In this book, Lawrence Sklar demonstrates the interdependence of science and philosophy by examining a number of crucial problems on the nature of space and time—problems that require for their resolution the resources of philosophy and of physics. The overall issues explored are our knowledge of the geometry of the world, the existence of spacetime as an entity over and above the material objects of the world, the relation between temporal order and causal order, and the problem of the direction of time.

Without neglecting the most subtle philosophical points or the most advanced contributions of contemporary physics, the author has taken pains to make his explorations intelligible to the reader with no advanced training in physics, mathematics, or philosophy.

The arguments are set forth step-by-step, beginning from first principles; and the philosophical discussions are supplemented in detail by nontechnical expositions of crucial features of physical theories. With an epilogue on the limits of human understanding, this volume hosts contributions from six or more diverse fields. It presumes only rudimentary background knowledge on the part of the reader. Time and again, through the prism of intellect, humans have tried to diffract reality into various distinct, yet seamless, atomic, yet holistic, independent, yet interrelated disciplines and have attempted to study it contextually.

Philosophers debate the paradoxes, or engage in meditations, dialogues and reflections on the content and nature of space and time. Physicists, too, have been trying to mold space and time to fit their notions concerning micro- and macro-worlds. Mathematicians focus on the abstract aspects of space, time and measurement.

The list is never-ending. Linguists, logicians, artists, evolutionary biologists, geographers etc. However, our endeavour into a world of such endless imagination is restrained by intellectual dilemmas such as: Can humans comprehend everything? Are there any limits? Can finite thought fathom infinity? We have sought far and wide among the best minds to furnish articles that provide an overview of the above topics. We hope that, through this journey, a symphony of patterns and tapestry of intuitions will emerge, providing the reader with insights into the questions: What is Space?

What is Time? Chapter [15] of this book is available open access under a CC BY 4. From Nobel prize-winner Roger Penrose, this groundbreaking book is for anyone "who is interested in the world, how it works, and how it got here" New York Journal of Books. What is the source of order in our universe? And what cosmic future awaits us? I will definitely recommend this book to science, physics lovers. Your Rating:. Your Comment:. Read Online Download. When I first studied this effect in I expected I would find a burst of emission during the collapse but that then the particle creation would die out and one would be left with a black hole that was truely black.

To my great surprise I found that after a burst during the collapse there remained a steady rate of particle creation and emission.

This was just what was required to make consistent the idea that a black hole had an entropy proportional to the area of its event horizon. This would reflect the enormous number of different ways in which it could be made. However, joint work with Jim Hartle and Gary Gibbons uncovered the deep reason. To explain it I shall start with the example of the Schwarzschild metric.

However, this is just caused by a bad choice of coordinates. One can choose other coordinates in which the metric is regular there.

The region on the right, marked 1 on the diagram is the asymptotically flat space in which we are supposed to live. There is another asymptotically flat region 3 on the left that seems to correspond to another universe that is connected to ours only through a wormhole. However, as we shall see, it is connected to our region through imaginary time. The null surface from bottom left to top right is the boundary of the region from which one can escape to the infinity on the right.

Thus it is the future event horizon. The epithet future being added to distinguish it from the past event horizon which goes from bottom right to top left. Let us now return to the Schwarzschild metric in the original r and t coordinates. I shall refer to such positive definite metrics as Euclidean even though they may be curved. On the right hand of the equation one has a path integral. The periodicity in imaginary time explained why the messy calculation of frequency mixing led to radiation that was exactly thermal.

However, this derivation avoided the problem of the very high frequencies that take part in the frequency mixing approach. It can also be applied when there are interactions between the quantum fields on the background.

The fact that the path integral is on a periodic background implies that all physical quantities like expectation values will be thermal. This would have been very difficult to establish in the frequency mixing approach. One can extend these interactions to include interactions with the gravitational field itself. One starts with a background metric g0 such as the Euclidean-Schwarzschild metric that is a solution of the classical field equations.

The linear term vanishes because the background is a solution of the field equations. The quadratic term can be regarded as describing gravitons on the background while the cubic and higher terms describe interactions between the gravitons.

The path integral over the quadratic terms are finite. There are non renormalizable divergences at two loops in pure gravity but these cancel with the fermions in supergravity theories. It is not known whether supergravity theories have divergences at three loops or higher because no one has been brave or foolhardy enough to try the calculation.

Some recent work indicates that they may be finite to all orders. More interesting than the higher order terms is the zeroth order term, the action of the background metric g0.

This is zero for vacuum solutions so one might think that the action of the Euclidean-Schwarzschild solution was zero. However, there is also a surface term in the action proportional to the integral of K, the trace of the second fundemental form of the boundary surface.

This confirms the relation between the mass and the period, or inverse temperature, that we already knew. However, one can go further. Why does one get this intrinsic gravitational entropy which has no parallel in other quantum field theories.

The reason is gravity allows different topologies for the spacetime manifold. One can fill in this boundary with metrics of at least two different topologies. One of course is the Euclidean-Schwarzschild metric. These two topologies have different Euler numbers. The Euler number of periodically identified flat space is zero, while that of the Euclidean-Schwarzschild solution is two.

