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← 17. Hubble's Law and the Big Bang (cont.)

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Ukazujem Revíziu 1 vytvorenú 07/17/2012 od Amara Bot.

  1. Professor Charles
    Bailyn: Okay,
  2. welcome back for more
    cosmology.
  3. What I want to do today is
    quickly review what we were
  4. doing about magnitudes and make
    a comment or two about the
  5. problem set,
    and then, go back and talk
  6. about the implications of the
    Hubble Law and the Hubble
  7. Diagram,
    which are formidable,
  8. to put it mildly.
    Okay, magnitudes.
  9. There's a couple of these
    magnitude equations.
  10. I'm just going to write them
    down.
  11. The first of them looks like
    this.
  12. And this equation is used--okay.
    So, this equation is used to
  13. relate magnitudes of two
    different objects to each other.
  14. So, we've got two different
    objects.
  15. And it can be used for either
    kind of magnitude--either
  16. absolute or apparent magnitude,
    just so long as you don't mix
  17. them.
    So, it's two different objects,
  18. but only one of the magnitudes.
    One kind of magnitude.
  19. And depending on which kind of
    magnitude you use,
  20. this brightness ratio--it's
    either the ratio of how bright
  21. it looks or the ratio of how
    bright it is--whatever's
  22. appropriate.
    Now, on the help sheet on the
  23. web, I have this equation in a
    somewhat different form,
  24. and it's important to realize
    that it's the exact same
  25. equation.
    Watch this.
  26. Let's see.
    Let's multiply both halves by -
  27. 2/5 which is - 0.4.
    So, this is -0.4
  28. (M_1 –
    M_2) = log
  29. (b_1 /
    b_2).
  30. And then, let's take 10 to the
    power of that.
  31. That gets rid of the log.
    And this is now the form that
  32. it is on the help sheet on the
    web.
  33. So, it's exactly the same
    equation, just expressed
  34. differently.
    And you can use either form,
  35. whichever is more convenient.
    Okay.
  36. The other equation looks like
    this.
  37. 5 log (D/10 parsecs).
    And this relates one object,
  38. but it relates both kinds of
    magnitude to each other.
  39. So, the first one is two
    different objects,
  40. but only one of the magnitudes.
    The other is one object and it
  41. relates the two different kinds
    of magnitudes to each other,
  42. and to the distance to the
    object.
  43. And as you can see,
    this--both of these equations,
  44. actually, have three unknowns.
    One, two, three.
  45. That means you've got to know
    two things in order to find out
  46. the third.
    And that brings me to the
  47. comment I want to make about
    problem 2a on the current
  48. problem set.
    You are asked in this problem
  49. to determine the difference
    between the absolute magnitude
  50. of one kind of star,
    called Type 1 Cepheid,
  51. so I label it C1.
    And another kind of star,
  52. Type 2 Cepheids,
    which I label C2.
  53. And if you're asked to--and
    this difference is called,
  54. I don't know,
    delta M_C or
  55. something like that.
    And having been asked to do
  56. this, the logical thing that you
    might try to do is say,
  57. all right, I'm going to use one
    or the other of these
  58. equations--I'm not sure,
    in advance, which,
  59. to compute this one.
    Then I'm going to compute this
  60. one.
    And I'm going to subtract the
  61. two, and that's going to give me
    the answer.
  62. That approach will fail.
    Okay?
  63. That doesn't work in this
    particular case,
  64. because you don't actually have
    enough information to compute
  65. either one of these things.
    You do have enough information
  66. to compute the difference.
    And let me just give you a very
  67. brief hint on how you might go
    about doing that.
  68. Let's see.
    Let me take a new piece of
  69. paper here.
    Write down
  70. m_C1 -
    M_C1 = 5 log
  71. (D_C1 /
    10parsecs).
  72. And now write the exact same
    equation down for C2,
  73. where the two different
    distances are the distances you
  74. get by assuming one or the other
    kinds of these magnitudes.
  75. This is equation one.
    This is equation two.
  76. Now, let's subtract.
    1 – 2.
  77. So, then, you get
    m_C1 -
  78. m_C2 -
    [M_C1 -
  79. M_C2]
    = 5 (log D_C1
  80. - log D_C2).
    Okay, now.
  81. Here's the trick.
    Turns out for reasons that you
  82. had better tell me--and TFs
    [teaching fellows],
  83. take note that we really want
    them to say why this is true,
  84. now that I've told them it is.
    This is zero.
  85. The two apparent magnitudes are
    the same.
  86. And so, that means that this
    side of the equation is what you
  87. want.
    It's the difference between
  88. these two magnitudes.
    And then, over here,
  89. you have to use one of these
    log rules: log (x) - log
  90. (y),
    if you remember back to
  91. eleventh grade,
    is log (x / y).
  92. And if you use that,
    it turns out that you have
  93. information elsewhere in the
    problem, which will tell you
  94. what you need to know about the
    distances.
  95. And so, in this way,
    you can solve for the
  96. difference without actually
    being able to determine either
  97. one of these two things.
    So, we'll just leave it at that
  98. for the moment.
    If you have problems there's
  99. the usual forum,
    there's the usual office hours,
  100. but ponder this.
    That's basically how the
  101. problem has to go.
    Okay?
  102. Problems with magnitudes?
  103. Okay.
    If you do have some,
  104. let us know,
    because this is going to be
  105. critical for solving,
    basically, problems for the
  106. whole rest of the class.
    All right.
  107. Yes go ahead.
    Student: What is the
  108. location--the last thing in
    green-- Professor Charles
  109. Bailyn: This?
    Student: --5 log--is it
  110. over log?
    Professor Charles
  111. Bailyn: This is log
    (D_C1 / 10
  112. parsecs) - log
    (D_C2 / 10
  113. parsecs).
    That's just subtracting the two
  114. right-hand sides of this
    equation.
