[SQUEAKING]
[RUSTLING]
[CLICKING]
SCOTT HUGHES: The key
textbook for this class
is Sean Carroll's textbook
on general relativity.
It's now almost 20 years old.
I would say-- I
think it's listed
on the website as required.
I would actually call it
sort of semi-required.
It is where I will tend to
post most of the readings
to the course.
It's a really good, complete
textbook for a one semester
course, which is what
we have, and I will not
be going through
the entire thing.
You can't in a one
semester course.
And I will, from time to time,
there will be a few topics that
I cannot go as in into as
deeply as I would like,
and if we had two semesters
maybe I'd be able to do
a little bit more.
And so for those
who are interested,
I will suggest readings in this.
My personal favorite supplement
to this is a textbook
by Bernard Schutz , A First
Course in General Relativity.
OK, so these are all things
where-- so the MIT bookstore,
I'm not sure how
much they carry.
They're all available
through Amazon,
and you can definitely
find these kind of things.
If you get Schutz's
textbook, definitely
get the second edition.
The first edition
contains errors.
There is actually a very
important geometric object
that we are going to
introduce in a couple of weeks
that his textbook has a
really clever derivation of.
I remember seeing that
and thinking, wow,
that's really clever.
The reason he was able do it so
simply is because it's wrong.
AUDIENCE: [LAUGH]
SCOTT HUGHES: It's corrected
in the second edition.
Another one is Gravitation by
Misner, Thorne and Wheeler.
This is sort of a bit of
personal history for me.
I was Thorne's graduate student,
and I used this textbook
when I learned this
subject originally.
I, frankly, do not
recommend this textbook
to somebody who is learning
the subject for the first time.
It's a good place for a
reference for certain things.
It's available in
the reading room.
Know where you can get
a copy and pick it up.
So first of all,
picking it up is
kind of a-- it's good exercise.
It's actually like a--
you know, it's a huge book.
It gravitates.
So it's a great
reference and has
a couple of good sections
in it for new students.
But I will indicate from
time to time, especially
some of the stuff we do in
the second half of the class,
my lectures are kind of inspired
by Misner, Thorne and Wheeler--
known as MTW.
And for those of you who have
sort of a more mathematically
minded approach to things,
General Relativity by Wald.
In the course syllabus I
call this the uber book.
This really is sort of
the self-contained book
that really pins down the
subject very, very well.
It's quite terse and formal,
but also very, very clear.
If you are a mathematically
minded thinker,
you will find this to be
a really good textbook
to refer to.
There is one
particular derivation
that I'm going to do in
about a month and a half
that I essentially take
from Wald's textbook
because it's just beautiful.
Other things in it--
to me, it's a little too terse
from my own personal taste,
but others may find
it to be pretty good.
Two other quick things.
So there are 11 problem
sets and your grade
is determined entirely
on these problems sets.
The syllabus gives the schedule
for when they will be posted
and when they are
to be handed in.
11 does not divide evenly
into 100, so what we do is we
have 10 that are worth 9%
of your grade and the 11th
is worth 10% of your grade.
Is that 11th one
really 1% longer?
I don't know.
Again, I'm taking the viewpoint
that you are graduate students,
or at least you're playing one
for the next hour and a half,
and if you're going to sweat
the difference of one percentage
point on that thing, come on.
This is about
learning the material,
don't worry about those
little details that much.
That is where all
of your assessment
is going to come from.
The very first time I lectured
this course I had a final exam,
and it just turned to be a
complete waste of my time
and the students' time.
You really cannot write--
you either write
problems that are
so easy that you can
do them in your sleep,
or they're so difficult
you can't do them
in the time period of an exam.
So we're just going to
stick with problem sets
and that's fine.
Let's move into, then,
the way the course
is going to be structured.
So my presentation
of this material--
the first half of the course
basically up to spring break
is essentially the
mathematical foundations
of general relativity.
There are several
choices that need to be
made when you're doing this.
This often, in
some universities,
if they have multiple
semester sequences, what
I'm going to cover
in this first half
of our semester in some places
goes for a full semester.
And what this means is there's
a couple of things that I just
cannot cover in
quite as much depth.
