Part 13 (1/2)

1. Cla.s.sical Mechanics: Cla.s.sical Mechanics:[image] . Electromagnetism: . Electromagnetism: d*F=*J;dF=0 d*F=*J;dF=0. Quantum mechanics:[image] . General relativity: . General relativity:[image] . .

2. I am referring here to the I am referring here to the fine structure constant, e fine structure constant, e2/hc, whose numerical value (at typical energies for electromagnetic processes) is about 1/137, which is roughly .0073.

3. Witten argued that when the Type I string coupling is dialed large, the theory morphs into the Heterotic-O theory with a coupling that's dialed small, and vice versa; the Type IIB at large coupling morphs into Witten argued that when the Type I string coupling is dialed large, the theory morphs into the Heterotic-O theory with a coupling that's dialed small, and vice versa; the Type IIB at large coupling morphs into itself itself, the Type IIB theory but with small coupling. The cases of the Heterotic-E and Type IIA theories are a little more subtle (see The Elegant Universe The Elegant Universe, Chapter 12, for details), but the overall picture is that all five theories partic.i.p.ate in a web of interrelations.

4. For the mathematically inclined reader, the special thing about strings, one-dimensional ingredients, is that the physics describing their motion respects an infinite dimensional symmetry group. That is, as a string moves, it sweeps out a two-dimensional surface, and so the action functional from which its equations of motion are derived is a two-dimensional quantum field theory. Cla.s.sically, such two-dimensional actions are conformally invariant (invariant under angle-preserving rescalings of the two-dimensional surface), and such symmetry can be preserved quantum mechanically by imposing various restrictions (such as on the number of s.p.a.cetime dimensions through which the string moves-the dimension, that is, of s.p.a.cetime). The conformal group of symmetry transformations is infinite-dimensional, and this proves essential to ensuring that the perturbative quantum a.n.a.lysis of a moving string is mathematically consistent. For example, the infinite number of excitations of a moving string that would otherwise have negative norm (arising from the negative signature of the time component of the s.p.a.cetime metric) can be systematically ”rotated” away using the infinite-dimensional symmetry group. For details, the reader can consult M. Green, J. Schwarz, and E. Witten, For the mathematically inclined reader, the special thing about strings, one-dimensional ingredients, is that the physics describing their motion respects an infinite dimensional symmetry group. That is, as a string moves, it sweeps out a two-dimensional surface, and so the action functional from which its equations of motion are derived is a two-dimensional quantum field theory. Cla.s.sically, such two-dimensional actions are conformally invariant (invariant under angle-preserving rescalings of the two-dimensional surface), and such symmetry can be preserved quantum mechanically by imposing various restrictions (such as on the number of s.p.a.cetime dimensions through which the string moves-the dimension, that is, of s.p.a.cetime). The conformal group of symmetry transformations is infinite-dimensional, and this proves essential to ensuring that the perturbative quantum a.n.a.lysis of a moving string is mathematically consistent. For example, the infinite number of excitations of a moving string that would otherwise have negative norm (arising from the negative signature of the time component of the s.p.a.cetime metric) can be systematically ”rotated” away using the infinite-dimensional symmetry group. For details, the reader can consult M. Green, J. Schwarz, and E. Witten, Superstring Theory Superstring Theory, vol. 1 (Cambridge: Cambridge University Press, 1988).

5. As with many major discoveries, credit deserves to be given to those whose insights laid its groundwork as well as to those whose work established its importance. Among those who played such a role for the discovery of branes in string theory are: Michael Duff, Paul Howe, Takeo Inami, Kelley Stelle, Eric Bergshoeff, Ergin Szegin, Paul Townsend, Chris Hull, Chris Pope, John Schwarz, Ashoke Sen, Andrew Strominger, Curtis Callan, Joe Polchinski, Petr Hoava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard DeWitt. As with many major discoveries, credit deserves to be given to those whose insights laid its groundwork as well as to those whose work established its importance. Among those who played such a role for the discovery of branes in string theory are: Michael Duff, Paul Howe, Takeo Inami, Kelley Stelle, Eric Bergshoeff, Ergin Szegin, Paul Townsend, Chris Hull, Chris Pope, John Schwarz, Ashoke Sen, Andrew Strominger, Curtis Callan, Joe Polchinski, Petr Hoava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard DeWitt.