  115. But then, you get to do the log
    thing and divide them instead,
  116. which is this to this on this
    log subtraction.
  117. Okay.
    Let me remind you why we're
  118. putting ourselves through this
    pain.
  119. Okay?
    Recall why we started doing
  120. this in the first place.
    The goal was to figure out how
  121. to measure the Hubble Diagram.
    The Hubble Diagram is this
  122. diagram of velocity,
    which you can measure by
  123. redshift versus distance.
    And the whole reason we
  124. embarked on this adventure in
    magnitudes was because that's a
  125. critical component in how you
    determine the distance.
  126. But if you've got a bunch of
    galaxies and you measure these
  127. two quantities for each one of
    them,
  128. what you discover--what Hubble
    discovered--what Edwin Hubble
  129. discovered many years ago,
    is that they line up.
  130. You get this perfect;
    well, not quite perfect,
  131. but close to it--this beautiful
    straight line,
  132. if you measure a bunch of these
    things.
  133. And the way you represent that
    straight line is with this
  134. equation, where H is
    Hubble's Constant.
  135. And so, that's the purpose of
    all this magnitude stuff,
  136. is to be able to determine the
    y-axis of this plot.
  137. What I now want to do is talk
    about the implications of this
  138. observational fact that galaxies
    line up on this line.
  139. It turns out,
    this is one of the most
  140. profound plots in all of
    astrophysics,
  141. and possibly all of science.
    Because what this implies is,
  142. first of all,
    that the Universe is expanding,
  143. and hence, it's the basis for
    the whole Big Bang Theory of
  144. cosmology.
    And by performing relatively
  145. simple calculations using this
    quantity H,
  146. you can determine the age of
    the Universe,
  147. and the ultimate fate of the
    Universe.
  148. Not bad for a relatively simple
    algebraic equation.
  149. Yes?
    Student: [Inaudible]
  150. Professor Charles
    Bailyn: Huh?
  151. Student: [Inaudible]
    Professor Charles
  152. Bailyn: Oh,
    fate.
  153. I'll get back to that.
    The big question in cosmology
  154. is, you know--the Universe is
    now expanding.
  155. The question is,
    will it continue to expand,
  156. in which case,
    the Universe just gets sparse
  157. and cold and boring,
    and expands forever until
  158. there's, you know,
    one pathetic hydrogen atom
  159. every cubic megaparsec of space.
    Or, alternatively,
  160. it could stop,
    slow down, and recollapse into
  161. something called a Big Crunch,
    which is sort of the Big Bang
  162. run backwards,
    and basically the whole thing
  163. turns into a massive black hole.
    These are--you know,
  164. this is ending in fire or in
    ice, I guess.
  165. And it can be computed in ways
    we'll describe later.
  166. Okay, so, here's what I want to
    do.
  167. I want to start understanding
    how this plot and this little
  168. equation gives you all these
    wonderful things.
  169. I'm going to go on for a little
    while, then we'll pause,
  170. and we'll do one of these Q
    & A sessions,
  171. because this is sort of the
    heart of the Big Bang Theory.
  172. And so, we'll do one of these
    things that we did when we were
  173. talking about relativity,
    where you talk to each other
  174. and come up with questions.
    So, if you've got questions
  175. along the way,
    by all means,
  176. ask them, but we will have a
    specific moment a little ways
  177. down the line where we actually
    pause and do this on purpose.
  178. So, everybody,
    keep thinking as we go along,
  179. what are your questions?
    What don't you understand or
  180. what questions could you ask to
    understand more than what I've
  181. just told you?
    Okay.
  182. Here we go.
    Imagine a one-dimensional
  183. Universe, just because it's easy
    for me to write down.
  184. And here's our one-dimensional
    Universe.
  185. It's all strung out on a line.
    Here's the line.
  186. And it's got a bunch of
    galaxies on it.
  187. Let's label these galaxies,
    A, B,
  188. C, D,
    E, and F.
  189. And these galaxies are spaced
    evenly, let us imagine.
  190. And we'll give them coordinates.
    So this is at 0,1,
  191. 2, and so forth.
    Okay?
  192. Now, next thing we're going to
    do: the Universe is going to
  193. double in size.
    So, we're just going to stretch
  194. the thing.
    The whole thing is going to get
  195. stretched.
    So, here's our Universe.
  196. And now A,
    B, C,
  197. D, E and F
    are further apart by a factor of
  198. two.
    A, B,
  199. C, D, E,
    F. And if A,
  200. we imagine stays at the same
    coordinate--if our coordinate
  201. system starts with A,
    this means B is now at 2.
  202. C is 4--6,8 and 10.
    Okay.
  203. And let us imagine that it
    takes one time unit--one year or
  204. something like that,
    for this doubling to take
  205. place.
    Now, we're going to ask,
  206. if you sit--if you live in
    galaxy A,
  207. if you live on planet A,
    and you observe the distance
  208. and velocity of all these other
    galaxies, what's it going to
  209. look like?
    So, observer on A.
  210. And so, we're going to observe
    a particular galaxy--one of
  211. these other galaxies.
    We're going to write down the
  212. distance.
    We'll choose the distance at
  213. the start, because it's going to
    change.
  214. Then we're going to evaluate
    how that galaxy has moved,
  215. and then the distance changed.
    And then, over here,
  216. we're going to get the
    velocity.
  217. The velocity is going to be the
    change in distance divided by
  218. the change in time,
    which we've defined to be one
  219. time unit.
    Okay?
  220. So, galaxy B.
    Galaxy B starts at a
  221. distance of 1 away from us,
    because it starts 1.
  222. A starts at 0.
    It moved from 1 to 2,
  223. and that gives it a change in
    distance of 1 and therefore its
  224. velocity over this time,
    it's changed in distance by 1,
  225. it's taken 1 time unit.
    The velocity is 1 divided by 1,
  226. equals 1.
    Okay?
  227. The algebra is easy.
    Okay, so, how about C?
