29 November 2001Astronomy 102, Fall 20011 Today in Astronomy 102: relativity and the Universe From last time: wormhole time machines General relativity. - презентация
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29 November 2001Astronomy 102, Fall Today in Astronomy 102: relativity and the Universe From last time: wormhole time machines General relativity and the Universe. The Universe is homogeneous, isotropic, and expanding; what is its structure? Figure: visible-light spectrum of the quasar 3C 273 (from Chaisson and McMillan, Astronomy Today ).
29 November 2001Astronomy 102, Fall How to build a wormhole time machine Start with a wormhole whose two mouths (called mouths A and B) are close together in space. Fix things up so that they stay the same distance apart in hyperspace. In Thornes description in the book, this is illustrated by two people reaching into each mouth to hold hands. Take mouth B on a trip at high speeds (approaching light speed), out a great distance, and then back to its former spot, without ever changing the distance between the two mouths in hyperspace.
29 November 2001Astronomy 102, Fall How to build a wormhole time machine (continued) Mouth A Mouth B B takes a trip at relativistic speeds …and returns to its original position
29 November 2001Astronomy 102, Fall How to build a wormhole time machine (continued) Because of time dilation, the trip will take a short time according to an observer travelling with mouth B, and a much longer time according to an observer who stays with the stationary mouth A. While B is gone, the observer at A can travel into the future (to the time when B returns) by passing through mouth A. After B returns, an observer at B can travel into the past (to the time when B left) by passing through mouth B. The length of time travel is thus the time lag between clocks fixed to A and B during Bs trip, and is thus adjustable by adjusting the details of the trip. Travel between arbitrary times is not provided!
29 November 2001Astronomy 102, Fall Odd features of time travel Paradoxes such as the matricide paradox come up! One could use a time machine, for example, for travelling back through time before ones birthday and killing ones mother. Does physics prevent one from being born and travelling back through time in the first place? Maybe. How is it that one can start with laws of physics like the Einstein field equation, that have cause and effect built in, and derive from them violations of cause and effect? Maybe not. What about quantum mechanics? Vacuum fluctuations, for instance, have no cause. If quantum behavior (associated with singularities) is inherent in the wormhole, one could still exist after committing paradoxical matricide.
29 November 2001Astronomy 102, Fall Alas, it may be impossible to build a stable wormhole time machine Geroch, Wald, and Hawking on self-destruction of wormhole time machines: Light leaving the origin during Bs trip, and entering mouth B as it is returning, can travel backwards in time, emerge from A, and meet itself in the act of leaving. It can do this as many times as it likes, even an infinite number of times. Since light can interfere constructively (all the peaks and troughs of the light wave lining up), a large positive energy density could be generated in the wormhole, which would collapse it. This process could take as little as seconds in the frame of reference of mouth A.
29 November 2001Astronomy 102, Fall A recipe for wormhole time-machine destruction Mirror A (past) B (future) Light emerging from mouth A Laser Aim a laser at mouth B; orient mirror so that light emerging from mouth A joins up with the beam aimed at mouth B. Immediately a huge number of photons emerges from mouth A, and the vast increase of energy inside it collapses the wormhole.
29 November 2001Astronomy 102, Fall Alas, it may be impossible to build a stable wormhole time machine (continued) Its also possible for this to happen with light created by vacuum fluctuations !. Since light has wave properties too, the probability that virtual photons from near A to travel to B and re- emerge from A pointed again at B is not zero, even if there is nothing to aim the photons that way. The interference may not be constructive, though, because the wormhole tends to defocus the light in the manner of a negative lens; therefore we do not know whether this is a fatal objection.
29 November 2001Astronomy 102, Fall Mid-lecture Break. Remember, homework #6 is due tomorrow, Friday, 30 November 2001, at 11 PM. Heres what you missed by not getting up before 5AM on Sunday, 11/18, to see the Leonid meteor shower. (From APOD; photo by Jerry Lodriguss.)
