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12. Stellar Mass Black Holes

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    Professor Charles
    Bailyn: Okay,
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    here's the plan for today.
    I want to do one last foray
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    into relativity theory.
    And this is going to be a
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    tricky one, so I hope you're all
    feeling mentally strong this
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    morning.
    If not, we--gosh,
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    we should have ordered coffee
    for everyone.
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    And, in so doing,
    I want to introduce one key
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    concept, and also answer at
    least three of the questions
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    that you guys have asked before
    in a more--in more depth,
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    and also relate the whole thing
    back to black holes.
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    And then, having done that,
    we'll have some more questions.
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    And then, having done that,
    I want to get back to
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    astronomy;
    that is to say,
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    to things in the sky that
    actually manifest these
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    relativistic effects.
    So, that's where we're going
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    today.
    And along the way,
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    as I said, we'll deal with some
    of the questions you've been
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    asking in a deeper kind of way.
    In particular--so, questions.
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    Watch out for the answers to
    these questions.
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    Somebody asked,
    "What's special about special
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    relativity, and what's general
    about general relativity?"
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    How do they relate?
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    So, we'll come back to that one.
    Somebody also asked,
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    "Why use the speed of light to
    convert time into space and vice
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    versa, to get them in the same
    coordinate system?"
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    So, why use c to convert
    time to space and vice versa?
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    And then, also,
    there was the question of,
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    you know, "What is the
    mathematical formulation of
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    general relativity?"
    So, how to express general
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    relativity in some kind of
    equation.
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    And we'll get to the key
    equation, which is something
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    called a metric,
    for general relativity,
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    and then we're going to stop.
    Because to go on from there is
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    fairly heavy calculus and we're
    just not going to do that.
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    But I want to get at least that
    far.
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    Okay, so let's go back to
    special relativity for a minute.
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    So, special relativity.
    Flat space-time, no gravity.
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    And you'll recall what happens.
    As you get close to the speed
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    of light, all sorts of things
    that you thought were kind of
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    constant and properties of
    objects,
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    like mass and length and
    duration, and duration of time,
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    and things like that,
    all start to get weird and
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    change.
    So, length, time,
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    mass, all these things,
    vary with the velocity of the
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    person doing the measuring.
    And so, you could ask the
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    question, is there anything that
    doesn't vary?
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    Is there anything that's an
    invariant?
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    And the answer is, yes.
    There are some things that
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    don't vary.
    So, some things are invariant.
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    And Einstein actually said
    later in his career that it's
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    actually the invariants that are
    important, not the things that
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    change.
    And so, he should have called
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    his theory invariant theory
    instead of relativity theory.
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    Think of what that would have
    done to pop philosophy.
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    Instead of saying,
    "everything is relative," all
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    this stuff, you would have had
    the exact same theory.
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    You would have called it
    invariance theory.
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    And the pop philosophy
    interpretation of this would be,
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    "some things never change."
    And it would have been a whole
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    different concept in three in
    the morning dorm room
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    conversations.
    Okay, so some things are
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    invariant, what things?
    Now, let me first give you a
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    little bit of a metaphor and
    then come back to how this
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    really works in space-time.
    Supposing you're just looking
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    at an xy-coordinate
    system and you have two points
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    in a two-dimensional space.
    So, here's a point and here's a
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    point.
    Now, if you arrange for some
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    kind of coordinate
    system--here's a coordinate
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    system.
    This is x,
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    this is y--and you ask
    how far apart these points are.
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    Well, you can do that--let's
    see.
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    They're separated in x
    by this amount here,
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    which we'll call delta [Δ]
    x. And they're separated
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    in y by this amount here,
    and that's Δ y.
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    But of course,
    those quantities depend on the
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    orientation of your coordinate
    system.
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    If I now take this coordinate
    system and I shift it like this,
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    now it's going to be totally
    different.
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    Now I'm going to have x
    look like this and I'm going to
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    have y,
    Δ y look like that.
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    So Δ y has gotten a
    whole lot smaller.
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    Δ x has gotten a whole
    lot bigger.
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    And all I did was twist the
    coordinate system.
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    Yeah?
    Student: [Inaudible.]
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    Professor Charles
    Bailyn: You still get the
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    same distance,
    thank you very much,
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    that's exactly right.
    The distance is the invariant.
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    The x-coordinate and the
    y-coordinate,
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    those vary with the coordinate
    system, but the distance is the
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    same.
    That's exactly the point.
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    And so, the quantity--let me
    summarize this.
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    For points on a 2-D space,
    Δ x varies.
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    Δ y varies.
    But there is a quantity that is
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    invariant and that is--well,
    let's call it (Δ x^(2))
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    + (Δ y^(2)),
    which is the distance squared,
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    (Δ D^(2)).
    And this is invariant.
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    It doesn't matter which way you
    twist things around,
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    that will--that quantity will
    remain the same.
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    So, now, imagine that you've
    got events in space-time.
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    So an event in space-time has
    three spatial coordinates and
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    one time coordinate.
    So, it's basically a point in a
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    four-dimensional space.
    And as your velocity changes,
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    the distance and time also
    change.
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    That's the equivalent of
    rotating the coordinate system.
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    But there is something that
    doesn't change,
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    and let me write that down.
    This is usually given the Greek
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    letter Tau [T]
    squared.
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    And this is equal to (Δ
    x^(2)) + (Δ
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    y^(2)) + (Δ
    Z^(2)) - c^(2)
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    (ΔT)^(2).
    And this is invariant.
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    This is an invariant interval,
    sometimes called proper
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    distance.
    And as you change your
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    velocity--as the space,
    as the mass,
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    as the time all change--this
    quantity,
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    describing the separation of
    two events--so,
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    this describes the separation
    of two events,
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    that quantity changes--doesn't
    change.
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    That quantity is invariant.
    Okay, so now,
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    this answers one of the
    questions that was asked before.
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    Why does one use c^(2)
    or c to transform the
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    space coordinate into the time
    coordinate and back?
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    It's because you need the
    c^(2) out here in order
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    to make this invariant.
    If you're calculating the
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    distance, if you use
    x^(2) plus 1/2
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    y^(2) or some other
    constant times y^(2),
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    you're not going to get
    something that's invariant.
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    And it's only when you use the
    c, here,
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    that you end up with something
    that's invariant.