  228. C starts 2 away.
    Its motion--it goes from 2 away
  229. to 4 away, and so,
    the change in its distance is
  230. 2.
    And since it takes 1 time unit
  231. to do, its velocity is 2/1,
    which is equal to 2.
  232. See, the algebra is simple,
    but I screw it up.
  233. Bethany is laughing at me and
    well she might,
  234. but I caught myself.
    All right.
  235. And so on down the line.
    I could repeat this simple
  236. exercise.
    D, E,
  237. F start at 3,4,
    and 5.
  238. Goes from 3 to 6,
    from 4 to 8,
  239. from 5 to 10,
    and their velocities are 3,4,
  240. and 5, respectively.
    And so, if I plot distance
  241. versus velocity,
    I'll get points lined up just
  242. like this.
    And so, basically,
  243. what happens is this.
    If you take a set of points on
  244. a coordinate system and you
    simply stretch the coordinate
  245. system,
    what happens is that the
  246. further away you start,
    the greater the stretch is.
  247. And so, there is a correlation
    between how far away you start
  248. and how fast the thing recedes
    from you.
  249. Now, this is true regardless of
    which point you sit on.
  250. Let's imagine that we sit on
    point--that the observer is on
  251. point E.
    So, on point E, we're
  252. now going to--which galaxy,
    initial distance,
  253. motion, velocity.
    So, this is the same
  254. plot--chart as before.
    E--let's look at
  255. F, starts at a distance
    of 1 because it's 1 unit away.
  256. The motion--okay,
    at the start it's--let's take a
  257. quick look at how this is set up
    here.
  258. Yes, E is at 4.
    So, at the start,
  259. the distance is between 4 and
    5, which is 1.
  260. And it goes to a distance of 8,
    which is where E ends
  261. up, and 10, which is where
    F ends up.
  262. So that is a change of 2.
    No a change of 1, right?
  263. Because it's gone from 1 away
    to 2 away, and the velocity is
  264. 1.
    If we look at D,
  265. it starts at a distance of 1,
    goes--starts at 4 to 3,
  266. that's one separation.
    Goes to 8 to 6,
  267. that's a separation of 2.
    Delta D is again equal
  268. to 1.
    Velocity is equal to 1.
  269. Let's look, for example,
    at B.
  270. That starts 3 away.
    And it starts from 4--B,
  271. C, yeah--from 4 to 1.
    And it goes from 8 to 2.
  272. So, that's a difference of 6
    here, a difference 3 there.
  273. And so delta D is equal
    to 3.
  274. And so, the velocity is once
    again 3.
  275. And so, you get the exact same
    plot with different galaxies,
  276. because you're sitting in a
    different place.
  277. So, it doesn't matter which
    galaxy you're sitting on.
  278. You see the exact same ratio of
    distance to velocity and you
  279. create the exact same Hubble
    flow no matter which galaxy you
  280. sit on in this little toy
    Universe.
  281. So, that's the key point--that
    if you take a coordinate system
  282. and you expand it,
    you naturally get this
  283. relationship between distance
    and velocity.
  284. Or to turn it around,
    if you observe this
  285. relationship between distance
    and velocity,
  286. then what you're looking at is
    a system in which all the
  287. coordinates are--in which you've
    simply stretched the coordinate
  288. system.
    Okay.
  289. Now, this gives rise to--this
    analogy, with these stretching
  290. one-dimensional lines gives rise
    to two questions people have,
  291. which I like to call
    un-questions,
  292. because they're actually
    questions that arise because of
  293. the analogy,
    not because of the way the
  294. Universe works.
    One question is,
  295. Q1: "Where is the center?"
    You know, here's your line.
  296. It's expanding.
    But somewhere in the middle
  297. here, around C or
    D in our thing,
  298. is the center away from which
    everything is expanding.
  299. So that's one question.
    And the second question is,
  300. "What is it expanding into?"
  301. You know, here you have a
    little Universe and it's moving
  302. outwards.
    And so, what's going on over
  303. here?
    What was there before the
  304. Universe moved into it?
    These kinds of questions.
  305. And those kinds of questions
    come about because this is
  306. actually a bad analogy--this
    straight line Universe.
  307. So, let me give you a slightly
    better one.
  308. We'll stick with the
    one-dimensional Universe,
  309. but now we'll do it this way.
    Here's a one-dimensional
  310. Universe.
    You have to stay on the line.
  311. So here's A,
    B, C, D, E, F,
  312. whatever.
    And it's going to expand.
  313. And it's going to expand into
    something that looks like this:
  314. A, B, C, D,
    E, F. And all of what we
  315. just did about the velocity,
    and so forth,
  316. remains the same.
    But, notice that this system is
  317. unbounded.
    There's no edge.
  318. There's no edge.
    There's no place where you can
  319. say, this is the end of the
    Universe, because if you
  320. traveled around it you'd just
    come back to where you were,
  321. and therefore,
    there's also no center.
  322. And where does it expand into?
    It expands into a dimension
  323. that, if you're a
    one-dimensional creature,
  324. you can't experience because
    the whole thing is being pushed
  325. out.
    But if you're forced to live on
  326. this circle you can't even--you
    have no comprehension of what it
  327. expands into.
    It expands into a higher
  328. dimension.
  329. But all of this stuff about,
    you know, velocity and
  330. distance, remains basically the
    same.
  331. Here's a two-dimensional
    analogy.
  332. Let's see, this is--so,
    I made a little diagram of the
  333. Old Campus [an area of the Yale
    campus].
  334. Here's Linsley–Chittenden [a
    classroom building]
  335. where we're sitting right now,
    yes?
  336. Here's the statue of Abraham
    Pierson.
  337. This is the gate between Durfee
    and Wright [two undergraduate
  338. dorm buildings].
    Here's Phelps Gate [a classroom
  339. building and the entrance to Old
    Campus].
  340. Here's Vanderbilt [a dorm
    building].