There will be two contributions to the action, a volume integral over the matter Lagrangian, plus the Einstein-Hilbert Lagrangian and a surface term. Thus the only net contribution to the surface term comes from the boundary at infinity. If the mass is non-zero there must be non-zero matter fields to create the mass. However, the entropy contributed by the background field will be zero. The situation is different however with the Euclidean-Schwarzschild solution.

The best one can do is choose the imaginary time coor- dinate of the Schwarzschild solution. Leaving out the surface term on the horizon reduces the action by one quarter the area of the horizon, which is just the intrinsic gravitational entropy of the black hole. This idea is anathema to most particle physicists who are a very conservative lot and want to make everything like Yang-Mills theory. They agree that the radiation from black holes seems to be thermal and independent of how the hole was formed if the hole is large compared to the Planck length.

But they would claim that when the black hole loses mass and gets down to the Planck size, quantum general relativity will break down and all bets will be off. However, I shall describe a thought experiment with black holes in which information seems to be lost yet the curvature outside the horizons always remains small. It has been known for some time that one can create pairs of positively and negatively charged particles in a strong electric field.

One way of looking at this is to note that in flat Euclidean space a particle of charge q such as an electron would move in a circle in a uniform electric field E. One gets a pair of positively and negatively charged particles accelerating away from each other pulled apart by the electric field. One then joins the upper half of the Minkowski space diagram to the lower half of the Euclidean space diagram. It tunnels through Euclidean space to get from one Minkowski space world line to the other.

Black holes can also carry electric charges so one might expect that they could also be pair created. However the rate would be tiny compared to that for electron positron pairs because the mass to charge ratio is times bigger.

This means that any electric field would be neutralized by electron positron pair creation long before there was a significant probability of pair creating black holes. However there are also black hole solutions with magnetic charges. But one might expect that they could be pair created in a strong magnetic field. In this case there would be no competition from ordinary particle creation because ordinary particles do not carry magnetic charges.

So the magnetic field could become strong enough that there was a significant chance of creating a pair of magnetically charged black holes. In Ernst found a solution that represented two magnetically charged black holes accelerating away from each other in a magnetic field.

The black hole moves on a circle in a curved Euclidean space just like the electron moves in a circle in flat Euclidean space. There is a complication in the black hole case because the imaginary time coordinate is periodic about the horizon of the black hole as well as about the center of the circle on which the black hole moves.

One has to adjust the mass to charge ratio of the black hole to make these periods equal. Physically this means that one chooses the parameters of the black hole so that the temperature of the black hole is equal to the temperature it sees because it is accelerating.. The temperature of a magnetically charged black hole tends to zero as the charge tends to the mass in Planck units.

Thus for weak magnetic fields, and hence low acceleration, one can always match the periods. Like in the case of pair creation of electrons one can describe pair creation of black holes by joining the lower half of the imaginary time Euclidean solution to the upper half of the real time Lorentzian solution.

The accelerating black hole solution is not asymptotically flat because it tends to a uniform magnetic field at infinity. But one can nevertheless use it to estimate the rate of pair creation of black holes in a local region of magnetic field. One could imagine that after being created the black holes move far apart into regions without magnetic field. One could then treat each black hole separately as a black hole in asymptotically flat space.

One could throw an arbitrarily large amount of matter and information into each hole. The holes would then radiate and lose mass. Thus they would eventually get back to their original state with the mass slightly bigger than the charge. One could then bring the two holes back together again and let them annihilate each other.

The annihilation process can be regarded as the time reverse of the pair creation. Thus it is represented by the top half of the Euclidean solution joined to the bottom half of the Lorentzian solution.

In between the pair creation and the annihilation one can have a long Lorentzian period in which the black holes move far apart, accrete matter, radiate and then come back together again. But the topology of the gravitational field will be the topology of the Euclidean Ernst solution. One might worry that the Generalized Second Law of Thermodynamics would be violated when the black holes annihilated because the black hole horizon area would have disappeared. However it turns out that the area of the acceleration horizon in the Ernst solution is reduced from the area it would have if there were no pair creation.

This is a rather delicate calculation because the area of the acceleration horizon is infinite in both cases. Nevertheless there is a well defined sense in which their difference is finite and equal to the black hole horizon area plus the difference in the action of the solutions with and without pair creation.

One can see from this thought experiment that one gets intrinsic gravitational en- tropy and loss of information when the topology of spacetime is different from that of flat Minkowski space. If the black holes that pair create are large compared to the Planck size the curvature outside the horizons will be everywhere small compared to the Planck scale. This means the approximation I have made of ignoring cubic and higher terms in the perturbations should be good.

Thus the conclusion that information can be lost in black holes should be reliable. One could imagine that particles and information could fall into these holes and get lost. Maybe that is where all those odd socks went. Quantities like energy and electric charge, that are coupled to gauge fields, would be conserved but other information and global charge would be lost. This would have far reaching implications for quantum theory. It is normally assumed that a system in a pure quantum state evolves in a unitary way through a succession of pure quantum states.

Instead the loss of information will mean that the final state after the black holes have disappeared will be what is called a mixed quantum state.