Those will be things
where, for those of you who
are interested in it, have that
kind of a mathematical thinking
of things, happy to push
you to additional readings.
We can dive in and look
at it a bit more depth.
My goal is to give
you just enough stuff
that we can do the most
important applications
of this subject.
And I'm an astrophysicist,
and to be blunt,
most of my really
interesting applications
tend to be things that have to
do with things like cosmology,
black holes, dense stars,
and things like that.
And so I want to get
enough formalism together
that we can get to
that part of things.
And so the goal of this
is that by the week right
before spring break what
we will do is "derive"--
I put that in quotes, and you'll
see why a little bit later--
the Einstein field
equations that govern
gravity in general relativity.
So there are several
things that we
could do that are not strictly
necessary to get there,
and just because
of time limitations
I'm going to choose to
elide a few of these topics.
The second half will
then be applications.
We will use everything we
derived in the first half
to see how general relativity
gives us a relativistic theory
of gravity.
We will begin applying it.
We'll see how Newton's law
is encoded in these field
equations.
We'll see how we go
beyond Newton's law,
get some of the classic
tests of general relativity,
and then start
looking at solving it
for more interesting systems.
Looking at the evolution
of the universe as a whole,
looking at the behavior
of black holes,
looking for gravitational
waves, constructing
the spacetime of neutron
stars, things like that.
So it's a fun semester.
It sort of works well to fit
these two things in like this.
And for those of you who are
interested in taking things
more deeply, there's a lot
of room to grow after this.
And it does look like-- so
people who particularly would
like to go a little bit more
detail on some of the math,
if you've looked
at the syllabus,
I have one of my
absolute favorite quotes
from a course evaluation
is put on there.
Where a student in
2007 or so wrote,
"The course was fine as it
was, but Professor Hughes
as an astrophysicist
tended to focus
on really mundane topics like
cosmology and black holes."
If you think those are
mundane topics, what can
I say, guilty as charged.
But for those you who do want
to take a different approach
these sorts of
things, we'll probably
alternate lecturing this course
in the future between someone
from the CTP who works
more in quantum gravity
and things related to that.
That'll be Netta Engelhardt
in spring of 2021.
We're now ready to
start talking about,
after doing all
this sort of prep,
we can actually talk about
some of the foundations
of the theory.
So before I dive in,
are there any questions?
All right.
What we're going to begin doing
for the first couple of weeks--
well, not really
first couple weeks.
The first couple of
lectures, is we're
going to begin by discussing
special relativity,
but we're going to do
special relativity using
mathematical language that
emphasizes the geometric nature
of this form of relativity.
What this does is it allows
us to introduce basically
the formalism, the notation,
all the different tools that
are important for when
things get more complicated.
When we apply a lot of these
tools to special relativity,
like we will be doing
in the first three
or so lectures of
this course, it's
kind of like swatting a
mosquito with a sledgehammer.
You really don't need that
much mathematical structure
to discuss special relativity.
But you're going to be
grateful for that sledgehammer
when we start talking about
strong field orbits of rotating
black holes, right.
And so the whole idea of this
is to develop the framework
in terms of a
physical system where
it's simple to understand
what is going on.
These are sort of
a way to introduce
the mathematical
tools in a place where
the physics is straightforward
and then kind of carry forward
from there.
I will caution that
as a consequence
of this many students find
these first three lectures to be
a little on the dull side.
So, sorry.
It's just some stuff that we
kind of have to get out there,
and then as we generalize
to more interesting
mathematical objects, more
interesting physical settings,
it gets more interesting.
All right, so let's dive in.
So the setting for everything
that we will be doing
is a geometric concept
known as spacetime.
So we give you a precise
mathematical definition
of spacetime.
A spacetime is a
manifold of events
that is endowed with a metric.
So, a wonderful mathematical
definition, and I've
written it in a way that
requires me to carefully define
three additional terms.
The concepts that
I've underlined here,
I've not defined
them yet precisely
exactly what I mean by them.
So let's go over
to the sideboard
and talk about what
exactly these are.
So a manifold-- if you
are a mathematician,
you might twitch a
little bit about the way
that I am going to
define this, and I
will point you to a reading
that does it a little bit more
precisely.