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    And so, if you think about
    these as representing the four
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    coordinates of the system,
    it's clear this one coordinate
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    is x, one is y,
    one is z, just as you
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    would expect.
    And then, there's this other
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    coordinate, which is c
    times T,
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    but it's negative so it has to
    be times the square root of -1.
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    So, the four coordinates in
    space-time can be thought of as
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    x, y, Z and i cT,
    if you want to think of it that
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    way.
    And the time coordinate is
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    imaginary, because when you
    square it, you have to end up
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    with a negative number.
    Don't worry about the details
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    of that.
    But the presence of the
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    c^(2) here is why you
    have to use c, in
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    particular, to get from time to
    space and back.
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    And that's necessary because
    this is the thing that doesn't
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    vary with velocity.
    All right.
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    So, this is actually kind of a
    weird expression.
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    Because unlike the distance
    between two points--and you'll
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    notice, these three terms put
    together,
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    that's just the distance
    squared, ordinarily,
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    but that is not invariant
    anymore,
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    because there's this other term
    here, which can vary.
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    Unlike the distance,
    this doesn't have to be
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    positive.
    You've got three different
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    cases here.
    This interval can be zero for
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    different points.
    In ordinary distance it can
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    only be zero if the two points
    are the same,
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    but this can be zero for
    different points,
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    for different events.
    It can be zero,
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    it can be negative,
    and it can be positive.
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    So, what does that mean?
    What happens when it's zero?
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    So, if the interval is zero,
    that means that the distance
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    between the events in
    light-years,
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    for example,
    is equal to the time separation
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    in years, because
    that's--because this term has to
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    be exactly equal to that term
    there,
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    in order for them to subtract
    out and get zero.
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    And the c^(2) converts
    from light-years to years and
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    back again.
    And so, what does that mean?
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    That means if you emit a photon
    at one event,
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    that same photon can,
    if it's going in the right
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    direction, be present at the
    second event.
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    So, if you ride along with
    light you'll see both--you'll
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    participate in both these
    events.
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    So, you sit at event one.
    You flash a light.
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    You ride along with the
    expanding light waves from that
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    event and you get to something
    one light-year away,
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    exactly a year later.
    And so, if the second event is
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    one light-year away in distance
    and a year later in time,
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    that same photon will be
    present at the second event,
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    as at the first event.
    So, things that have one of
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    these intervals of zero are
    separated by an appropriate
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    amount so that the same ray of
    light can participate in both of
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    them.
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    So, if the interval--let's keep
    that up there for a minute.
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    If the interval is negative,
    what does that mean?
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    The distance is less than the
    light travel time.
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    So, the photon is already past
    the second event.
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    So, if you were to emit a ray
    of light at event number one,
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    it would have passed the second
    event by the time it occurred.
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    The photon has already gone by.
    And similarly,
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    if the interval is positive,
    then the light photon hasn't
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    reached event two--hasn't yet
    reached event two.
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    Now, this is important,
    because this means that you
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    can't communicate from event one
    to event two.
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    So, if you're at event two,
    you don't know what happened at
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    event one.
    Because even if you'd sent out
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    a signal, a radio signal or
    whatever, it would not have
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    reached you by the time event
    two takes place.
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    So you can't communicate from
    event one to event two.
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    And similarly,
    you can't travel from event one
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    and reach event two,
    because you'd have to go faster
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    than the speed of light to do
    it.
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    These kinds of intervals,
    these negative intervals,
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    these are called time-like,
    because the time term is larger
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    than the distance turn.
    And these kinds of intervals
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    are called space-like intervals.
    And you can only travel or
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    communicate over time-like
    intervals.
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    Yes?
    Student: What are these
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    so-called events?
    Professor Charles
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    Bailyn: So,
    they're events--you can think
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    of them as points in space-time.
    So, they have a particular
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    position in space and a
    particular point in time.
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    So, they can be described by
    four numbers,
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    three spatial coordinates and a
    time coordinate.
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    They can be anything.
    You know, turning on a light,
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    doing anything you want to do.
    Receiving a photon,
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    whatever it is.
    But they are points in a
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    four-dimensional space-time and
    therefore require four numbers
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    to describe them.
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    And you can only get from one
    to another if they're separated
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    from a time-like,
    that is to say,
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    a negative, interval.
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    Okay, so this expression,
    which I'll write down again,
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    this is called a metric.
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    And the particular metric that
    I've written down here is the
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    metric for flat space.
    Because remember,
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    this is special relativity.
    There's no masses,
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    no curvature of space,
    none of that stuff,
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    yet.
    This is the metric for flat
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    space with no mass present.
    And there are many other
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    metrics possible.
    Any time you add mass or do
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    other things,
    you get different kinds of
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    metrics more complicated than
    this.
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    So, what's special about
    special relativity is that you
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    use the metric appropriate for
    flat space as opposed to the
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    many other different kinds of
    metrics that you can use in
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    general relativity,
    which has a much more general
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    form for the metric.
    I should say,
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    you can also write this down.
    You can write down the spatial
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    terms here in polar coordinates.
    Remember polar coordinates?
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    Polar coordinates,
    you describe the position in
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    space, instead of with x,
    y, Z, you describe it with
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    a radius, a distance from zero,
    and some angles.
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    And it turns out,
    that's convenient to do so.
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    Let me write this down in polar
    coordinates, or in polar.
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    That's an r.
    Let me write that explicitly.
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    This is an Omega,
    that's some angle.
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    And then the T thing
    remains the same.
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    And I've pulled a little bit of
    a notational fast one on you
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    here.
    I've gone away from the deltas
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    and I've written these down as
    d.
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    This is the differential
    d.
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    Those of you who have taken
    some calculus will remember
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    this.
    If this were a calculus-based
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    course I would explain why I did
    that, but I'm not going to.
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    So, just allow me this slight
    of hand, here.
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    For technical reasons,
    these have to be differential.
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    Yes?
    Student: But you do need
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    a second angle term for the
    [inaudible]
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    Professor Charles
    Bailyn: I do need a second
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    angle term.
    I should say--good point.
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    You need two angles and a
    distance in three space.
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    This capital Omega here is
    actually--Omega squared is
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    actually sin θ,
    d θ,
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    d φ,
    which is the correct form.
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    And so you could write out both
    terms here, but in fact,
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    this one isn't going to change.
    But you're absolutely right.
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    In principle,
    you need two angles.
