic tubes of

space-time that act as tunnels from one point to

another. But before you

climb into one, there is a problem: wormholes snap

shut in an instant

unless propped open by a supply of something called

exotic matter.

Unlike the familiar stuff found on Earth, which always

feels the pull of

gravity, this exotic matter has repulsive gravity,

halting the

wormhole's collapse. "We don't know whether such matter

exists and if it

is stable," says Päs.

Päs confesses that the scenario his team has examined

also requires

exotic matter to warp the fifth dimension, but he

still maintains that

it is more plausible than the other scenarios.

What sets their

space-time apart from wormholes is that the hypothetical

exotic matter

is hidden away in the higher-dimensional bulk instead

of roaming around

the brane. If it exists, this might explain why we have

never seen it.

Understandably, the idea is not without its critics.

Sydney Deser of

Brandeis University in Waltham, Massachusetts, is

convinced, as was

Einstein, that time machines are not possible and

does not like

"unphysical" exotic matter. He believes that concealing

it in the bulk,

as Päs's team suggests, is little better than a scenario

in which it is

out in the open. "It's only a matter of degree," he says.

Päs, however, points out that the kind of space-times

he and his

colleagues have considered can do away with a number

of problems that

have plagued general relativity. For instance,

faster-than-light

connections between far-flung parts of the cosmos

would have allowed

heat to flow back and forth across the early universe.

This would have

evened out any temperature variations, explaining the

uniformity that

cosmologists observe. This could provide an alternative

to the theory of

inflation, in which cosmologists believe space-time was

stretched

unimaginably fast after the big bang, and which would

also account for

the evenness in temperature. While the majority of

cosmologists believe

in inflation, no one has explained the detailed

physics behind it.

Others do not find asymmetrically warped space-times

so plausible. "I

certainly think the idea is interesting, but I have

some worries," says

Tony Padilla of the University of Barcelona in Spain.

"For a start, I

think it is premature to claim that these space-times

are 'natural'. One

needs to examine their stability first, and in this

case I would expect

the solution to be unstable, although I could be wrong."

Padilla concedes that it is possible that we may one

day find a stable

brane universe with the properties described by

Päs's team. "I'm just

not convinced we are there yet," he says.

John Cramer of the University of Washington in

Seattle agrees that the

work outlines some interesting ideas. "The scheme,

however, requires

asymmetrically warped brane universes - and our

universe may not be one

of these," he cautions. "Nevertheless, it's a

fascinating proposal."

Of course, if time travel is possible in the way

Päs envisages, it may

be accessible only to special particles like sterile

neutrinos and

gravitons, and therefore won't cause much havoc in

the everyday

universe. Päs takes a pragmatic view of all this.

As long as the

possibility of time machines remains, he believes it

is worth exploring

experimentally. "Even if time travel is not possible,

by manipulating

particles like sterile neutrinos we can explore the

physics that

intervenes to prevent it," he says.

?Anything travelling through the fifth dimension

can move faster than

the speed of light as we know it?

The first answers might come soon courtesy of the

MiniBoone neutrino

experiment at Fermilab. It could confirm the existence

of sterile

neutrinos and short cuts in extra dimensions as early as

this year. Then

again, if time travel really is true, maybe the answer

has already been

published.

From issue 2552 of New Scientist magazine, 22 May 2006,

page 34

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