But for the purposes
of our class,
a manifold is
essentially just a set
or a collection of points
with well understood
connectedness properties.
What I mean by
that, is I'm going
to talk about manifolds
of space and time.
I haven't defined an event
yet, but I'm about to,
but I'm going to say that
there's a bunch of events that
happen at this place and at this
time and a bunch of events that
happen at this place
and at this time.
And the manifold, the
spacetime, gives me
some notion of how I
connect the events over here
to the events over here.
A manifold is a
tautological concept.
It's all about how one connects
one region to another one.
So if you're working
on a manifold that
lives on the surface
of a donut, you
have a particular topology
associated with that.
If it lives on the
surface of a sphere
you have a different
topology associated with it.
If you would like to see
more careful and more
rigorous discussion
of this, this
is one of the places where
Carroll is very good.
So go into Carroll, at
least in the edition
that I have it's
on pages 54 to 62.
He introduces a bit of
additional mathematical
machinery, discusses things
with a little more rigor
than I'm doing here--
this isn't rigorous at all,
so significantly more rigor
than I'm doing here.
Those of you are into that,
that should be something
that you enjoy.
So an event.
This is when and where
something happens.
Could be anything.
From our point of view,
the event essentially
is going to be the
fundamental notion
of a coordinate in space time.
We will actually,
in many cases--
actually, that's bad word
choice, I should say.
Coordinates are
actually sort of labels
that we attach to events.
We are going to
be free to adjust
those labels, but the underlying
geometric idea that the event
is here, that's independent of
the coordinates that we choose.
So we will label these
things with coordinates.
But the event itself exists
independent of these labels.
Just to give an
example, there might
be one event which is I
punch myself in the head.
And so I'm very
egotistic, so I will say
this event happened at time 0--
x, y, and z equals 0.
Because I define this
corner of my skull
as the origin the
coordinate system,
and I always think whatever's
happening right now
is the origin of time.
Those of you out in the
room are also egocentric
and you would say whatever,
I'm going to call that--
let's say you are
at y of 3 meters
and I'm going to put the floor
as the origin of my z-axis,
so z of 1.7 meters or whatever.
You will come with your
own independent labeling
of these things.
So you're all
familiar with the idea
that we can just change
coordinate systems.
I'm going to harp on
this a bit, though,
because there's going to be a
really important distinction we
make between
geometrical objects that
live in this
manifold of spacetime
and how we represent them using
labels that might be attached
to coordinate systems.
And I'm going to
come back to this
when we start talking about some
additional geometric objects
in just a couple of minutes.
So the last object that I
have introduced into here
is one that we
will begin talking
about in a lot more detail
in the next lecture,
but let me put it
into here right away.
And so that is the metric.
Metric comes from a
root meaning to measure,
and what this is
is it's something
it gives me a notion of distance
between events in the manifold.
For physics to work,
this has to exist, right?
But it's worth knowing
that the idea of a manifold
is in some way more
primitive than this.
You can have a manifold
without any notion
of a metric attached to it.
And if that's the case--
people like to joke
that if you don't
know what the difference
is between a metric
with a manifold
and without a man--
excuse me, a manifold with a
metric and without a metric,
feel free to drink
coffee out of a donut.
Because topologically, those
are the exact same thing,
but their geometry, which is
encoded in the metric, which
tells me how the different
points on that manifold
are arranged, are
rather different.
What this basically
does is it's going
to give me a mathematical object
that enforces or really conveys
the idea that different
events in this manifold
have a particular
distance between them.
So without this, a
manifold has no notion
of distance encoded in it.
So the two things
together really make
this concept come to life.
You can see a lot
more, like I said.
You can get more information
about many of these concepts
from the readings in
Carroll, Wald's textbook
also goes into quite a lot
of detail about this stuff.
So this is the venue.
This is the setting in which we
are going to talk about things.
And just cutting
forward roughly 2
and 1/2 months'
worth of lectures,
what we are going to find is
that part of Einstein's genius
is that it turns out that
this notion of the metric
ends up encoding gravity.
So that's kind of
where we're going
to end up going with things.