  341. This is Harkness Tower and
    here's Starbucks.
  342. Okay, that's all that's
    important, right?
  343. So, you following me with that?
    And what I did was,
  344. I took this little picture and
    I took it to the Xerox machine
  345. and I blew it up by 20%.
    So, here's the exact same
  346. diagram blown up by 20%.
    So, now, supposing we're
  347. sitting in Linsley-Chittenden,
    which we happen to be doing,
  348. and the Universe expanded by
    20%--or, our little corner of
  349. the Universe expanded by 20%,
    here's what would happen.
  350. Now, notice what's happening.
    Every object in the Universe is
  351. moving away from us.
    See?
  352. Here's where Harkness was,
    and now it's moved a little
  353. further, in a straight line away
    from us.
  354. Here is Pierson,
    and he's moved a little
  355. further, straight away from us.
    And here's Phelps,
  356. and it's moved a little
    further, straight away from us.
  357. And let me erase those lines,
    because what I want to
  358. demonstrate is that if you're
    anywhere else in this Universe,
  359. the exact same thing happens.
    Here we're now sitting on that
  360. statue, and Linsley-Chittenden
    is moving away from us.
  361. Harkness is moving away from us.
    Phelps is moving away from.
  362. Starbucks is moving away from
    us, and so forth.
  363. Similarly, if you're sitting in
    Starbucks, waiting for students
  364. to come by or something,
    the exact same thing happens.
  365. And now, because the distances
    are greater, you can see the
  366. effect that the velocity is
    greater at greater distances.
  367. If I'm looking down at
    Vanderbilt, it moves straight
  368. away from me,
    but only a little bit.
  369. If I'm looking all the way
    across Old Campus,
  370. this gate moves a lot away from
    me.
  371. And so, once again,
    you have a situation in which
  372. the further away
    some--everything is moving
  373. straight away from you,
    but the further away it is,
  374. the faster it's moving away,
    right?
  375. And that's just a consequence
    of the fact that you have taken
  376. this geometry and expanded it.
    And so, wherever you sit in an
  377. expanding geometry,
    every object you see will be
  378. moving directly away from you.
    And the further away it is,
  379. the faster it will be moving,
    which is Hubble's law.
  380. Oh, and one other thing about
    this nice analogy,
  381. here.
    Let us imagine for a second
  382. that this tiny piece of a tiny
    Universe is actually not a flat
  383. plane,
    but is sitting on a curved
  384. surface, which is curved all the
    way round into a big ball.
  385. That's actually not so hard to
    imagine because it's true.
  386. This sits on the surface of the
    Earth.
  387. And so, what is happening when
    this thing blows up by 20% is,
  388. basically, somebody has taken a
    valve to the Earth and has blown
  389. the Earth up by a factor of 20%.
    And that would have this effect.
  390. And it would have the exact
    same effect everywhere else on
  391. the surface of the Earth.
    And the Earth,
  392. you know--where is the center
    of the surface of the Earth?
  393. You can answer the question:
    "Where is the center of the
  394. Earth?"
    But you can't answer the
  395. question of where is the center
    of the surface of the Earth.
  396. Because wherever you sit,
    whether you're sitting at
  397. Starbucks or in Phelps Gate or,
    you know, in Los Angeles
  398. somewhere or wherever,
    if they blow the Earth up by
  399. 20% you're going to see this
    exact same effect.
  400. Everything will be moving away
    from you.
  401. The further away something is
    the faster it will be moving.
  402. So that's the one--yes,
    go ahead.
  403. Student: Someone in the
    back apparently [inaudible]
  404. Professor Charles
    Bailyn: Yeah,
  405. talk.
    Student: If something's
  406. far enough away from you would
    it appear to be moving at the
  407. speed of light?
    Professor Charles
  408. Bailyn: Yes,
    yes, good question.
  409. If something's far away from
    you, will it appear to be moving
  410. at the speed of light?
    Yes, it will,
  411. and that's one of the
    fundamental differences between
  412. the motion of an object due to
    what's called the Hubble Flow,
  413. due to the expansion,
    and ordinary motion of objects.
  414. Now, if something's moving
    faster than the speed of light,
  415. of course, you can't see it,
    because the light from that is
  416. redshifted down to greater than
    infinite wavelengths.
  417. So, the photons don't have any
    energy left.
  418. But, let me come back to the
    question after I do one more
  419. thing.
    That's a good question.
  420. Here's the thing I want to do.
    So, we've had the
  421. one-dimensional case,
    the circle.
  422. We've had the two-dimensional
    case, the expanding sphere.
  423. Of course, what we want is the
    three-dimensional space.
  424. Okay, here we are in three
    dimensions.
  425. Someone is expanding the
    Universe, so everywhere we look,
  426. everything is going away from
    us, and the further away it is,
  427. the faster it's going.
    What's it expanding into?
  428. Well, that, we have a little
    more trouble visualizing,
  429. right?
    Because in one dimension,
  430. you can visualize this circle
    expanding onto the plane.
  431. In two dimensions,
    you can imagine this spherical
  432. surface expanding.
    In three dimensions,
  433. we can't imagine what it's
    expanding into.
  434. That's beyond us.
    And so, having had this failure
  435. of the imagination,
    what do you do?
  436. You resort to mathematics.
    That's what we always do.
  437. And so, imagine that every
    object has a position,
  438. which is denoted by three
    coordinates,
  439. three spatial coordinates
    x, y,
  440. and z. But now,
    let's imagine that every
  441. object's position has this
    coordinate system times a scale
  442. factor,
    which is a function of T.
  443. So, it's a scale factor times a
    coordinate position.
  444. And there are two ways that
    things can change their
  445. position.
    One is, they can move;
  446. they can change their x,
    y, z position.
  447. This is the equivalent of
    somebody walking across the Old
  448. Campus.
    You walk from Starbucks to
  449. Phelps Gate, or something like
    that, and you change your x,
  450. y, z coordinate position by
    moving through space.