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    Okay, why have I done this?
    Excellent question.
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    I ask myself a question at this
    point.
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    Where am I going?
    What I want to do now is write
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    down a different metric.
    A metric that actually involves
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    curved space and the presence of
    a mass.
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    And this is something called
    the Schwarzschild metric.
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    Remember Schwarzschild?
    He had a radius.
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    And this is the appropriate
    metric for the presence of a
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    single point mass at the center
    of the coordinate system,
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    at R = 0.
    That's why I put it into polar
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    coordinates, because the
    presence of the mass is going to
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    change the space-time as a
    function of distance--from
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    radial distance from where the
    mass is.
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    And so, it's much more
    convenient for the Schwarzschild
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    metric to use this in polar
    coordinates.
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    So, here's the Schwarzschild
    metric, (d T^(2)),
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    that's the--this is the
    interval,
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    is equal to (dR) / (1 -
    R_s /
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    R).
    So, that's just like the flat
  • 20:31 - 20:34
    term, except with something in
    the denominator there.
  • 20:34 - 20:38
    Plus R^(2),
    d Omega squared,
  • 20:38 - 20:42
    that's just like the flat
    metric.
  • 20:42 - 20:48
    And then the--whoops this had
    better be - c^(2) (1 -
  • 20:48 - 20:54
    R_S / R)
    (cT) ^(2).
  • 20:54 - 20:57
    Where R_S is
    the Schwarzschild radius,
  • 20:57 - 21:01
    which we've had before,
    which is 2GM /
  • 21:01 - 21:04
    c^(2).
    Okay, so this is just like the
  • 21:04 - 21:08
    flat metric with two exceptions.
    It's got a term in the radial
  • 21:08 - 21:12
    part of this 1 -
    R_S / R.
  • 21:12 - 21:17
    And it's got that same term,
    but this time in the numerator,
  • 21:17 - 21:23
    in the time term here.
    Now, what do you do with
  • 21:23 - 21:28
    this--with such an equation?
    Well, we've done--in special
  • 21:28 - 21:31
    relativity, we've dealt with
    these kinds of things.
  • 21:31 - 21:33
    What you do is you start taking
    the limiting cases.
  • 21:33 - 21:37
    You say, okay,
    what happens when it's getting
  • 21:37 - 21:39
    really close to flat on the one
    hand,
  • 21:39 - 21:42
    and what's happening when it's
    getting very seriously
  • 21:42 - 21:48
    relativistic on the other hand?
    So let's do that.
  • 21:48 - 21:54
    If R_S /
    R goes to zero,
  • 21:54 - 21:58
    then the metric turns into the
    flat metric.
  • 21:58 - 22:01
    Because if R_S
    / R = 0,
  • 22:01 - 22:04
    this term disappears because
    it's 1 - 0, and it just cancels.
  • 22:04 - 22:08
    This term disappears and you
    recover the flat metric.
  • 22:08 - 22:13
    This happens in two cases.
    If the mass goes to zero,
  • 22:13 - 22:18
    then R_S goes
    to zero, and you recover the
  • 22:18 - 22:22
    flat metric.
    Or if R gets really big,
  • 22:22 - 22:27
    then R_S /
    R goes to zero,
  • 22:27 - 22:31
    and again, you recover the flat
    metric.
  • 22:31 - 22:36
    So in--there are two situations
    where Schwarzschild metric
  • 22:36 - 22:40
    blends smoothly into the
    ordinary flat space.
  • 22:40 - 22:44
    One is if the mass is zero,
    that's not surprising.
  • 22:44 - 22:46
    If the mass is zero then
    space-time isn't curved.
  • 22:46 - 22:48
    Or alternatively,
    if you're really,
  • 22:48 - 22:51
    really far away from the mass.
    If R is much,
  • 22:51 - 22:53
    much bigger than the
    Schwarzschild radius,
  • 22:53 - 22:56
    you're way out there.
    There's no gravitational effect.
  • 22:56 - 23:01
    Space-time remains flat.
    So, these the--this is the
  • 23:01 - 23:05
    limiting case where you recover
    special relativity.
  • 23:05 - 23:09
    Now, the other case is when
    R gets close to the
  • 23:09 - 23:12
    Schwarzschild radius and
    approaches it.
  • 23:12 - 23:16
    So then 1 -
    R_S / R
  • 23:16 - 23:20
    approaches zero,
    because these two are going to
  • 23:20 - 23:25
    get closer and closer together.
    1 - 1 = 0.
  • 23:25 - 23:28
    What happens then?
    So, this is now,
  • 23:28 - 23:30
    first of all--in physical terms
    you're getting really close to
  • 23:30 - 23:33
    the Schwarzschild radius.
    So now, what happens to the
  • 23:33 - 23:39
    metric if you do that?
    The dR term gets very
  • 23:39 - 23:46
    big, because it's got a zero in
    the denominator.
  • 23:46 - 23:54
    The dT term gets really
    small, because it's got that
  • 23:54 - 24:00
    thing that's going to zero in
    the numerator.
  • 24:00 - 24:02
    Fine.
    What does that mean?
  • 24:02 - 24:05
    Well, remember,
    this is the negative term.
  • 24:05 - 24:08
    This is the positive term.
    So the positive term is getting
  • 24:08 - 24:10
    really, really big.
    The negative term is getting
  • 24:10 - 24:14
    really, really small.
    And that means that all
  • 24:14 - 24:21
    intervals are gradually becoming
    space-like.
  • 24:21 - 24:27
  • 24:27 - 24:30
    What do I mean by that?
    Well, the negative term is
  • 24:30 - 24:32
    getting small.
    One of the positive terms is
  • 24:32 - 24:35
    getting big.
    So the sum of those tends to be
  • 24:35 - 24:37
    positive.
    It's becoming more and more
  • 24:37 - 24:39
    positive.
    Positive intervals are these
  • 24:39 - 24:42
    space-like intervals,
    and you can't communicate or
  • 24:42 - 24:45
    travel across space-like
    intervals.
  • 24:45 - 24:48
    When you get all the way to the
    Schwarzschild radius,
  • 24:48 - 24:51
    this blows up completely,
    becomes infinite.
  • 24:51 - 24:54
    This becomes zero,
    and there are no time-like
  • 24:54 - 24:57
    intervals.
    There are no time-like
  • 24:57 - 25:00
    intervals that cross the event
    horizon.