The idea that the
mathematical structure
that tells me how far
apart two events are
is intimately connected to
the properties of gravity.
It's pretty cool, and
it is something that--
physics is an
experimental science.
All of our measurements
are consistent with it.
So that's cool.
So everything I have said
so far, nothing but math.
Nothing but definitions.
So let's start working with
a particular form of physics.
So we are going to begin, as I
said, with special relativity.
This is the simplest
theory of spacetime
that is compatible with
physics as we know it.
Not fully compatible, but
does a pretty good job.
And we'll see that it
turns out to correspond
to general relativity
when there is no gravity.
So to set this up
we need to have
some kind of a way of
labeling our events.
And so I'm going to introduce
kind of a conceptual--
you almost think of
it as scaffolding,
which we're going to use to
build a lot of our concepts
around.
And in this one--
I mean that in a kind
of an abstract sense.
There's going to be
sort of the edifice
that we use to help us
build the building that's
going to be the mathematics
of general relativity.
But this one really is
kind of like a scaffolding.
Because what I want
to introduce here
is a notion of what is called
an inertial reference frame.
So I'm going to
sketch this quickly,
and I'm going to post
to the course website
a chapter from an
early draft textbook
by Roger Blandford
and Kip Thorne.
So when I talk about the
inertial reference frame,
I want you to sort of
visualize in your head
a lattice of clocks--
clocks and measuring rods--
that allows us to label,
in other words, to assign
coordinates any event
that happens in spacetime.
So just sort of, in
your head, imagine
that there's this grid of little
clocks and measuring rods,
and a mosquito
lands on your head.
It's right near a particular
rod and a particular clock.
It bites you, that's an event.
You slap it, that's
another event.
And the measuring
rods and events
are what allows
you to sort of keep
track of the ordering
of those events
and where they happen
in this four dimensional
manifold of spacetime.
So I'm going to
require this lattice
to have a certain
set of properties.
First, I'm going to say that
this lattice moves freely
through spacetime.
What do I mean by moving freely?
I mean no forces act on
it, it does not rotate,
it is inertial.
Every clock and every
measuring rod has no inertia,
no force is acting on it at all.
You look at that, you might
think to yourself, well,
why don't you make it at rest?
Well, I did.
It's at rest in
respect to someone,
but we might have a different
observer who's coming along
who has no forces acting on her,
and she's moving relative to me
at three quarters
the speed of light.
It's not at rest with
respects to that observer.
That's actually kind
of the key here.
So this inertial reference
frame is at rest with respect
to someone who feels no forces
but not to all observers.
I'm going to require that
my measuring rods are
orthogonal to each other.
So they define an orthogonal
coordinate system,
and I am also going to require
that the little markings
on them that tell me
where things happen
are uniformly ticked.
In other words, I'm
going to just make sure
that the spacing
between tick marks
here is exactly the same as
the spacing between tick marks
here.
You may sort of think well,
that's a result or an idea
worthy of the journal Duh, but
it's important to specify this.
You want to make sure that
the standard you are using
to define length is the same
in this region of spacetime
as it is over in this
region of spacetime.
When we start getting
into general relativity,
we start to see there can be
concerns about this coming
about, so it's worth
spelling it out and making
it clear at the beginning.
I'm also going to require
that my clocks tick uniformly.
We're going to make
this lattice that
fills all of spacetime
using the best
thing that Swiss
engineers can make for us,
we want to make sure that one
second, an interval of one
second, is the same
here in this classroom
as it is somewhere off
in the Andromeda Galaxy.
We want to make sure that there
is no evolution to the time
standard when we do this.
Finally, I'm going to
synchronize all these clocks
with each other in
the following way.
This is going to use the
Einstein synchronization
procedure.
This is the first place
where a little bit of physics
is actually beginning to
finally enter our discussion.
I'll comment that--
I'm going to go through
what this procedure is
in just a second.
But an Easter egg here.
Whenever you see a name
that has Einstein in it,
your ears should
perk up a little.
Because it probably means
this is something important.
This is, after all, a
course in relativity,
and that tends to be the
things that end up mattering.