  451. So, changes in position,
    which is to say,
  452. velocity, can be accomplished
    in two ways.
  453. One is motion through the
    coordinate system--that is to
  454. say, changing your x,
    y, and z.
  455. This is called peculiar motion.
    That's a jargon.
  456. And it's called peculiar
    because one object can have a
  457. different peculiar motion from
    every other.
  458. It's peculiar in the
    old-fashioned sense meaning
  459. specific to one object.
    So, you have a peculiar motion
  460. that's all your own.
    I have one that's all mine as
  461. we move through x,
    y, and z.
  462. But the other is just the
    effect of the change.
  463. And in particular,
    in the case of the current
  464. Universe, the increase in the
    scale factor.
  465. And these two kinds of velocity
    are conceptually different from
  466. each other.
    Because you don't have to do
  467. anything to change your position
    in this way.
  468. You just sit there.
    You don't expend any energy.
  469. You don't have any requirement
    to expend energy or to exert a
  470. force or to do any of these
    things that we ordinarily do to
  471. change our position.
    You just sit there and the
  472. Universe expands you,
    or expands your position.
  473. And that's why,
    going back to your question,
  474. that's why it's possible for
    this kind of velocity to turn
  475. out to be greater than the speed
    of light,
  476. if it's far enough away,
    whereas, it's not possible
  477. here.
    What the effect of having this
  478. kind of velocity be faster than
    the speed of light does,
  479. is it makes the object
    impossible to see,
  480. because photons coming off them
    would be redshifted into
  481. oblivion.
    And so, you can't actually see
  482. them.
    And this imposes a kind of
  483. cosmic event horizon,
    similar in kind to the event
  484. horizons around black holes.
    You can't see events on the
  485. other side of these horizons.
    All right, let me go backwards
  486. in time.
    Back in time to when this scale
  487. factor A of T is
    equal to 0.
  488. So, if the scale factor is
    expanding, you know,
  489. the Old Campus,
    whatever, is expanding,
  490. and you reverse time and you
    think about what happened long
  491. ago, it must have been smaller.
    So this A factor must
  492. have been smaller.
    And if you go back a sufficient
  493. amount in time,
    you go back to the point where
  494. A is equal to 0.
    So then, what does the Old
  495. Campus look like?
    Looks like this, right?
  496. It's all been--imagine I take a
    Xerox machine and I de-magnify
  497. the thing down to 0% of its
    original size,
  498. or 0 plus epsilon,
    perhaps.
  499. What does that little diagram
    look like?
  500. It looks like a tiny little dot.
    A single point,
  501. except it isn't a single point.
    What it is, is it's many points
  502. superposed on each other.
    And not only is the Old Campus
  503. in there--imagine you've taken
    this sphere that is the Earth
  504. and collapsed its radius down to
    0.
  505. Not only the Old Campus is
    there, but so is the rest of New
  506. Haven, so is Connecticut,
    so is Los Angeles,
  507. so is the whole rest of the
    surface of the Earth.
  508. All of the points that will
    eventually make the surface of
  509. the Earth are superposed on one
    another.
  510. And yet, the whole geometry of
    the Earth, of the Old Campus,
  511. whatever, is already somehow
    encoded in that point.
  512. Because, you know,
    you're taking 0 times x,
  513. y, z for every
    point.
  514. But then, if you increase this
    so that it becomes epsilon or
  515. some non-zero number,
    times x,
  516. y, z, then you
    already get the geometry of the
  517. Old Campus and everything else
    on the surface of the Earth.
  518. So, it is wrong to think of the
    Big Bang as starting at a point
  519. and expanding into space.
    That's kind of the impression
  520. that word the Big Bang gives
    you.
  521. And you think,
    naturally enough,
  522. of an explosion,
    where something at a point
  523. explodes into empty space.
    But that's not right.
  524. All the space,
    all the empty space is
  525. contained in that point.
    It's all in there.
  526. It's just, it's all multiplied
    by 0, so it all comes out to be
  527. on top of each other in the same
    place.
  528. How are we doing?
    Wonderful.
  529. Okay, that's the essence of the
    Big Bang--the whole idea of the
  530. Big Bang--that what is happening
    is that the whole coordinate
  531. system of the Universe is
    multiplied by this constant.
  532. And that constant changes in
    time.
  533. It gets bigger.
    And that fact is inferred from
  534. the observations of these
    galaxies.
  535. That you observe galaxies,
    and that they're moving away
  536. from us.
    But more than that--that
  537. there's a linear relationship
    between how far away from us
  538. they are and how fast they're
    moving.
  539. That implies this kind of
    coordinate expansion.
  540. And if you run it in reverse,
    it implies an origin to the
  541. Universe at some specific time
    in the past.
  542. All right.
    Let's have questions.
  543. I'll tell you what.
    Talk to each other.
  544. Come up with good questions,
    and remember how we do this.
  545. When you come up with a
    question--a question can be
  546. either to explain some of this
    or to expand upon it.
  547. When you're ready with a
    question, put your hand up.
  548. I'll answer a few of them while
    other people are getting their
  549. questions together.
    And then we'll answer as many
  550. of them as we have time for in
    the remainder of the class.
  551. So, talk to each other.
    Talk amongst yourselves.
  552. Come up with a question,
    any question.
  553. All questions are good,
    and see what you can do.
  554. I'll come around and try and
    answer some of these.
  555. Yeah – so,
    talk to each other by all
  556. means.
    Yes?
  557. Student: There is a
    center of the Universe,
  558. isn't there?
    Professor Charles
  559. Bailyn: No.
    Student: Well,
  560. I mean, like,
    there's got to be a point in
  561. the Universe at which,
    like, all other points--or all
  562. other points that are furthest
    away from that point are
  563. expanding equally fast,
    right?
  564. Professor Charles
    Bailyn: No.