6. The diligent reader might argue that the Inflationary Multiverse also entwines time in a fundamental way, since, after all, our bubble's boundary marks the beginning of time in our universe; beyond our bubble is thus beyond our time. While true, my point here is meant more generally-the multiverses discussed so far all emerge from a.n.a.lyses that focus fundamentally on processes occurring throughout s.p.a.ce. In the multiverse we will now discuss, time is central from the outset. The diligent reader might argue that the Inflationary Multiverse also entwines time in a fundamental way, since, after all, our bubble's boundary marks the beginning of time in our universe; beyond our bubble is thus beyond our time. While true, my point here is meant more generally-the multiverses discussed so far all emerge from a.n.a.lyses that focus fundamentally on processes occurring throughout s.p.a.ce. In the multiverse we will now discuss, time is central from the outset.

7. Alexander Friedmann, Alexander Friedmann, The World as s.p.a.ce and Time The World as s.p.a.ce and Time, 1923, published in Russian, as referenced by H. Kragh, in ”Continual Fascination: The Oscillating Universe in Modem Cosmology,” Science in Context Science in Context 22, no. 4 (2009): 587612. 22, no. 4 (2009): 587612.

8. As an interesting point of detail, the authors of the braneworld cyclic model invoke an especially utilitarian application of dark energy (dark energy will be discussed fully in As an interesting point of detail, the authors of the braneworld cyclic model invoke an especially utilitarian application of dark energy (dark energy will be discussed fully in Chapter 6 Chapter 6). In the last phase of each cycle, the presence of dark energy in the braneworlds ensures agreement with today's observations of accelerated expansion; this accelerated expansion, in turn, dilutes the entropy density, setting the stage for the next cosmological cycle.

9. Large flux values also tend to destabilize a given Calabi-Yau shape for the extra dimensions. That is, the fluxes tend to push the Calabi-Yau shape to grow large, quickly running into conflict with the criterion that extra dimensions not be visible. Large flux values also tend to destabilize a given Calabi-Yau shape for the extra dimensions. That is, the fluxes tend to push the Calabi-Yau shape to grow large, quickly running into conflict with the criterion that extra dimensions not be visible.

Chapter 6: New Thinking About an Old Constant.

1. George Gamow, George Gamow, My World Line My World Line (New York: Viking Adult, 1970); J. C. p.e.c.k.e.r, Letter to the Editor, (New York: Viking Adult, 1970); J. C. p.e.c.k.e.r, Letter to the Editor, Physics Today Physics Today, May 1990, p. 117.

2. Albert Einstein, Albert Einstein, The Meaning of Relativity The Meaning of Relativity (Princeton: Princeton University Press, 2004), p. 127. Note that Einstein used the term ”cosmologic member” for what we now call the ”cosmological constant”; for clarity, I have made this subst.i.tution in the text. (Princeton: Princeton University Press, 2004), p. 127. Note that Einstein used the term ”cosmologic member” for what we now call the ”cosmological constant”; for clarity, I have made this subst.i.tution in the text.

3. The Collected Papers of Albert Einstein The Collected Papers of Albert Einstein, edited by Robert Schulmann et al. (Princeton: Princeton University Press, 1998), p. 316.

4. Of course, some things Of course, some things do do change. As pointed out in the change. As pointed out in the notes notes to Chapter 3, galaxies generally have small velocities beyond the spatial swelling. Over the course of cosmological timescales, such additional motion can alter position relations.h.i.+ps; such motion can also result in a variety of interesting astrophysical events such as galaxy collisions and mergers. For the purpose of explaining cosmic distances, however, these complications can be safely ignored. to Chapter 3, galaxies generally have small velocities beyond the spatial swelling. Over the course of cosmological timescales, such additional motion can alter position relations.h.i.+ps; such motion can also result in a variety of interesting astrophysical events such as galaxy collisions and mergers. For the purpose of explaining cosmic distances, however, these complications can be safely ignored.