Even when they end up
being really easy to--
things we look at now and
kind of see as fairly obvious,
it's important, OK, and we often
attach Einstein's name to this.
So this Einstein
synchronization procedure, this
takes advantage of the fact--
and by the way, we're not going
to teach special relativity
in this course, 8.962.
I assume you have already
studied special relativity
and you're all experts in this,
and so I can freely borrow
from its important results.
This procedure takes
advantage of the fact
that the speed of light is
the same to all observers,
no matter what
inertial reference
frame they might be in.
So the speed of light
is a key invariant.
It connects--
because it's a speed,
it connects space and
time, and because it
is the same to all observers,
it defines a particular standard
for relating space
and time that is
going to have important
invariant meaning associated
with it.
Just to remind you what
this means-- let's see,
do I have a laser
pointer with me?
I might, but don't
worry about it.
I have a pretend laser pointer.
My chalk's a laser pointer.
I point my laser
pointer at the wall
and you all see it dashing
across the room at 300,000
kilometers per second.
I then start jogging at half the
speed of light, as is my wont,
and continuing to point that,
you guys measure the light
going across the room.
You still measure 300,000
kilometers per second.
I, on the other hand,
measure the light
coming out of my
laser pointer and I
get 300,000
kilometers per second.
So just because my
laser pointer is
moving at half the speed
of light according to you
doesn't mean the light
that's coming out of it
is boosted to a higher speed.
If you studied
special relativity
you'll know that its energy
is boosted to a higher energy
level, but the speed of
light is always just C.
So we're going to
take advantage of that
to come up with a way of
synchronizing our clocks.
The way it works is this.
So let's look at a two
dimensional slice of my lattice
here.
Time, and let's make
this be the x-axis.
And so I will have, let's say,
this is where clock one exists
and this is where
clock two exists.
Let's go into the
reference frame
that is at rest with
respect to this lattice.
We want to synchronize
clock one with clock two.
So as time marches on,
these guys stand still.
So here's the path
in spacetime--
the world line, traced
out by clock one.
Here's the clock path in
spacetime, the world line,
traced out by clock two.
So let's say-- let's
call this event--
let's say that this event
happens at a time t1e.
t1 is when clock one
emits a pulse of light.
So this light will just follow
a little trajectory a bit
through spacetime.
This goes out and strikes
clock two at which point
it is bounced back to clock one.
Let's ignore this point
for just a second.
So let's just say for the
moment that this is then
reflected back and then it
is received back at clock one
at a time t1r.
Let's make this a
little bit neater.
t1e, t1r.
Clock one receives
the reflected pulse.
So the moment at
which it bounces,
we'll call that t2b, that is
the moment at which the light
bounces off of clock two.
And the way we
synchronize our clocks
is just by requiring
that this be
equal to the average of the
emission and the reception
time.
Totally trivial idea, right?
All I'm saying is I'm
going to just require that
in order for clock
one and clock two
to be synchronized
to one another,
let's make sure that when I
bounce light between any two
pairs of clocks they
are set such that when
the light bounces
it's the midway point
between the total light--
the halfway of the
total light travel
time that the light moves along.
Very, very simple concept.
So there's nothing
particularly deep
here, but notice Einstein's
name is attached to this.
And I don't say that to be--
I'm not being sarcastic.
It really points
to the fact that we
are using one of the
most fundamental results
of special relativity
in designing
how these clocks work in this
inertial reference frame.
Believe it or not, this thing,
this really simple concept, it
comes back to sort of bite us
on the butt a little bit later
in this course.
Because when we throw
gravity into the mix,
we're going to
learn that gravity
impacts the way light
travels through spacetime.
We're going to get to some
objects where gravity is
so strong that light
cannot escape them.
And we're going to find
that our perhaps most naive
ways of labeling time in the
spacetime of such objects
kind of goes completely haywire.
Fundamentally, the reason
why time is going haywire
when you have really
strong gravity
is because we use
light as our tool
for synchronizing
all of our clocks,
and if the way that light
moves in your space time
is affected, the way
you're going to label time
is going to be affected.
So this is a very
simple concept.
