This episode is
sponsored by Audible. The special theory of
relativity tells us that one person's past
may be another's future. When time is relative,
paradoxes threaten. Today, we peer deeper
into Einstein's theory to find that the immutable
ordering of cause and effect emerges when we discover the
causal geography of spacetime. Recently, we've been
talking about the weirdness of spacetime in the vicinity of
a black hole's event horizon. Very soon, we'll be
dropping below that horizon to peer at the interior
of the black hole. There, space and
time switch roles, but to truly understand
that bizarre statement, we need to think a little bit
more about how the flow of time is described in relativity. Today, we're going to look at
the amazing geometric structure that time, or more accurately
causality, imprints on the fabric of spacetime. First, let's recap a little bit
of Einstein's special theory of relativity. There are two previous
episodes in particular that will be useful here if you
find you need more background. Special relativity tells us that
our experience of both distance and time are, well, relative. If I accelerate my rocket ship
to half the speed of light, the distance I need to
travel to a neighboring star shrinks dramatically
from my point of view. An observer I leave behind
with an amazing telescope, observes me traveling the
entire original distance but will perceive my
clock as having slowed. The combination of this length
contraction and time dilation allows both moving and
stationary observers to agree on how much
older every one looks at the end of the journey. Everyone agrees on the
number of ticks that occurred on everyone else's clock. They just don't agree on the
duration of all of those ticks. Reminder-- time measured
by a moving observer on their own clock is
called proper time, but counting those clock ticks
isn't the best way for everyone to agree on spacetime
relationships. There's this thing called
the spacetime interval that relates observer dependent
perspectives on the length and duration of any journey that
all observers will agree on, even if they don't
agree on the delta x and delta t of that journey. We've talked about it before,
but it's a tricky concept to understand intuitively. But we want that intuition
because, more than proper time, the spacetime interval
defines the flow of causality. In relativity, 3D
space and 1D time become a 4D entity
called spacetime. To preserve our
sanity, we represent this on a spacetime diagram
plotting time and only one dimension of space. We'll see our causal geometry
emerge plain as day, even in this simplified picture. There is no standing still
on a spacetime diagram. If I don't move
through space, I still travel forward in time
at a speed of exactly one second per second according
to my proper time clock. Motion at a constant velocity
appears as a sloped line, and the time axis is scaled
so that the speed of light is a 45 degree line. Now, let's say we have a
group of spacetime travelers. They start at the origin,
where x and t equals 0. They race away to the left
and the right for five seconds according to their own watches. They all travel at
different speeds, some close to the speed of
light, but never faster. The path they cut
through spacetime is called their world line. My world line is
only through time, and the tick marks
on the time axis correspond to my own
proper time clock ticks. The faster a traveler moves,
the longer their world line. That's not just because
of their speed, though. To me, their clocks tick slow. They time their journey
on these slow clocks, so I perceive them
traveling for longer. Accounting for this, we find
that our spacetime travelers are arranged on a curve
that looks like this. This shape is a hyperbola. Drawing a connecting line at
the tick of every traveler's proper time clock gives a
set of nested hyperbola, but these aren't
just a pretty pattern. These curves are kind
of the contours defining the gradient of causality down
which time flows, and etched into spacetime by the equations
of special relativity. To understand why,
we need to see how these proper time contours
appear to other spacetime travelers. Instead of doing
that with equations, we can see it with geometry. First, we need to draw
the spacetime diagram from the perspective of
one of the other travelers. To transform the
diagram, we need to figure out what they see
as their space and time axes. Time is easy. They see themselves
as stationary, so their time axis is just their
own constant velocity world line. And their x-axis? Well, from my stationary
point of view, I define my x-axis as a long
string of spacetime events at different distances, but
that all occur simultaneously at time t equals 0. To observe those points,
I just wait around until their light
had time to reach me. At every future
tick of my clock, a signal arrives from
the left and the right, and I use that to build up a
set of simultaneous events, defining my t equals 0 x-axis. Our traveler does
the same thing, but from my point of
view, their clock is slow, so I see them register
signals at a different rate. At the same time,
they're moving away from the signals
coming from the left and towards the ones
originating on the right, affecting which signals are
seen at a given instant. The traveler infers a set
of simultaneous events that, to me, are
not simultaneous, but there is no preferred
reference frame. Their sloped x-axis
is right for them. Even just doing
this graphically, we see that the
traveler's x-axis is rotated by the same
angle as their time axis. That comes from
insisting that we all see the same speed
of light, 45 degrees on the spacetime diagram. Moving between these
reference frames is now a simple
matter of squaring up our traveler's axes. In fact, we grid up the
diagram with a set of lines parallel to these new axes
and square up everything while maintaining our
intersection points. My world line is now
speeding off to the left, while our traveler
is motionless. We just performed a
Lorentz transformation, but using geometry
rather than math. This transformation allows you
to calculate how properties, like distance, time, velocity,
even mass and energy, shift between reference frames. But check out what
happens if I attach pins to all of the intersections
when I transform between frames. They trace out hyperbola. Those intersections represent
locations of spacetime events relative to the origin. They will always land on the
same hyperbola, no matter the observer's reference frame. I told you that
these contours show where clocks moving
from the origin reach the same proper time
count, but more generally, each represents a single value
for the spacetime interval. The delta x and
delta t of the event at the end point of a
traveler's world line might change depending
on who is watching, but the hyperbolic contour
that they landed on, the spacetime
interval, will not. This is because the spacetime
interval itself comes directly from the Lorentz transformation,
as the only measurement of spacetime separation that
is unchanging or invariant under that transformation. Now, we can finally get to