  • 25:00 - 25:05
    That's why you can't get out.
    This takes us back to the basic
  • 25:05 - 25:13
    principle of black holes.
    So, cannot communicate or
  • 25:13 - 25:20
    travel over space-like
    intervals.
  • 25:20 - 25:25
  • 25:25 - 25:31
    And so, you can't cross
    R equals the
  • 25:31 - 25:36
    Schwarzschild radius.
    All right.
  • 25:36 - 25:39
    Let's see.
    Let me write the thing down
  • 25:39 - 25:40
    again, here, for you.
  • 25:40 - 25:57
  • 25:57 - 26:01
    Okay, so that's the metric
    we're worrying about here.
  • 26:01 - 26:07
    And now, let's think about what
    happens inside the Schwarzschild
  • 26:07 - 26:09
    radius.
    R less than
  • 26:09 - 26:12
    R_Sch
    warzschild.
  • 26:12 - 26:18
    That means the dR term
    becomes negative,
  • 26:18 - 26:24
    because 1 -
    R_S / R.
  • 26:24 - 26:27
    If R_S is
    bigger than R then this
  • 26:27 - 26:29
    term is--this term is greater
    than 1,
  • 26:29 - 26:32
    and this whole thing is less
    than zero, and the signs change.
  • 26:32 - 26:38
    And the dT term becomes
    positive.
  • 26:38 - 26:42
    So, that means this is the
    time-like term,
  • 26:42 - 26:46
    where this one is the
    space-like term,
  • 26:46 - 26:48
    because it's positive.
  • 26:48 - 26:53
  • 26:53 - 26:56
    That's what I meant three,
    four, five lectures ago,
  • 26:56 - 26:59
    when I said that inside the
    Schwarzschild radius,
  • 26:59 - 27:02
    when you're inside the
    Schwarzschild radius,
  • 27:02 - 27:04
    space and time reverse.
    It's a sign change in the
  • 27:04 - 27:06
    metric.
    That's what it means.
  • 27:06 - 27:16
    And you can only travel along
    negative intervals.
  • 27:16 - 27:19
  • 27:19 - 27:23
    That means you have to move in
    R the same way outside
  • 27:23 - 27:27
    the Schwarzschild radius you
    have to move in T.
  • 27:27 - 27:30
    But notice it's only the radial
    term.
  • 27:30 - 27:33
    This term hasn't changed.
    You could go around in circles,
  • 27:33 - 27:36
    but whatever you do,
    you still have to move,
  • 27:36 - 27:40
    as it turns out,
    toward the center of the thing
  • 27:40 - 27:43
    in radius, in order to have a
    time-like interval.
  • 27:43 - 27:51
    And so, motion in R is
    required for inside the
  • 27:51 - 27:59
    Schwarzschild radius,
    whereas motion in T is
  • 27:59 - 28:02
    required outside.
  • 28:02 - 28:09
  • 28:09 - 28:11
    So space and time reverse.
  • 28:11 - 28:16
  • 28:16 - 28:19
    All of which is very nice,
    but I've left out
  • 28:19 - 28:22
    something--I've left something
    out,
  • 28:22 - 28:24
    which is the fact--inside the
    event horizon,
  • 28:24 - 28:27
    how do you know that this is
    still the metric?
  • 28:27 - 28:30
    One could invent some function
    that looks just like this
  • 28:30 - 28:32
    outside the Schwarzschild
    radius,
  • 28:32 - 28:34
    but then looks like something
    else inside the Schwarzschild
  • 28:34 - 28:36
    radius.
    And because no communication
  • 28:36 - 28:40
    across the Schwarzschild radius
    is possible, you'd never be able
  • 28:40 - 28:43
    to test it.
    And so, this is how one gets
  • 28:43 - 28:46
    away with doing non-testable
    physics.
  • 28:46 - 28:48
    You say, well,
    we're just going to assume that
  • 28:48 - 28:51
    the metric hasn't changed.
    Why should it change?
  • 28:51 - 28:53
    After all, it's the same
    equation.
  • 28:53 - 28:56
    But inside the Schwarzschild
    radius you can't actually test
  • 28:56 - 28:58
    this.
    Outside the Schwarzschild
  • 28:58 - 29:01
    radius you can test it,
    because you see whether objects
  • 29:01 - 29:05
    behave as they ought to behave
    in a space that's curved in this
  • 29:05 - 29:08
    particular way--in a space-time
    that's curved in this particular
  • 29:08 - 29:12
    way.
    And so, this is what I meant
  • 29:12 - 29:18
    by, five classes ago,
    by saying space and time
  • 29:18 - 29:23
    reverse.
    These two quantities reverse
  • 29:23 - 29:28
    their signs.
    All right, that's as far as we
  • 29:28 - 29:32
    can go, because the next thing
    that one would want to do is,
  • 29:32 - 29:36
    you find out what the equation
    is for finding out how things
  • 29:36 - 29:39
    move in these curved
    space-times.
  • 29:39 - 29:42
    Basically, you remember,
    things go from one event to
  • 29:42 - 29:45
    another in the shortest possible
    path, that's the equivalent of a
  • 29:45 - 29:48
    straight line.
    That means if you integrate
  • 29:48 - 29:50
    over dT,
    it has to be minimized.
  • 29:50 - 29:53
    So, you minimize this integral.
    That tells you how things move.
  • 29:53 - 29:57
    We're not going to do that.
    Sighs of relief?
  • 29:57 - 30:03
    And because--for obvious
    reasons.
  • 30:03 - 30:06
    So, this is as far as we can
    go, just to write down the
  • 30:06 - 30:09
    metric here.
    So, let me know pause for
  • 30:09 - 30:13
    questions, and then we're going
    to go back and talk about
  • 30:13 - 30:17
    astronomy--about things in the
    sky that actually exhibit these
  • 30:17 - 30:20
    relativistic behaviors.
    Yes?
  • 30:20 - 30:22
    Student: You were
    talking before about intervals,
  • 30:22 - 30:23
    and how all the intervals are
    negative.
  • 30:23 - 30:27
    What exactly is one interval?
    [Inaudible]
  • 30:27 - 30:28
    Professor Charles
    Bailyn: Sorry?
  • 30:28 - 30:30
    Student: What exactly is
    one interval?