Again, I sort of emphasize
these first couple lectures
you're swatting a mosquito
with a sledgehammer,
but we're setting
up this edifice
because this will
come back and it's
important to bear
this in mind when
things get a little
more interesting later
in the course.
So let's start setting up
some geometrical objects here.
Pardon me, let me do
one other thing really
quickly before I start setting
up some geometrical objects.
So when I sketched this thing
called a spacetime diagram
here, I should have talked
a little bit about the units
that I'm going to
use to describe
the ticking of my clocks and
the spacing of tick marks
on my measuring rods.
What we will generally
do in this course
is choose the basic
unit of length
to be the distance light travels
in your basic unit of time.
While you parse that sentence,
what that's basically saying is
suppose I set my clock so
that they tick every second.
Well, if my clocks
tick once per second,
then my basic unit of length
will be the light second.
Do you want to put this
into more familiar units?
That's about 300,000 kilometers.
One of my personal favorites--
if the time unit
is 1 nanosecond,
the length unit is, of course,
one I will call it LNS--
Light Nanosecond.
Students who are in
8.033 with me are not
allowed to answer this
question, does anyone
know what one light
nanosecond is?
Actually, this is a
little bit ridiculous.
But to within far greater
than a percent accuracy
it is one foot.
The English unit that comes
on these asinine rulers
that those of us educated
the United States
learned in all of our European
friends sneer at us about.
The speed of light is
to incredible precision
1 foot per nanosecond.
To be fair, let's make
that a wiggly equal.
So what this means is
that in the units that I'm
going to be working
with, if I then
want to express the speed
of light in these units--
So C is one light time
unit per time unit, which
we are just going to call 1.
So we will generally set the
speed of light equal to 1.
Just bear in mind what
this essentially means
is that you can think of,
if you want to then convert
to your favorite meters per
second, furlongs per fortnight,
whatever it is that you're
most comfortable with,
C is effectively a
conversion factor then.
And so what this means
is that when we do this
all velocities that we measure
are going to be dimensionless.
Really what we're doing
is we're measuring them
as fractions of
the speed of light.
Now, with this system
of units defined,
let's talk about a
geometric object.
So let's imagine
that O is an observer
in the inertial
reference frame that I
defined a few moments ago.
This is a mouthful to say,
it's even more of a mouthful
to write, so I'm going to
typically abbreviate this IRF.
So o observes two
events, which I
will label P and Q. I'll just
go to a clean board for this.
So let's say here in
spacetime-- so imagine
I've got coordinate axes
that have been drawn,
it's going to be
three dimensional,
for simplicity I'm not going
to actually write them out.
Let's say I've got event P
here and event Q over here.
Now if we were just doing
Euclidean geometry in three
space, you guys have all
known that once you've
got two events written
down on a plane
or in a three dimensional
space, something
like that, you can define the
displacement vector from one
to the other.
We're going to the same
thing in spacetime.
So let's call delta x--
and I'm going to make a comment
on notation in just a moment--
this is the displacement
in spacetime from P to Q.
We're going to
define the components
of this displacement vector
as seen by O. So when
I write equals with a dot on it
that means the geometric object
that I've written on
the left hand side
is given according
to the specified
observer by the following
set of complements, which
I'm about to write down.
So this looks like so.
So, couple things that I want to
emphasize that I'm introducing
here, bits of
notation that we're
going to use over and
over again in this term.
Notice I am using an
over arrow to denote
a vector in spacetime.
Different texts,
different professors
use slightly different
notations for this.
Those of you who took 8.033
at MIT with Sal Vitale he
preferred to write a little
under tilde when he wrote that.
For us working in four
dimensional space time,
it is going to be of
paramount importance to us,
and so we're going to use this
over arrow which you probably
have all seen for ordinary
three dimensional vectors.
For us it's going to represent
a four dimensional vector.
Now in truth we're not going
to use it all that much
after the first couple
of weeks of the class,
our first couple lectures even.
Occasionally we'll
bust it out, but we
will tend to use a more
compact notation in which we
say delta x, that displacement
factor has the components
delta x mu, where mu lies
is in either t, x, y, and z,
or 0, 1, 2, 3.