5. There is one complication that does not affect the essential idea I've explained but which does come into play when undertaking the scientific a.n.a.lyses described. As photons travel to us from a given supernova, their number density gets diluted in the manner I've described. However, there is another diminishment to which they are subject. In the next section, I'll describe how the stretching of s.p.a.ce causes the wavelength of photons to stretch too, and, correspondingly, their energy to decrease-an effect, as we will see, called There is one complication that does not affect the essential idea I've explained but which does come into play when undertaking the scientific a.n.a.lyses described. As photons travel to us from a given supernova, their number density gets diluted in the manner I've described. However, there is another diminishment to which they are subject. In the next section, I'll describe how the stretching of s.p.a.ce causes the wavelength of photons to stretch too, and, correspondingly, their energy to decrease-an effect, as we will see, called reds.h.i.+ft reds.h.i.+ft. As explained there, astronomers use reds.h.i.+ft data to learn about the size of the universe when the photons were emitted-an important step toward determining how the expansion of s.p.a.ce has varied through time. But the stretching of photons-the diminishment of their energy-has another effect: It accentuates the dimming of a distant source. And so, to properly determine the distance of a supernova by comparing its apparent and intrinsic brightness, astronomers must take account not just of the dilution of photon number density (as I've described in the text), but also the additional diminishment of energy coming from reds.h.i.+ft. (More precisely still, this additional dilution factor must be applied twice; the second red s.h.i.+ft factor accounts for the rate at which photons arrive being similarly stretched by the cosmic expansion.) 6. Properly interpreted, the second proposed answer for the meaning of the distance being measured may also be construed as correct. In the example of earth's expanding surface, New York, Austin, and Los Angeles all rush away from one another, yet each continues to occupy the same location on earth it always has. The cities separate because the surface swells, not because someone digs them up, puts them on a flatbed, and transports them to a new site. Similarly, because galaxies separate due to the cosmic swelling, they too occupy the same location in s.p.a.ce they always have. You can think of them as being st.i.tched to the spatial fabric. When the fabric stretches, the galaxies move apart, yet each remains tethered to the very same point it has always occupied. And so, even though the second and third answers appear different-the former focusing on the distance between us and the location a distant galaxy had eons ago, when the supernova emitted the light we now see; the latter focusing on the distance now between us and that galaxy's current location-they're not. The distant galaxy is now, and has been for billions of years, positioned at one and the same spatial location. Only if it moved Properly interpreted, the second proposed answer for the meaning of the distance being measured may also be construed as correct. In the example of earth's expanding surface, New York, Austin, and Los Angeles all rush away from one another, yet each continues to occupy the same location on earth it always has. The cities separate because the surface swells, not because someone digs them up, puts them on a flatbed, and transports them to a new site. Similarly, because galaxies separate due to the cosmic swelling, they too occupy the same location in s.p.a.ce they always have. You can think of them as being st.i.tched to the spatial fabric. When the fabric stretches, the galaxies move apart, yet each remains tethered to the very same point it has always occupied. And so, even though the second and third answers appear different-the former focusing on the distance between us and the location a distant galaxy had eons ago, when the supernova emitted the light we now see; the latter focusing on the distance now between us and that galaxy's current location-they're not. The distant galaxy is now, and has been for billions of years, positioned at one and the same spatial location. Only if it moved through through s.p.a.ce rather than solely ride the wave of swelling s.p.a.ce would its location change. In this sense, the second and third answers are actually the same. s.p.a.ce rather than solely ride the wave of swelling s.p.a.ce would its location change. In this sense, the second and third answers are actually the same.

7. For the mathematically inclined reader, here is how you do the calculation of the distance-now, at time For the mathematically inclined reader, here is how you do the calculation of the distance-now, at time t tnow-that light has traveled since being emitted at time t temitted. We will work in the context of an example in which the spatial part of s.p.a.cetime is flat, and so the metric can be written as ds ds2=c2dt2 a a2(t)dx2, where a(t) a(t) is the scale factor of the universe at time is the scale factor of the universe at time t t, and c c is the speed of light. The coordinates we are using are called is the speed of light. The coordinates we are using are called co-moving co-moving. In the language developed in this chapter, such coordinates can be thought of as labeling points on the static map; the scale factor supplies the information contained in the map's legend.