  • 30:30 - 30:31
    [Inaudible]
    Professor Charles
  • 30:31 - 30:32
    Bailyn: Oh,
    an interval.
  • 30:32 - 30:35
    So, what I'm doing is I'm
    taking two events,
  • 30:35 - 30:39
    each of which is one of these
    points in space-time,
  • 30:39 - 30:42
    and I'm asking,
    "What is the interval between
  • 30:42 - 30:44
    them?"
    What is--you could measure the
  • 30:44 - 30:47
    distance between them,
    you could measure the time from
  • 30:47 - 30:49
    one event to another.
    But as it turns out,
  • 30:49 - 30:52
    those aren't invariants.
    And so, there's this other
  • 30:52 - 30:55
    thing, the metric,
    which is invariant.
  • 30:55 - 31:00
    And so, that's a measure--an
    invariant measure of how
  • 31:00 - 31:05
    separated these two events are.
    So, you take two events and you
  • 31:05 - 31:08
    ask yourself,
    "Are they separated by zero,
  • 31:08 - 31:11
    a positive quantity or a
    negative quantity?"
  • 31:11 - 31:13
    Where by separation,
    I mean, this curious
  • 31:13 - 31:16
    combination of space and time.
    Student: So the
  • 31:16 - 31:19
    intervals are before the metric,
    before the interval?
  • 31:19 - 31:21
    Professor Charles
    Bailyn: Yeah,
  • 31:21 - 31:23
    it's an interval--think of--let
    me go back to the analogy I
  • 31:23 - 31:26
    started with.
    Here's--in two spatial
  • 31:26 - 31:31
    dimensions, x and
    x, here are two points.
  • 31:31 - 31:33
    And depending on how I set the
    coordinate system up,
  • 31:33 - 31:36
    the x-distance--the
    x difference between them
  • 31:36 - 31:38
    and the y difference
    between them can change,
  • 31:38 - 31:41
    but the distance is always the
    same.
  • 31:41 - 31:47
    So now, I've got two points
    with--each with and x,
  • 31:47 - 31:53
    y, Z and a T. And
    depending on how I change my
  • 31:53 - 31:59
    velocity or my coordinates the
    particular values of x,
  • 31:59 - 32:04
    y, Z,
    and T can change,
  • 32:04 - 32:10
    but this (Δ T^(2)) defined
    by--I'm in flat space now,
  • 32:10 - 32:12
    right?
    This separation,
  • 32:12 - 32:16
    this interval between those two
    points, this is the
  • 32:16 - 32:20
    invariant--in the same way that
    the distance between two points
  • 32:20 - 32:23
    doesn't change if you change the
    coordinate system,
  • 32:23 - 32:27
    even though the x and
    y separations do.
  • 32:27 - 32:29
    Student: [Inaudible]
    Professor Charles
  • 32:29 - 32:32
    Bailyn: It gives you--no.
    Well, it combines these four
  • 32:32 - 32:35
    things into one thing that
    doesn't change.
  • 32:35 - 32:40
    That's the point.
    Yes?
  • 32:40 - 32:44
    Student: Is there a way
    that you can--like as an
  • 32:44 - 32:49
    interval from zero basically
    describes two events that appear
  • 32:49 - 32:51
    simultaneous?
    Professor Charles
  • 32:51 - 32:53
    Bailyn: As the--yeah
    exactly.
  • 32:53 - 32:55
    Student: So,
    is that then collapsed into a
  • 32:55 - 32:57
    Newtonian theory,
    things with two--things appear
  • 32:57 - 32:59
    simultaneously if they happen at
    the same time -- Professor
  • 32:59 - 33:03
    Charles Bailyn: Well okay.
    So, there's two different ways
  • 33:03 - 33:06
    things can appear,
    quote, simultaneously.
  • 33:06 - 33:11
    One is if they are two
    different events in space-time
  • 33:11 - 33:16
    and light travels from one,
    and they're exactly separated
  • 33:16 - 33:20
    by--the amount of time between
    them is the same as the distance
  • 33:20 - 33:23
    between them if you multiply by
    c^(2).
  • 33:23 - 33:26
    They can also appear
    simultaneously if they are the
  • 33:26 - 33:31
    same point as each other.
    And then everything goes to
  • 33:31 - 33:34
    zero.
    And it's only in that second
  • 33:34 - 33:40
    case that it--that the Newtonian
    concept of simultaneous kicks
  • 33:40 - 33:43
    in.
    Simultaneous is usually taken
  • 33:43 - 33:47
    to mean that the time separation
    is zero.
  • 33:47 - 33:49
    Two things happen
    simultaneously when they happen
  • 33:49 - 33:52
    at the same time.
    Student: [Inaudible]
  • 33:52 - 33:56
    Professor Charles
    Bailyn: On Earth the--the
  • 33:56 - 34:01
    distances and the velocities and
    the gravitational fields are
  • 34:01 - 34:05
    never so strong that you have
    any trouble--that the Δ T
  • 34:05 - 34:11
    changes significantly depending
    on what your point of view is.
  • 34:11 - 34:14
    So, in our everyday life,
    we have a strong concept of
  • 34:14 - 34:16
    simultaneity.
    It's two things that happen at
  • 34:16 - 34:17
    the same time.
    Turns out, though,
  • 34:17 - 34:20
    that if you move at close to
    the speed--if you observe two
  • 34:20 - 34:23
    events to be at the same time
    and I'm moving at close to the
  • 34:23 - 34:26
    speed of light,
    I don't observe those two
  • 34:26 - 34:28
    events at the same time,
    even though you do.
  • 34:28 - 34:32
    And so, at that point,
    you have to abandon the
  • 34:32 - 34:37
    Newtonian concept that Δ T = 0
    tells you that two events are
  • 34:37 - 34:40
    simultaneous.
    And the whole concept of
  • 34:40 - 34:42
    simultaneity takes on a
    different task.
  • 34:42 - 34:47
    Other questions, yes?
    Student: [Inaudible]
  • 34:47 - 34:48
    Professor Charles
    Bailyn: Okay,
  • 34:48 - 34:51
    so these units can be any units
    of length you like provided
  • 34:51 - 34:53
    that--any units of length you
    like,
  • 34:53 - 35:02
    provided the time units are
    related to it by c.
  • 35:02 - 35:04
    That is to say,
    if your distance units are
  • 35:04 - 35:07
    light-years, your time units
    have to be years.