  565. Well, that's true of every
    point in the Universe.
  566. That's true of every point in
    the Universe.
  567. If you're sitting on the statue
    of Pierson, everything at the
  568. same distance is moving away
    from you by the same amount.
  569. That's true of every point.
    Student: Oh, I see.
  570. Professor Charles
    Bailyn: Yeah.
  571. Student: So,
    wait, so, like,
  572. is it, like,
    in four dimensions,
  573. then?
    Professor Charles
  574. Bailyn: Imagine a
    three-dimensional thing
  575. expanding into the fourth
    dimension the way the surface of
  576. a sphere expands into three.
    Student: Okay, thank you.
  577. Student: If you switched
    from the expansion to getting
  578. close together,
    would it precipitate the point
  579. at which the expansion
    [inaudible]
  580. Professor Charles
    Bailyn: Oh we'll talk about
  581. that later.
    You can imagine that gravity is
  582. going to slow the expansion
    down.
  583. But at the moment I'm just
    thinking about running backwards
  584. in time.
    Student: Okay,
  585. and at some point you got to do
    expansion [inaudible]
  586. Professor Charles
    Bailyn: Yeah,
  587. or could.
    Yeah?
  588. Student: I don't
    think--so have you had like one
  589. of those fake event horizons
    where something could be
  590. like--appear to be moving faster
    than the speed of light?
  591. Could you then like--in like a
    normal event horizon you can't
  592. pass information between them
    [inaudible]
  593. Professor Charles
    Bailyn: Right.
  594. Student: But could you,
    like relay information?
  595. Professor Charles
    Bailyn: Yes,
  596. but by the time you had done
    that--that is to say,
  597. the Universe is expanding
    faster than the information
  598. would get to you.
    So, you know,
  599. by the time you've moved your
    information from here to here,
  600. the distance from here to here
    has expanded.
  601. And so, it's not actually
    getting any closer.
  602. Okay, let's have a few of these.
    Yes, go ahead.
  603. Student: Well I
    have--first, just simply,
  604. I didn't understand why it's 0
    times the coordinate,
  605. why there's a point at all and
    why there's just nothing.
  606. Why there's not [inaudible]
    Professor Charles
  607. Bailyn: Well,
    okay.
  608. So, if A is equal to 0
    here, then all events are at
  609. 0,0, 0.
    So, that point of 0,0,
  610. 0 is occupied,
    if you want to think of it that
  611. way.
    It's the point--the point in
  612. question is the one at 0,0,
    0.
  613. And if you have anything in
    here times 0,
  614. that's where you're going to
    end up.
  615. And as you go backwards and run
    this scale factor down to 0 as
  616. you think about going back in
    time,
  617. everything's going to end up
    closer and closer to 0.
  618. And then, at the moment when
    this equals 0,
  619. it's all piled on top of
    itself.
  620. Does that make sense?
    Yeah go ahead.
  621. Student: Everything has
    to be equidistant [Inaudible]
  622. Professor Charles
    Bailyn: No,
  623. no, no, no, no.
    Imagine-- Student:
  624. [Inaudible]
    Professor Charles
  625. Bailyn: Oh,
    in the other dimension.
  626. Student: Yes.
    Professor Charles
  627. Bailyn: Yeah,
    okay.
  628. In the higher dimension,
    that's true.
  629. Well, okay, let me back off.
    That's true if you have
  630. constant curvature.
    You could imagine something
  631. that, you know,
    looks like this,
  632. that also expands in exactly
    the same way.
  633. And what creates the curvature
    of space-time?
  634. Gravity.
    And so, we're going to end up
  635. in a situation where we're going
    to be able to determine what the
  636. curvature of space is by seeing
    how much mass there is.
  637. So, this actually relates back
    to part two of the course.
  638. Student: So,
    is that why time could move
  639. differently?
    Professor Charles
  640. Bailyn: Huh?
    Student: Time could move
  641. differently.
    Professor Charles
  642. Bailyn: Time could move
    differently, too.
  643. Yep.
    Yes?
  644. Student: Isn't--in the
    one-dimensional circle and you
  645. said that thing's expanding to a
    higher dimension--what does that
  646. mean?
    Professor Charles
  647. Bailyn: Okay.
    So, imagine you're a
  648. one-dimensional creature.
    You can only go on a line.
  649. But your line,
    unbeknownst to you,
  650. is curved, so that it ends on
    itself.
  651. That then--the line expands.
    The circle expands.
  652. What do you think is happening?
    You think suddenly all the
  653. distances are greater,
    but you can't perceive what
  654. it's expanding into,
    because that's a second
  655. dimension, and you're
    constrained to move on the line.
  656. Similarly, if you're an ant,
    or something moving on the
  657. surface of the Earth,
    and you have no understanding
  658. that there's up and down,
    and the Earth suddenly
  659. increases in size,
    you have no perception of the
  660. dimension you're moving into.
    Student: So then,
  661. they just keep moving into
    higher dimensions?
  662. Professor Charles
    Bailyn: Well,
  663. our perception is of a
    three-dimensional Universe.
  664. So, if you say,
    okay, what happens when the
  665. scale factor gets bigger?
    You can write it down
  666. mathematically into all these
    calculations,
  667. and what you have to imagine,
    if you want to have a
  668. conceptual understanding of
    this, is that there's some
  669. fourth spatial dimension into
    which this three-dimensional
  670. space is moving into.
    But we can't perceive that.
  671. So, it's better to do it
    mathematically,
  672. because it's hard on the brain
    to imagine that fourth
  673. dimension.
    And there's basically two ways
  674. to think about this.
    One is by analogy.
  675. You think of the
    one-dimensional creature and
  676. what it thinks about the second
    dimension,
  677. or the two-dimensional
    creature--what it thinks about
  678. the third dimension.
    And then, you just,
  679. sort of, take this leap of
    faith into us moving into the
  680. fourth.