29 November 2001Astronomy 102, Fall General relativity and the Universe It was recognized soon after Einsteins invention of the general theory of relativity in 1915 that this theory provides the best framework in which to study the large-scale structure of the Universe: Gravity is the only force known (then or now) that is long- ranged enough to influence objects scales much larger than typical interstellar distances. All the other forces (electricity, magnetism and the nuclear forces) are shielded in large accumulations of material or are naturally short ranged. And theres lots of matter around to serve as gravitys source. Einstein himself worked on this application of GR, rather than on stellar applications like black holes. The results wind up having a lot in common with black holes, though.
29 November 2001Astronomy 102, Fall Basic structure of the Universe The Universe contains: Planets Earths diameter = km Stars Suns diameter = km Planetary systems Solar system diameter = km Star clusters, interstellar clouds Typical distance between stars = a few light years (ly) = km Galaxies Diameter of typical galaxy = a hundred thousand ly = km Typical distance between galaxies = a million light years (Mly) = km It was long thought by most scientists that the nebulae we now call galaxies were simply parts of the Milky Way. In the early 1920s Edwin Hubble measured their distances and proved otherwise.
29 November 2001Astronomy 102, Fall The Universe is full of galaxies, and to observe them is to probe the structure of the Universe. This is part of the northern HST Deep Field (NASA/STScI). Only a few stars are present; virtually every dot is a galaxy. If the whole sky were imaged with the same sensitivity and the galaxies counted, wed get about 100 billion.
29 November 2001Astronomy 102, Fall General relativity and the structure of the Universe The Universe is not an isolated, distinct object like those weve dealt with hitherto. In its description we would be interested in the large-scale patterns and trends in gravity (or spacetime curvature). Why? These trends would tell us how galaxies and groups of galaxies – the most distant and massive things we can see – would tend to move around in the Universe, and why we see groups the size we do: this study is called cosmology. They might also tell us about the Universes origins and fate: this is called cosmogony. By large-scale, we mean sizes and distance large compared to the typical distance between galaxies.
29 November 2001Astronomy 102, Fall General relativity and the structure of the Universe (continued) To solve the Einstein field equation for the Universe one needs to apply what is known observationally about the Universe, as initial conditions or boundary conditions. The solutions will tell us the conditions for other spacetime points or conditions. And in the early 1920s observations (by Edwin Hubble, again) began to suggest that the distribution of galaxies, at least in the local Universe, is isotropic and homogeneous on large scales. Isotropic = looks the same in all directions from our viewpoint. Homogeneous = looks the same from any viewpoint within the Universe. These facts serve as useful boundary conditions for the field equation.
29 November 2001Astronomy 102, Fall Isotropy of the Universe on large scales: modern measurements Here positions are marked for galaxies lying within a 6 o x6 o patch of the sky: the galaxies are essentially randomly, and uniformly, distributed. (From F. Shu, The Physical Universe. ) For scale: 0.5 o (size of the full moon)
29 November 2001Astronomy 102, Fall Isotropy of the Universe on large scales: modern measurements (continued) Isotropy on the scale represented by these circless diameter means that approximately the same numbers of galaxies are contained within them, no matter where on the sky we put them, which is evidently true in this picture.
29 November 2001Astronomy 102, Fall Homogeneity of the Universe on large scales: modern measurements The Las Campanas Redshift Survey, showing the positions, out to distances of about 3x10 9 light years along the line of sight, of almost 24,000 galaxies in six different thin stripes on the sky. Data from stripes in the northern and southern sky are overlaid. Again, the galaxies and their larger groupings tend to be randomly distributed through volume on large scales. From Huan Lin, U. Toronto. Huan Lin, U. Toronto
29 November 2001Astronomy 102, Fall Homogeneity of the Universe on large scales: modern measurements (continued) Homogeneity on the scale of these circless diameter means that approximately the same numbers of galaxies are contained within them, no matter where we put them within the Universes volume, which is evidently true in this picture. Hubble actually did it a bit differently: he observed that fainter galaxies (same as brighter ones, but further away) appear in greater numbers than brighter galaxies, by the amount that would be expected if the galaxies are distributed uniformly in space.