  • 35:07 - 35:11
    If your distance units are
    meters, then your time units are
  • 35:11 - 35:14
    some kind of meter,
    light-meter-second thing.
  • 35:14 - 35:18
    And so, the only restriction on
    the units you use is that the
  • 35:18 - 35:22
    time and the space units have to
    be convertible into each other
  • 35:22 - 35:24
    through c^(2).
    Or, alternatively,
  • 35:24 - 35:28
    another way of saying it is,
    you use any units you like,
  • 35:28 - 35:32
    and as long as you express the
    speed of light in those units.
  • 35:32 - 35:35
    If you have a time unit and a
    space unit, if you're in--if
  • 35:35 - 35:38
    you're measuring your space in
    furlongs and your time in
  • 35:38 - 35:41
    fortnights,
    as long as your c is in
  • 35:41 - 35:45
    furlongs per fortnight,
    it's going to come out okay.
  • 35:45 - 35:47
    So, as long as it's convertible.
  • 35:47 - 35:50
  • 35:50 - 35:56
    Other questions?
    Okay, if you don't get all the
  • 35:56 - 36:00
    details and nuance of what I've
    said this period,
  • 36:00 - 36:04
    don't worry too much about it.
    I just wanted to get the
  • 36:04 - 36:08
    concept of the metric out there
    and show you how,
  • 36:08 - 36:13
    if you look at that equation
    these concepts of space and time
  • 36:13 - 36:16
    reversing,
    and so forth,
  • 36:16 - 36:19
    have a kind of mathematical
    consequence, as well as just
  • 36:19 - 36:21
    spouting words.
    And if you get,
  • 36:21 - 36:23
    sort of, the basic outline of
    the argument,
  • 36:23 - 36:24
    that's fine.
  • 36:24 - 36:28
  • 36:28 - 36:39
    Okay, back to actual things in
    actual--that actually exist.
  • 36:39 - 36:47
    So, what I want to talk about
    now is evidence for general
  • 36:47 - 36:54
    relativity from astronomical
    objects--real black holes,
  • 36:54 - 36:58
    stuff like that.
    Now, one of the curious things
  • 36:58 - 37:01
    about this is that when Einstein
    thought all this stuff up,
  • 37:01 - 37:05
    he thought it up from basically
    these philosophical concerns
  • 37:05 - 37:08
    about mass – that the inertial
    mass turned out to always be
  • 37:08 - 37:13
    equal to the gravitational mass.
    Why would that be?
  • 37:13 - 37:16
    And there wasn't,
    when he thought it up,
  • 37:16 - 37:22
    a great body of evidence for
    his theory in the real world.
  • 37:22 - 37:25
    This is in contrast to special
    relativity.
  • 37:25 - 37:27
    Special relativity,
    there were all these
  • 37:27 - 37:29
    experiments that needed to be
    explained.
  • 37:29 - 37:32
    General relativity--very,
    very little.
  • 37:32 - 37:35
    In fact, when Einstein first
    put forward the theory in 1917,
  • 37:35 - 37:39
    there was only one thing that
    had ever been observed that
  • 37:39 - 37:42
    actually showed an effect of
    general relativity,
  • 37:42 - 37:46
    and that was the orbit of
    Mercury, which you're reading
  • 37:46 - 37:49
    about for this week's problem
    set.
  • 37:49 - 37:56
    So, just going back a little
    bit, in the nineteenth century,
  • 37:56 - 38:03
    people had observed the orbits
    of planets in great detail.
  • 38:03 - 38:06
    And they found out that two of
    the planets were moving in ways
  • 38:06 - 38:10
    they couldn't quite explain.
    There were very small
  • 38:10 - 38:13
    deviations from the predictions
    orbit.
  • 38:13 - 38:16
    In particular,
    the orbit of Uranus was a
  • 38:16 - 38:20
    little weird.
    And that was quickly explained
  • 38:20 - 38:24
    by the presence of an
    unknown--hitherto unknown
  • 38:24 - 38:27
    planet,
    which was also exerting a
  • 38:27 - 38:31
    gravitational force on Uranus
    and pulling it out of the orbit
  • 38:31 - 38:35
    that it should have been,
    by a very small amount,
  • 38:35 - 38:38
    because the gravitational force
    of another planet is very small
  • 38:38 - 38:41
    compared to that of the Sun.
    But by the middle of the
  • 38:41 - 38:43
    nineteenth century they could
    measure such things.
  • 38:43 - 38:48
    And they therefore predicted
    the presence of this other
  • 38:48 - 38:53
    planet, of Neptune,
    and they calculated where it
  • 38:53 - 38:58
    should be.
    And some guy went off and
  • 38:58 - 39:04
    observed in that spot and found
    it--predicted presence of
  • 39:04 - 39:10
    Neptune and discovered it in the
    predicted place.
  • 39:10 - 39:13
    Big triumph!
    Everybody--if they had had
  • 39:13 - 39:16
    Nobel Prizes back then,
    they would have won it for
  • 39:16 - 39:19
    this, for sure.
    And then, there was a whole big
  • 39:19 - 39:21
    kerfuffle because they couldn't
    decide whether the French guy
  • 39:21 - 39:24
    had done it before the English
    guy, or vice versa.
  • 39:24 - 39:28
    And they argued with each other
    for decades about who gets the
  • 39:28 - 39:30
    credit.
    But in scientific terms,
  • 39:30 - 39:32
    there was a prediction,
    and the prediction was
  • 39:32 - 39:34
    verified.
    Excellent news.
  • 39:34 - 39:42
    Now, there was also a problem
    with the orbit of Mercury--also
  • 39:42 - 39:48
    perturbed, from what you would
    expect.
  • 39:48 - 39:50
    And having had this big triumph
    in the Outer Solar System,
  • 39:50 - 39:52
    they figured,
    well, we know how to deal with
  • 39:52 - 39:54
    this.
    There's got to be another
  • 39:54 - 39:58
    planet in there.
    So, they predict the presence
  • 39:58 - 40:04
    of a planet called Vulcan,
    which then disappears from the
  • 40:04 - 40:10
    scientific literature until it's
    resurrected by Star Trek.
  • 40:10 - 40:14
    But Vulcan--the concept of
    Vulcan was, this was going to be
  • 40:14 - 40:17
    a planet that's closer to the
    Sun than Mercury.