    Or, you have to write things
  681. down mathematically.
    Those are the only two options
  682. you have.
    Actually picturing this is not
  683. going to happen.
    Yes?
  684. Student: If we were to
    speed up this process would we
  685. be able to see things getting
    farther away from us?
  686. Professor Charles
    Bailyn: Ah,
  687. interesting question.
    Would we, if we cranked up this
  688. process some,
    would we actually,
  689. you know, be able to take out
    our tape measure and notice that
  690. the distance between here and
    Starbucks has increased?
  691. No.
    Here's why.
  692. We are not--we--that is to say,
    you, me, our bodies,
  693. whatever--are not participating
    in the expansion of the Universe
  694. at the moment.
    Why?
  695. Because the molecules in our
    body are being held together by
  696. other forces--chemical forces.
    We are being--the Earth is held
  697. together by gravitational forces
    and those forces have stopped
  698. the expansion of the Universe
    locally,
  699. but not globally.
    And that's why you have to go
  700. out and measure this stuff with
    galaxies.
  701. Otherwise, you'd be able to,
    you know, measure it with this.
  702. Now, of course,
    then, there's the problem,
  703. the ruler also expands,
    all this kind of stuff.
  704. But, in fact,
    what happens is that
  705. locally--local objects,
    by which I mean,
  706. our own galaxy and anything
    smaller, are held together by
  707. other forces.
    Picture a balloon expanding,
  708. so that's the two-dimensional
    case, this balloon expands.
  709. And put a bunch of leather
    patches on the balloon.
  710. So, those leather patches don't
    expand, but the distance
  711. between--because they're held
    together by something else--but
  712. the distance between them does.
    Yes?
  713. Student: So,
    what causes the expansion?
  714. Professor Charles
    Bailyn: What causes the
  715. expansion?
    Oh, that's theology.
  716. No, no, seriously,
    it's an initial condition.
  717. Something at the start,
    at T equals 0,
  718. where this A is 0.
    So, you're starting with
  719. A of--with this scale
    factor of 0, but the derivative
  720. isn't 0.
    It's expanding,
  721. A is increasing.
    Why?
  722. Why is it increasing by the
    particular amount it is?
  723. You have to think of that as
    being a parameter of the
  724. Universe.
    It's one of the things about
  725. the--sort of like the speed of
    light.
  726. Why is that the quantity it is?
    Or why is the gravitational
  727. constant the value it is?
    It's one of the parameters that
  728. governs our Universe.
    Why those particular parameters?
  729. That's not quite a science
    question, and we don't know.
  730. You can turn it around.
    You can make the following
  731. interesting point.
    If it wasn't expanding at
  732. approximately the rate it is,
    we wouldn't be here to notice
  733. it.
    Because if it was expanding a
  734. whole lot slower,
    then it would never have gotten
  735. big enough for stars to condense
    out.
  736. If it was expanding a whole lot
    faster, all the atoms would be
  737. spread out so far that,
    again, stars could never form.
  738. So, you can make this
    interesting argument that while
  739. you don't know why it has this
    particular value,
  740. it's very important that it
    does, because otherwise we
  741. wouldn't be here to observe it.
    This is a form of argumentation
  742. called the anthropic principle.
    It is highly debated among
  743. scientists whether this is a
    scientific argument or not.
  744. But it's amusing either way.
    So--yes?
  745. Student: So if
    derivative A is positive,
  746. then A should have been
    0 for that, right?
  747. Professor Charles
    Bailyn: Well,
  748. you go back before A and
    you're in trouble.
  749. So - Student: What's
    that mean?
  750. Professor Charles
    Bailyn: Okay.
  751. So, this is another one of
    these un-questions.
  752. It's usually phrased as,
    well, what happened before the
  753. Big Bang, right?
    That's the equivalent question.
  754. And, again, that's a
    theological question.
  755. It's like asking the question:
    "What's going on inside an
  756. event horizon?"
    You can write down equations.
  757. You can talk about it.
    But it's un-testable by its
  758. very nature.
    And so, something– This,
  759. by the way, is why the
    Catholics like the Big Bang
  760. Theory so much.
    In fact, the mathematics,
  761. the relativistic mathematics
    that describe the Big Bang and
  762. the expansion of the Universe,
    were worked out by a man named
  763. Lemaitre, who was a Catholic
    priest, who was a Jesuit.
  764. They love this because it gives
    you a creation moment.
  765. It kind of gives you a
    scientifically verified creation
  766. moment.
    John Paul was very enthusiastic
  767. about astrophysics.
    He used to throw big
  768. conferences in the Vatican,
    give after-dinner speeches.
  769. There's a thing called the
    Vatican Astrophysical
  770. Observatory.
    They have a bunch of Jesuits.
  771. They run a telescope in Arizona.
    They do research into this.
  772. And so, you know,
    if you want to see science and
  773. religion converge,
    you want to run away from
  774. biology as fast you can and talk
    to us about cosmology.
  775. And what I don't understand is
    why the, sort of,
  776. fundamentalist type worry so
    much about biology,
  777. where all the science is
    dead-set against them.
  778. Whereas, in the case of
    astrophysics--now,
  779. of course, it is possible,
    if you take an atheistic point
  780. of view,
    to come up with all kinds of
  781. clever ways to avoid this
    creation event.
  782. But again, it stops being
    science at a certain point.
  783. It becomes another odd kind of
    theology, and we'll talk about
  784. that, perhaps,
    a little bit.
  785. But if you want a place where
    science turned out to be the
  786. congruent to,
    at least,
  787. certain kinds of
    non-fundamentalist religious
  788. beliefs, you're way better off
    in astrophysics than you are in
  789. biology.
    Yeah, go ahead.
  790. Student: Back to the
    fourth dimension,
  791. is that a time dimension or a
    space dimension?
  792. Professor Charles
    Bailyn: No,
  793. that's a fourth spatial
    dimension that I'm talking about
  794. in this case.