The special characteristic of the trajectory followed by light is that ds ds2=0 (equivalent to the speed of light always being c) along the path, which implies that (equivalent to the speed of light always being c) along the path, which implies that[image] , or, over a finite time interval such as that between , or, over a finite time interval such as that between[image] . The left side of this equation gives the distance light travels across the static map between emission and now. To turn this into the distance through real s.p.a.ce, we must rescale the formula by today's scale factor; therefore, the total distance the light traveled equals . The left side of this equation gives the distance light travels across the static map between emission and now. To turn this into the distance through real s.p.a.ce, we must rescale the formula by today's scale factor; therefore, the total distance the light traveled equals[image] . If s.p.a.ce were not stretching, the total travel distance would be . If s.p.a.ce were not stretching, the total travel distance would be[image][image] , as expected. When calculating the distance traveled in an expanding universe, we thus see that each segment of the light's trajectory is multiplied by the factor , as expected. When calculating the distance traveled in an expanding universe, we thus see that each segment of the light's trajectory is multiplied by the factor[image] , which is the amount by which that segment has stretched, since the moment the light traversed it, until today. , which is the amount by which that segment has stretched, since the moment the light traversed it, until today.

8. More precisely, about 7.12 10 More precisely, about 7.12 1030 grams per cubic centimeter. grams per cubic centimeter.

9. The conversion is 7.12 10 The conversion is 7.12 1030 grams/cubic centimeter = (7.12 10 grams/cubic centimeter = (7.12 1030 grams/cubic centimeter) (4.6 10 grams/cubic centimeter) (4.6 104 Planck ma.s.s/gram) (1.62 10 Planck ma.s.s/gram) (1.62 1033 centimeter/Planck length) centimeter/Planck length)3 = 1.38 10 = 1.38 10123 Planck ma.s.s/cubic Planck volume. Planck ma.s.s/cubic Planck volume.

10. For inflation, the repulsive gravity we considered was intense and brief. This is explained by the enormous energy and negative pressure supplied by the inflaton field. However, by modifying a quantum field's potential energy curve, the amount of energy and negative pressure it supplies can be diminished, thus yielding a mild accelerated expansion. Additionally, a suitable adjustment of the potential energy curve can prolong this period of accelerated expansion. A mild and prolonged period of accelerated expansion is what's required to explain the supernova data. Nevertheless, the small non-zero value for the cosmological constant remains the most convincing explanation to have emerged in the more than ten years since the accelerated expansion was first observed. For inflation, the repulsive gravity we considered was intense and brief. This is explained by the enormous energy and negative pressure supplied by the inflaton field. However, by modifying a quantum field's potential energy curve, the amount of energy and negative pressure it supplies can be diminished, thus yielding a mild accelerated expansion. Additionally, a suitable adjustment of the potential energy curve can prolong this period of accelerated expansion. A mild and prolonged period of accelerated expansion is what's required to explain the supernova data. Nevertheless, the small non-zero value for the cosmological constant remains the most convincing explanation to have emerged in the more than ten years since the accelerated expansion was first observed.

11. The mathematically inclined reader should note that each such jitter contributes an energy that's inversely proportional to its wavelength, ensuring that the sum over all possible wavelengths yields an infinite energy. The mathematically inclined reader should note that each such jitter contributes an energy that's inversely proportional to its wavelength, ensuring that the sum over all possible wavelengths yields an infinite energy.

12. For the mathematically inclined reader, the cancellation occurs because supersymmetry pairs bosons (particles with an integral spin value) and fermions (particles with a half [odd] integral spin value). This results in bosons being described by commuting variables, fermions by anticommuting variables, and that is the source of the relative minus sign in their quantum fluctuations. For the mathematically inclined reader, the cancellation occurs because supersymmetry pairs bosons (particles with an integral spin value) and fermions (particles with a half [odd] integral spin value). This results in bosons being described by commuting variables, fermions by anticommuting variables, and that is the source of the relative minus sign in their quantum fluctuations.

13. While the a.s.sertion that changes to the physical features of our universe would be inhospitable to life as we know it is widely accepted in the scientific community, some have suggested that the range of features compatible with life might be larger than once thought. These issues have been widely written about. See, for example: John Barrow and Frank Tipler, While the a.s.sertion that changes to the physical features of our universe would be inhospitable to life as we know it is widely accepted in the scientific community, some have suggested that the range of features compatible with life might be larger than once thought. These issues have been widely written about. See, for example: John Barrow and Frank Tipler, The Anthropic Cosmological Principle The Anthropic Cosmological Principle (New York: Oxford University Press, 1986); John Barrow, (New York: Oxford University Press, 1986); John Barrow, The Constants of Nature The Constants of Nature (New York: Pantheon Books, 2003); Paul Davies, (New York: Pantheon Books, 2003); Paul Davies, The Cosmic Jackpot The Cosmic Jackpot (New York: Houghton Mifflin Harcourt, 2007); Victor Stenger, (New York: Houghton Mifflin Harcourt, 2007); Victor Stenger, Has Science Found G.o.d? Has Science Found G.o.d? (Amherst, N.Y.: Prometheus Books, 2003); and references therein. (Amherst, N.Y.: Prometheus Books, 2003); and references therein.