  • 40:17 - 40:20
    That's why they haven't been
    able to find it,
  • 40:20 - 40:22
    because it's too near the Sun
    to be easily observed.
  • 40:22 - 40:27
    And it's going to pull on
    Mercury in such a way that it's
  • 40:27 - 40:31
    going to explain the problems
    with the orbit of Mercury.
  • 40:31 - 40:37
    And so, they then look for
    Vulcan in the predicted place,
  • 40:37 - 40:39
    and they find it.
  • 40:39 - 40:42
  • 40:42 - 40:47
    And then somebody else finds it.
    And they find it many times and
  • 40:47 - 40:51
    each time it's different--all
    different.
  • 40:51 - 40:54
    And it gradually becomes clear
    that everybody's fooling
  • 40:54 - 40:56
    themselves.
    That there's no--this is a
  • 40:56 - 40:59
    really hard observation to make,
    right?
  • 40:59 - 41:01
    Because the thing is right near
    to the Sun.
  • 41:01 - 41:07
    And so, it turns out that all
    of this is wrong.
  • 41:07 - 41:12
    None of these observations are
    really any good.
  • 41:12 - 41:18
    It's not repeatable--so,
    not really.
  • 41:18 - 41:21
    And so, after some attempts to
    find Vulcan--and then,
  • 41:21 - 41:24
    they rule out the presence of
    Vulcan in various places.
  • 41:24 - 41:27
    So, then the people calculating
    the orbits have to go back and
  • 41:27 - 41:28
    say, well, if Vulcan isn't
    there,
  • 41:28 - 41:32
    maybe there are two or three
    planets combining together to do
  • 41:32 - 41:35
    the thing that we originally
    wanted Vulcan to do.
  • 41:35 - 41:37
    This gets sort out of control
    after a while.
  • 41:37 - 41:39
    And at a certain point,
    people just kind of give up,
  • 41:39 - 41:41
    and they say,
    well, it's a great big mystery
  • 41:41 - 41:44
    about Mercury.
    And after a while,
  • 41:44 - 41:48
    after that, people kind of even
    stopped caring.
  • 41:48 - 41:51
    Because, you know,
    we know Newton's laws worked.
  • 41:51 - 41:54
    This is just some weirdness
    about Mercury that we don't
  • 41:54 - 42:00
    understand.
    And then, when Einstein creates
  • 42:00 - 42:10
    his new theory of gravity,
    he then computes in the new
  • 42:10 - 42:16
    theory of Mercury's orbit.
  • 42:16 - 42:19
  • 42:19 - 42:26
    And he now gets something that
    agrees with the observations,
  • 42:26 - 42:30
    without the need for a new
    planet.
  • 42:30 - 42:36
  • 42:36 - 42:40
    And so, what happened was,
    Mercury's orbit is a little
  • 42:40 - 42:42
    different from the Newtonian
    prediction.
  • 42:42 - 42:45
    The general relativity
    prediction is a little bit
  • 42:45 - 42:48
    different in just the same way
    to explain this problem that
  • 42:48 - 42:52
    people had been trying to solve
    for fifty years unsuccessfully.
  • 42:52 - 42:57
    And so, this was the first
    verification,
  • 42:57 - 43:04
    empirical verification,
    of general relativity.
  • 43:04 - 43:07
    And if you think about it,
    you would expect that Mercury
  • 43:07 - 43:10
    would be the place you would
    find this out.
  • 43:10 - 43:13
    For Mercury,
    R_S /
  • 43:13 - 43:17
    R, this is the
    Schwarzschild radius of the Sun
  • 43:17 - 43:21
    because that's what's doing the
    gravitating,
  • 43:21 - 43:24
    is the biggest in the Solar
    System.
  • 43:24 - 43:28
  • 43:28 - 43:31
    Because the R,
    the distance from Mercury to
  • 43:31 - 43:33
    the Sun, is the smallest of any
    of the planets in the Solar
  • 43:33 - 43:38
    System.
    And so, the relativistic
  • 43:38 - 43:48
    effects, the general relativity
    effects, are relatively large.
  • 43:48 - 43:51
    But, you know,
    this is still a really small
  • 43:51 - 43:53
    number.
    This is 3 kilometers,
  • 43:53 - 43:55
    the Schwarzschild radius of the
    Sun.
  • 43:55 - 43:56
    Mercury is way out there
    somewhere.
  • 43:56 - 44:00
    So, even though this is the
    most--this is the biggest
  • 44:00 - 44:03
    relativistic effect in the Solar
    System, it still isn't that
  • 44:03 - 44:06
    huge.
    Let me just remind you what
  • 44:06 - 44:09
    this effect is.
    Here's the Sun.
  • 44:09 - 44:14
    Mercury's going around the Sun.
    And it's going around in a
  • 44:14 - 44:18
    slightly elliptical orbit.
    I'm going to draw a very
  • 44:18 - 44:20
    elliptical orbit,
    here, but it's really not that
  • 44:20 - 44:22
    big.
    And there's a point in the
  • 44:22 - 44:25
    orbit where it is closest to the
    Sun.
  • 44:25 - 44:28
    That point is called the
    perihelion.
  • 44:28 - 44:34
    "Peri" for close,
    "helios" for Sun--of Mercury.
  • 44:34 - 44:37
    And in the Newtonian theory,
    you should have exactly the
  • 44:37 - 44:40
    same orbit every time.
    You should come back and the
  • 44:40 - 44:44
    perihelion should be in the same
    place in each successive orbit.
  • 44:44 - 44:47
    The orbit doesn't move or
    doesn't change.
  • 44:47 - 44:51
    But, in general relativity,
    the perihelion moves.
  • 44:51 - 44:55
    So, after a while the
    perihelion will be here.
  • 44:55 - 44:59
    The whole orbit will kind of
    tip this way,
  • 44:59 - 45:03
    and it'll look like this.
    So, this is the perihelion
  • 45:03 - 45:03
    later.
  • 45:03 - 45:09
  • 45:09 - 45:13
    And it looks like that.
    And the angle which the
  • 45:13 - 45:17
    perihelion makes with the Sun
    has changed.
  • 45:17 - 45:21
    This angle is called the angle
    of the perihelion.
  • 45:21 - 45:25
    And this precesses.
    So this is called the
  • 45:25 - 45:28
    precession of the perihelion.