    Student: So,
  795. in other words,
    spatial dimension [Inaudible]
  796. Professor Charles
    Bailyn: It's expanding into
  797. a dimension which we don't
    perceive.
  798. Or, at least,
    that's one way of
  799. conceptualizing it.
    We don't get to deal with that
  800. dimension.
    So, again, it's a concept.
  801. It's not necessarily a physical
    thing.
  802. Yes?
    Student: Back to that
  803. one point, I was just wondering,
    is that like anything that will
  804. ever [Inaudible]
    Professor Charles
  805. Bailyn: Right.
    Student: [inaudible]
  806. and anything that will ever
    expand?
  807. Professor Charles
    Bailyn: Yeah,
  808. yeah, yeah.
    It has to be.
  809. Think of all the points that
    now exist in the Universe.
  810. Multiply them by 0,
    and they're all on top of each
  811. other.
    Yes?
  812. Student: So if you were
    to go really,
  813. really fast in a straight line
    in any direction,
  814. would it eventually end at a
    [Inaudible]
  815. Professor Charles
    Bailyn: So,
  816. this is the question of what is
    the curvature of the Universe.
  817. Is it, in fact,
    like this analogy,
  818. with the circle or the surface
    of the sphere.
  819. If you go forever,
    do you come back to where you
  820. are?
    That depends on what the
  821. overall curvature of the
    Universe turns out to be.
  822. That depends on how much mass
    it contains, because mass curves
  823. space.
    And so, that is a question
  824. that's empirically answerable.
    We think the answer is no.
  825. In any case,
    though, it would take you
  826. longer, at the speed of light,
    than the age of the Universe to
  827. accomplish it.
    So, even if it were positively
  828. curved, in this sense that it
    rejoined itself,
  829. you wouldn't actually be able
    to make that voyage.
  830. Student: [Inaudible]
    Professor Charles
  831. Bailyn: No.
    Student: [Inaudible]
  832. Professor Charles
    Bailyn: Well,
  833. you might, eventually,
    unless the Universe is
  834. expanding faster than you can
    move.
  835. This goes back to the question
    of can parts of it expand faster
  836. than the speed of light,
    and the answer is yes.
  837. So, yeah.
    You take off on this voyage to
  838. the back of your own head.
    Going in that direction,
  839. you're going to end up
    here--except the Universe is
  840. expanding faster than you can
    go,
  841. and so, you don't necessarily
    end up back there.
  842. Yes?
    Student: How does the
  843. Big Crunch happen in the
    expansion?
  844. Professor Charles
    Bailyn: Oh,
  845. one of the things that we
    haven't talked about is the
  846. expansion rate constant.
    Might it be that the Universe
  847. is expanding,
    but perhaps slowing down?
  848. Why would it slow down?
    It's full of mass.
  849. Gravity slows things down.
    And so, in principle,
  850. if you've got enough mass,
    the Universe will stop,
  851. turn around,
    and fall backwards.
  852. And you can compute from--we
    will compute,
  853. next period,
    how fast you have to go to make
  854. that happen.
    Or contrary,
  855. how dense the Universe has to
    be to stop it.
  856. And so, what you expect is the
    Universe is slowing down.
  857. Because you know you don't know
    how much it's slowing down,
  858. whether it will stop the
    expansion or not,
  859. but you expect it to be slowing
    down.
  860. The punch line of this whole
    part of the course,
  861. in order to anticipate where
    we're going,
  862. is that, in fact,
    we can measure the change in
  863. the expansion rate,
    and it's speeding up,
  864. not slowing down.
    This is very disturbing,
  865. because it means there's some
    kind of pervasive cosmic
  866. anti-gravity,
    which we call dark energy,
  867. because we don't know what it
    is.
  868. And that's basically the punch
    line of this whole section of
  869. the course.
    But you could imagine that
  870. there is a change in the
    expansion rate,
  871. either positive or negative,
    and that, therefore,
  872. various different outcomes are
    possible.
  873. Yeah?
    Student: Would a
  874. hypothetical of the Big Crunch
    be sort of different from Big
  875. Bang in that the Big Bang sort
    of expanded the whole geometry
  876. of the Universe where as the Big
    Crunch being motivated by
  877. gravity would only cause the
    object [inaudible]
  878. Professor Charles
    Bailyn: No,
  879. no, no, no, no.
    It turns out it--you might
  880. imagine that,
    but you'd be wrong.
  881. It turns out that this scale
    factor--you can write down a
  882. differential equation for the
    scale factor,
  883. and that it's the scale factor
    that stops, turns around and
  884. comes back.
    Now, it is the case that when
  885. it's going out,
    the particular objects the
  886. Universe contains are different
    because of the peculiar motions.
  887. It starts out much more evenly
    spread, and then gradually stuff
  888. accumulates due to gravity,
    and you get--when you're coming
  889. back, you get a lot of
    individual points,
  890. rather than a smooth
    distribution of matter.
  891. But that has to do with
    peculiar motions,
  892. not the overall motion of the
    Universe.
  893. One more and then we got to
    continue this next time.
  894. Yes, go ahead.
    Student: Before the Big
  895. Bang, what happens when the
    minus--negative number did
  896. [Inaudible]
    Professor Charles
  897. Bailyn: Right.
    So, before the Big Bang,
  898. can you put a negative number
    into that A?
  899. This gets you into exactly the
    same kind of mathematical
  900. difficulty as moving inside an
    event horizon.
  901. You can actually write down a
    metric for the whole Universe
  902. and you change all the signs
    when that happens.
  903. And you get into a very similar
    kind of problem.
  904. So, mathematically,
    you could compute a bunch of
  905. stuff.
    The physical meaning of that is
  906. not--first of all,
    the fact that that mathematics
  907. actually works in the real world
    is impossible to determine,
  908. and the physical nature of
    consequences of that are
  909. questionable.
    All right, stop.