14. Based on the material covered in earlier chapters, you might immediately think the answer is a resounding yes. Consider, you say, the Quilted Multiverse, whose infinite spatial expanse contains infinitely many universes. But you need to be careful. Even with infinitely many universes, the list of different cosmological constants represented might not be long. If, for example, the underlying laws don't allow for many different cosmological constant values, then regardless of the number of universes, only the small collection of possible cosmological constants would be realized. So, the question we're asking is whether (a) there are candidate laws of physics that give rise to a multiverse, (b) the multiverse so generated contains far more than 10 Based on the material covered in earlier chapters, you might immediately think the answer is a resounding yes. Consider, you say, the Quilted Multiverse, whose infinite spatial expanse contains infinitely many universes. But you need to be careful. Even with infinitely many universes, the list of different cosmological constants represented might not be long. If, for example, the underlying laws don't allow for many different cosmological constant values, then regardless of the number of universes, only the small collection of possible cosmological constants would be realized. So, the question we're asking is whether (a) there are candidate laws of physics that give rise to a multiverse, (b) the multiverse so generated contains far more than 10124 different universes, and (c) the laws ensure that the cosmological constant's value varies from universe to universe. different universes, and (c) the laws ensure that the cosmological constant's value varies from universe to universe.

15. These four authors were the first to show fully that by judicious choices of Calabi-Yau shapes, and the fluxes threading their holes, they could realize string models with small, positive cosmological constants, like those found by observations. Together with Juan Maldacena and Liam McAllister, this group subsequently wrote a highly influential paper on how to combine inflationary cosmology with string theory. These four authors were the first to show fully that by judicious choices of Calabi-Yau shapes, and the fluxes threading their holes, they could realize string models with small, positive cosmological constants, like those found by observations. Together with Juan Maldacena and Liam McAllister, this group subsequently wrote a highly influential paper on how to combine inflationary cosmology with string theory.

16. More precisely, this mountainous terrain would inhabit a roughly 500-dimensional s.p.a.ce, whose independent directions-axes-would correspond to different field fluxes. More precisely, this mountainous terrain would inhabit a roughly 500-dimensional s.p.a.ce, whose independent directions-axes-would correspond to different field fluxes. Figure 6.4 Figure 6.4 is a rough pictorial depiction but gives a feel for the relations.h.i.+ps between the various forms for the extra dimensions. Additionally, when speaking of the string landscape, physicists generally envision that the mountainous terrain encompa.s.ses, in addition to the possible flux values, all the possible sizes and shapes (the different topologies and geometries) of the extra dimensions. The valleys in the string landscape are locations (specific forms for the extra dimensions and the fluxes they carry) where a bubble universe naturally settles, much as a ball would settle in such a spot in a real mountain terrain. When described mathematically, valleys are (local) minima of the potential energy a.s.sociated with the extra dimensions. Cla.s.sically, once a bubble universe acquired an extra dimensional form corresponding to a valley that feature would never change. Quantum mechanically, however, we will see that tunneling events can result in the form of the extra dimensions changing. is a rough pictorial depiction but gives a feel for the relations.h.i.+ps between the various forms for the extra dimensions. Additionally, when speaking of the string landscape, physicists generally envision that the mountainous terrain encompa.s.ses, in addition to the possible flux values, all the possible sizes and shapes (the different topologies and geometries) of the extra dimensions. The valleys in the string landscape are locations (specific forms for the extra dimensions and the fluxes they carry) where a bubble universe naturally settles, much as a ball would settle in such a spot in a real mountain terrain. When described mathematically, valleys are (local) minima of the potential energy a.s.sociated with the extra dimensions. Cla.s.sically, once a bubble universe acquired an extra dimensional form corresponding to a valley that feature would never change. Quantum mechanically, however, we will see that tunneling events can result in the form of the extra dimensions changing.