29 November 2001Astronomy 102, Fall Back to general relativity and the structure of the Universe … Einstein and de Sitter (late 1910s and 1920s, Germany), Friedmann (1922, USSR), Lemaitre (1927, Belgium), and Robertson and Walker (1935, US/UK) produced the first solutions of the field equations for an isotropic and homogeneous Universe. The types of solutions they found: Collapse, ending in a singularity. Expansion from a singularity, gradually slowing and reversing under the influence of gravity, ending in a collapse to a singularity. This, and the previous outcome, are for universes with total kinetic energy (energy stored in the motions of galaxies) less than the gravitational binding energy. They are called closed universes.
29 November 2001Astronomy 102, Fall General relativity and the structure of the Universe (continued) Expansion from a singularity, that gradually slows, then stops. (Total kinetic energy = gravitational binding energy.) This is generally called a marginal, or critical, Universe. Expansion from a singularity, that continues forever (total kinetic energy greater than gravitational binding energy). This is called an open universe. Model 1 is of course a lot like what we now call black hole formation, since it ends in a singularity. Note that models 2-4 all involve expansion from a singularity, so the creation and development of the Universe must be rather like black hole formation running in reverse.
29 November 2001Astronomy 102, Fall Einstein doesnt like it. All the solutions have these features in common: singularities, and dynamic behavior: the structure given by the solutions is different at different times between singularities ( at which time doesnt exist, of course). Einstein thought that singularities such as these indicated that there were important physical effects not accounted for in the field equation. He also thought that the right answer would involve static behavior: large-scale structure should not change with time. He also saw how he could fix the field equation to eliminate singular and dynamic solutions: introduce an additional constant term, which became known as the cosmological constant, to represent the missing, unknown, physical effects.
29 November 2001Astronomy 102, Fall The field equation and the cosmological constant: hieroglyphics (i.e. not on the exam or homework) The field equation under a particularly simple set of assumptions and conditions for a homogeneous and isotropic Universe: The same equation modified by Einstein (1917): Typical distance between galaxies Mass per unit volume Spacetime curvature Cosmological constant A certain positive value of leads a static solution.
29 November 2001Astronomy 102, Fall Edwin Hubble strikes again: the Universe expands. Then, in 1929, Hubble made his third great contribution to cosmology; he observed that: distant galaxies are always seen to have redshifted spectra. Thus they all recede from us. the magnitude of this Doppler shift for any given distant galaxy is in direct proportion to the distance to this galaxy: with V = velocity and D = distance to galaxy, where H 0 = 20 km/sec/Mly according to the most recent measurements by the Hubble Space Telescope. This means that the Universe is expanding. (Hubbles Law)
29 November 2001Astronomy 102, Fall Relation between galaxy redshift and distance Visible-light spectra (left) of several different galaxies (right). The extent of the redshift, denoted by the horizontal yellow arrows, and the distance to each galaxy (in the center) increase from top to bottom. 1 parsec = 3.26 ly. (Figure: Chaisson and McMillan, Astronomy Today.)
29 November 2001Astronomy 102, Fall Einstein gives up. The Universe is observed to be expanding; it is not static. Thus the real Universe may be described by one of the dynamic solutions to the original Einstein field equation. Of the four types we discussed above, the last three spend at least part of their time, if not all, expanding. Thus there appeared to be no point in Einsteins cosmological constant, so he let it drop, calling it my greatest blunder. Thereafter he began trying to show that the singularities in the dynamic solutions simply wouldnt be realized. His effort resulted in the steady-state model of the Universe, which well describe later. This isnt the last well see of the cosmological constant, though.
29 November 2001Astronomy 102, Fall Possibilities for the structure of the expanding Universe All three expanding solutions predict that the matter in the universe was concentrated at earlier times, and that the expan- sion started as an explosion of this concentration: the Big Bang. Typical distance between galaxies Time Unbound (open) Bound (closed) Marginal (total kinetic energy = binding energy) Big BangBig Squeeze
29 November 2001Astronomy 102, Fall Possibilities for the structure of the expanding Universe (continued) Typical distance between galaxies Time Unbound (open) Marginal (total kinetic energy = binding energy) To tell which solution describes the real Universe, we need to measure the acceleration of the galaxies as well as their velocities. Astronomers have been working on this for decades. Bound (closed)