  • 45:28 - 45:31
  • 45:31 - 45:34
    And it's measured in some angle
    per time.
  • 45:34 - 45:37
    Because the question is,
    "How long does it take for the
  • 45:37 - 45:40
    perihelion to precess across
    some angle?"
  • 45:40 - 45:45
    And the key number for Mercury
    is 43 arc seconds.
  • 45:45 - 45:47
    Remember arc seconds?
    Those are small angles.
  • 45:47 - 45:55
    Per century – a really small
    movement, but something that can
  • 45:55 - 46:01
    be measured, and had been
    measured.
  • 46:01 - 46:03
    And it's not surprising that
    this is small,
  • 46:03 - 46:06
    because the relativistic
    effects are going to be small,
  • 46:06 - 46:08
    because the Schwarzschild
    radius of the Sun is really
  • 46:08 - 46:11
    small compared to the size of
    the orbit of Mercury.
  • 46:11 - 46:18
    But this was observed before
    Einstein made his theory.
  • 46:18 - 46:22
    Nobody understood it.
    Einstein came up with his
  • 46:22 - 46:24
    theory.
    It turned out it predicted a
  • 46:24 - 46:28
    precession of the perihelion in
    a way that Newton didn't,
  • 46:28 - 46:30
    and it turned out to work out
    precisely.
  • 46:30 - 46:34
    So, that was good.
    And at the time Einstein
  • 46:34 - 46:38
    published the theory,
    this was the only piece of
  • 46:38 - 46:44
    evidence that it was correct.
    Pretty small empirical
  • 46:44 - 46:52
    verification.
    And so, let's just write down
  • 46:52 - 46:59
    the fable, here.
    This is Einstein and the
  • 46:59 - 47:06
    precession of the perihelion.
    And there are two versions of
  • 47:06 - 47:08
    the moral.
    Sometimes in textbooks,
  • 47:08 - 47:11
    you know, they make a big deal
    out of this.
  • 47:11 - 47:13
    They say, oh,
    there was this terrible problem
  • 47:13 - 47:15
    with Mercury,
    and then Einstein came along
  • 47:15 - 47:18
    with this great new theory,
    solved that problem.
  • 47:18 - 47:21
    In the same way that they say,
    there was this terrible problem
  • 47:21 - 47:23
    with the speed of light being
    constant from all frames,
  • 47:23 - 47:26
    and Einstein came along with
    special relativity and solved
  • 47:26 - 47:30
    that problem.
    That's a misreading of history.
  • 47:30 - 47:32
    This was a by-product of
    Einstein.
  • 47:32 - 47:35
    It wasn't that there was a
    problem with the data and he
  • 47:35 - 47:40
    went out to try and fix the
    theory to conform with the data.
  • 47:40 - 47:46
    There was very little data.
    So, the moral here is aesthetic
  • 47:46 - 47:53
    considerations,
    aesthetic--perhaps you want to
  • 47:53 - 48:01
    call this philosophical,
    considerations can lead to a
  • 48:01 - 48:08
    good new theory because he
    didn't really do it to explain
  • 48:08 - 48:11
    the data.
    This is, however,
  • 48:11 - 48:14
    the only time I can think of
    where this actually happened
  • 48:14 - 48:17
    this way.
    Every other major advance in
  • 48:17 - 48:23
    science came about because the
    observers or the experimenters
  • 48:23 - 48:30
    had a problem--but not G.R.
    Only for general relativity.
  • 48:30 - 48:33
    Now, subsequent to that,
    between 1917,
  • 48:33 - 48:37
    when this theory was
    promulgated, and now,
  • 48:37 - 48:44
    there have been a variety of
    tests of general relativity
  • 48:44 - 48:50
    using astronomical objects.
    You always have to use
  • 48:50 - 48:53
    astronomical--or almost always
    have to use astronomical objects
  • 48:53 - 48:55
    to test this,
    because you need really strong
  • 48:55 - 48:58
    gravitational fields,
    and it's hard to produce a
  • 48:58 - 49:00
    really strong gravitational
    field in the laboratory.
  • 49:00 - 49:03
    You're kind of limited to what
    the Earth provides you with,
  • 49:03 - 49:06
    and that isn't such a strong
    gravitational field.
  • 49:06 - 49:07
    We computed,
    at some point,
  • 49:07 - 49:10
    the Schwarzschild radius of the
    Earth, R /
  • 49:10 - 49:13
    R_Sch
    warzschild is the relevant
  • 49:13 - 49:15
    quantity,
    and it's really small from the
  • 49:15 - 49:17
    Earth's gravitational field.
    So, it's hard to see these
  • 49:17 - 49:20
    things.
    So, all these tests tend to be
  • 49:20 - 49:23
    astronomical in nature.
    Since then, a variety of tests
  • 49:23 - 49:28
    of G.R., and the punch line is
    that it passes all of them.
  • 49:28 - 49:32
    G.R. is still a good theory.
    There's no contradictory data.
  • 49:32 - 49:36
    But the tests still aren't as
    strong as you might have hoped
  • 49:36 - 49:40
    they would be.
    And we'll talk about some of
  • 49:40 -
    those on Thursday.
Titre:
12. Stellar Mass Black Holes
Description:

Frontiers/Controversies in Astrophysics (ASTR 160)

One last key concept in Special Relativity is introduced before discussion turns again to black celestial bodies (black holes in particular) that manifest the relativistic effects students have learned about in the previous lectures. The new concept deals with describing events in a coordinate system of space and time. A mathematical explanation is given for how space and time reverse inside the Schwarzschild radius through sign changes in the metric. Evidence for General Relativity is offered from astronomical objects. The predicted presence and subsequent discovery of Neptune as proof of General Relativity are discussed, and stellar mass black holes are introduced.

00:00 - Chapter 1. Invariance in Special Relativity
10:10 - Chapter 2. Invariant Intervals and the Schwarzschild Metric
21:01 - Chapter 3. Schwarzschild Sign Changes and Space-Time Reversals
36:27 - Chapter 4. Evidence for General Relativity in Astronomy

Complete course materials are available at the Open Yale Courses website: http://open.yale.edu/courses

This course was recorded in Spring 2007.

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Équipe:
Veduca
Projet :
Fronteiras/Controvérsias na Astrofísica - Yale
Durée:
49:42
Amara Bot edited Anglais subtitles for 12. Stellar Mass Black Holes
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