This can be regarded as an ensemble of different pure quantum states each with its own probability. But because it is not with certainty in any one state one can not reduce the probability of the final state to zero by interfering with any quantum state. This means that gravity introduces a new level of unpredictability into physics over and above the uncertainty usually associated with quantum theory.

I shall show in the next lecture we may have already observed this extra uncertainty. It means an end to the hope of scientific determinism that we could predict the future with certainty. It seems God still has a few tricks up his sleeve. Quantum Cosmology S. Hawking In my third lecture I shall turn to cosmology. Cosmology used to be considered a pseudo-science and the preserve of physicists who may have done useful work in their earlier years but who had gone mystic in their dotage.

There were two reasons for this. The first was that there was an almost total absence of reliable observations. Indeed, until the s about the only important cosmological observation was that the sky at night is dark. However, in recent years the range and quality of cosmological observations has improved enormously with developments in technology.

There is, however, a second and more serious objection. Cosmology can not predict anything about the universe unless it makes some assumption about the initial conditions.

Without such an assumption, all one can say is that things are as they are now because they were as they were at an earlier stage. Yet many people believe that science should be concerned only with the local laws which govern how the universe evolves in time. They would feel that the boundary conditions for the universe that determine how the universe began were a question for metaphysics or religion rather than science. The situation was made worse by the theorems that Roger and I proved.

These showed that according to general relativity there should be a singularity in our past. At this singularity the field equations could not be defined.

Although many people welcomed this conclusion, it has always profoundly disturbed me. In quantum theory it is a principle that anything can happen if it is not absolutely forbidden. Once one allows that singular histories could take part in the path integral they could occur any where and predictability would disappear completely. If the laws of physics break down at singularities, they could break down any where. The only way to have a scientific theory is if the laws of physics hold everywhere including at the begining of the universe.

One can regard this as a triumph for the principles of democracy: Why should the begining of the universe be exempt from the laws that apply to other points. To implement the idea that the laws of physics hold everywhere, one should take the path integral only over non-singular metrics. But these are the completion in some suitable topology of the set of smooth paths with well defined action. Similarly, one would expect that the path integral for quantum gravity should be taken over the completion of the space of smooth metrics.

In the case of black holes we saw that the path integral should be taken over Euclidean, that is, positive definite metrics. This meant that the singularities of black holes, like the Schwarzschild solution, did not appear on the Euclidean metrics which did not go inside the horizon.

Instead the horizon was like the origin of polar coordinates. The action of the Euclidean metric was therefore well defined. One could regard this as a quantum version of Cosmic Censorship: the break down of the structure at a singularity should not affect any physical measurement. It seems, therefore, that the path integral for quantum gravity should be taken over non-singular Euclidean metrics. But what should the boundary conditions be on these metrics.

There are two, and only two, natural choices. The first is metrics that approach the flat Euclidean metric outside a compact set. The second possibility is metrics on manifolds that are compact and without boundary.

Natural choices for path integral for quantum gravity 1. Asymptotically Euclidean metrics. Compact metrics without boundary. The first class of asymptotically Euclidean metrics is obviously appropriate for scat- tering calculations. All measurements are made at infinity where one has a flat background metric and one can interpret small fluctuations in the fields as particles in the usual way.

That is why one does a path integral over all possible histories for the interaction region, that is, over all asymptotically Euclidean metrics. However, in cosmology one is interested in measurements that are made in a finite region rather than at infinity. We are on the inside of the universe not looking in from the outside. To see what difference this makes let us first suppose that the path integral for cosmology is to be taken over all asymptotically Euclidean metrics.

The first would be from connected asymptotically Euclidean metrics. The second would be from disconnected metrics that consisted of a compact spacetime containing the region of measurements and a separate asymptotically Euclidean metric. But they will affect measurements in cosmology that are made in a finite region. Indeed, the contribu- tions from such disconnected metrics will dominate over the contributions from connected asymptotically Euclidean metrics.

Thus, even if one took the path integral for cosmology to be over all asymptotically Euclidean metrics, the effect would be almost the same as if the path integral had been over all compact metrics. It therefore seems more natural to take the path integral for cosmology to be over all compact metrics without boundary, as Jim Hartle and I proposed in In the rest of this lecture I shall show that this no boundary proposal seems to account for the universe we live in.

This investigation of time and space is motivated by gaps in our current understanding: by the lack of definitions, by our failure to appreciate the nature of these entities, by our inability to pin down their properties. The author's approach is based on two key ideas: The first idea is to seek the geo-historical origins of time and space concepts.

A thorough investigation This investigation of time and space is motivated by gaps in our current understanding: by the lack of definitions, by our failure to appreciate the nature of these entities, by our inability to pin down their properties. A thorough investigation of a diversified archaeological corpus, allows him to draft coherent definitions; it furthermore gives clues as to whether time and space were discovered or invented.

The second idea is to define the units before trying to define space and time. The results presented here are unexpected: Time and space were not discovered in nature, but they were invented; time is not a phenomenon and space has no materiality; they are only concepts. This runs contrary to the opinion of most scientific and the philosophical authorities, although one would seek in vain for a theoretical validation of the conventional position.