why this thing is so important and what it really represents. It may seem counter-intuitive
that an event very close to the origin
in both space and time can be separated
from that origin by the same spacetime interval
as an event that is very distant in both space and time. The hyperbolic shape seems
to demand that, but remember, it takes the same
amount of proper time to travel from the origin to
a nearby near-future event compared to a distant far future
event on the same contour. From the point of
view of a particle communicating some
causal influence, those points are equivalent. The spacetime interval
tracks this causal proximity. We can think of these
lines as contours on a sort of causal geography. The way I define the
spacetime interval, it becomes increasingly negative
in the forward time direction, so we can represent this as a
valley dropping away from me here at the origin. I naturally slide through
time by the steepest path, straight down. I can change that path
by expanding energy to change my velocity,
although doing so realigns the
contours so I always slide down the steepest path. There's no point
anywhere downhill that I can't reach as
long as I can get close enough to the speed of light. In fact, the nearest
downhill contour defines the forward light
cone for anyone anywhere on the spacetime diagram. But uphill is impossible
as long as the cosmic speed limit is maintained. Breaking that speed limit and
sliding uphill are equivalent. To reverse the direction of
your changing spacetime interval is to reverse the
direction of causality, to travel backwards in time. The spacetime diagram
we looked at today was for a flat or
Minkowski space, in which faster
than light travel is the only way to flip
your space time interval. But in the crazy curved
space within a black hole, it gets flipped for you. We'll soon see how this
requirement of a forward causal evolution leads to
some incredible predictions when we try to calculate
the sub event horizon interval of spacetime. A big thank you to Audible for
sponsoring today's episode, and also for making it possible
for me to research spacetime while riding crowded
New York subways. Lately, I've been zoning
out to Audible books from two other New Yorkers. Janna Levin's "Black
Hole Blues" is a wonderful take
on the new window that gravitational waves
are opening on our universe. Also, Caleb Scharf's
"Gravity's Engines" gets into my favorite
space things of all-- quasars, and especially
how important they are in the evolution
of the universe. Check them out, for
free if you like, at audible.com/spacetime
for your free 30 day trial. "Space Time" is possible
only through your support. Watching is, of course, a huge
help, so thanks for tuning in. But an extra thanks is warranted
to our Patreon supporters who throw in a few
bucks each month to help us cover the costs. And an extra, extra
thanks to David Nicholas, who's supporting us
at the big bang level. David, we're naming an
entire galaxy after you. It's a beautiful barred spiral
galaxy in the Fornax cluster. It'll be called David. We skipped comments last week
because I was at the beach, so today, we're tackling
both phantom singularity and quasars. Michael Lloyd asks, "Is the
calculated infinite density of the core of a
black hole an artifact of the limitations of three
dimensional mathematics?" Well, maybe, sort of. One way out of the
mathematical singularity at the center of black
holes is with string theory, which proposes that particles
that we see in regular 4D spacetime result from
oscillations within many more coiled dimensions,
so-called strings. One idea is that the
inside of an event horizon is composed of a ball of raw
strings, a so-called fuzzball, and that no infinite
density exists. We'll get back to
this another time. Jose Hernandez says that,
for a mathematician, infinity is just a number. For a physicist,
it means madness. Not true-- everyone goes mad
thinking about infinities. Mathematician Georg Cantor
invented set theory, the mathematics we use to study
different types of infinity. He was in and out of sanitariums
throughout his later life. Joan Eunice asks whether there's
a spot near a quasar where a stable orbit could be created,
and what would time dilation be like there? Well, the smallest stable
orbit around a black hole is the so-called innermost
stable circular orbit. It's three times the
Schwarzschild shield radius for a non-rotating black hole. Below that, accreting material
spirals into the black hole very quickly, and
yeah, time dilation would be significant there. We actually do see the
effect of time dilation in some of the
x-ray light coming from right near the black hole. Ion atoms, orbiting
at around 10 times the Schwarzschild shield radius,
undergo an extremely energetic electron transition
that produces X-rays at a very particular frequency,
the ion K-alpha emission line. We see that these
x-rays are stretched out as they climb out of the black
hole's gravitational well. That gravitational
redshift is the same thing as gravitational time dilation. Mike Cammiso asks
whether nuclear fusion occurs inside accretion disks. Well, although quasar
accretion disks can reach some pretty
crazy temperatures, they aren't particularly dense. Stars are so good
at fusion, in part, because they're the cause
of creating high densities. That said, it may be that
parts of the accretion disks sometimes become
gravitationally unstable and collapse, in
which case you might get some weird stardust-like
activity and some fusion. But accretion disks
are very poorly understood because they're
too small to take images of, so this is all speculation. Bikram Sao asks how
large the original star must have been to produce
a supermassive black hole. Well, the answer is
probably very large, but nowhere near the mass of
the SMBHs that we see today. These giant black
holes have been growing since the dawn
of time by creating gas and by merging with
other black holes. The original seed
black holes may have been left over by
the deaths of an insanely large first generation of stars,
perhaps thousands of times the mass of the sun. But by now, some of those have
grown to billions of times the mass of the sun. Cinestar Productions
has a story for us. "When my dad was in
college, he needed one of those easy
classes for credit, so he took a class on
quasars and black holes in the universe. He was not a science student. He took the class
on astrophysics because he thought
it would be easy. Facepalm." I hear you. And to all my students in
Astronomy 101 this semester, no, we're not learning
about the star signs. Yes, it's going to be
harder than you thought. Yes, there is a curve. No, watching "Space Time"
doesn't count as extra credit, but it can't hurt. Right?
I love this channel so much. I think it occupies that middleground of science communication between entry level material and strictly academic level unfiltered research paper type material.
This is definitely the best physics channel out there. No BS, super accurate info and presented in a very understandable way. These guys need more funding!