When we set up a
problem, we need
to make a mapping to what
the numerical correspondences
between--
I need to tell you that mu
equals 0 corresponds to time,
and mu equals 1
corresponds to x.
We'll switch to other
coordinate systems
and I'll have to be careful
to say, almost always,
mu equals 0 will be time.
But what the other
three correspond to,
that depends on the
coordinate system.
It might be a radius, it
might be a different angle,
some things like that.
Again, just sort of being
a little overly cautious
and careful defining these.
I will note, though, that
generally Greek indices
in most textbooks,
they tend to be used
to label spacetime indices.
And then there
are times when you
might want to just
sort of imagine
you've chosen a
particular moment in time,
and you want to look at what
space looks like at that time.
And so you might then
go down to Latin indices
to pick out spatial components
at some moment in time.
We'll see that come up
from time to time, just
want you to be aware
there is this distinction.
And as you read other
textbooks, there
are a few others that are used.
Always just check,
usually in some
of the introductory
chapters of the textbook,
they will define these
things very carefully.
Wald is an example of someone
who actually does anything
a little bit different.
He tends to use lower
case Latin letters
from the top of the alphabet
to denote spacetime indices,
and those from i, j, k, he uses
them to denote Latin indices.
If you're old
enough to get this,
this is often called by some of
us who grew up in the Dark Ages
it is sometimes called
the Fortran convention.
If you've ever programmed in
Fortran, you know why that is.
If you didn't, please
don't bother learning it,
it's really not worth the
brain cells it would take.
So we've got this
geometric object
that is viewed by observer o.
Let's now think about what this
looks like from the viewpoint
of a different observer.
A different inertial observer.
Let's say somebody comes
dashing through the room here,
and observer o sees them running
across the room at something
like 87% of the speed of light.
You know, since,
as I have assumed,
you are all experts
in special relativity,
that they will measure intervals
of time and intervals of space
differently than
observer o does.
So here's event P. Here's
event Q. Here is delta x.
This is all as measured
by observer o bar.
Something which I really
want to strongly emphasize
at this point is that this
P, this Q, and this delta x,
notice I haven't put
bars on any of them.
I haven't put primes
or anything like that.
It is the exact same P and Q
and delta x as this over here.
That is because P, Q, and delta
x are geometric objects whose
meaning transcends the
particular inertial reference
frame used to define the
coordinates at which P exists,
at which Q exists, and that
then defined the delta x.
These geometric objects
exist independent
of the representation.
If I can use an
intuitive example--
if I take and I hold--
let's be careful the pose
I do with this-- let's
say I stick my arm out.
I say that my arm is
pointing to the left, right?
You guys will look
at this and say
your arm is pointing
to the right,
because you're using a slightly
different system of coordinates
to orient yourself in this room.
We're both right.
We have represented this
geometric object, my arm,
in different ways.
But me calling this
pointing to the left
and you calling it pointing
to the right doesn't change
the basic nature of my arm.
It doesn't mean
that my blood cells
changed because of something
like this happening.
This has an
independent existence.
In the same way,
this delta x, it
is the displacement,
these two events.
This might be mosquito
lands on my head,
this might be me
smacking it with my hand,
or flying near my head and
me smocking with my hand.
Those are events that
exist independent of how
we choose to represent them.
So the key thing is we
preserve that notion
of the geometric object's
independent existence.
What does change is
the representation
that the two observers use.
So I'm going to jump to
this Greek index notation.
And so what I'm going to say
is that according to observer o
bar, they are going to represent
this object by a collection
of components that are not the
same as the complaints that
are used by observer o.
And to keep my
notation consistent,
let's play a little o
underneath this arrow.
So this is just sort
of shorthand for delta
x is represented according
to o by those components.
Delta x is represented according
to o bar by these components.
And again, since
I'm assuming you all
are experts in
special relativity,
we already know how to
relate the barred components
to the unbarred components.
They're related by a
Lorentz transformation.
So what we would say is delta
x is zero bar component, or t
bar, if you prefer.
It's given by gamma--
I will define gamma in just a
moment, you can probably guess.
So I'm imagining
an observer that
is just moving along the x-axis
or the coordinate one axis.
So what went on in
that transformation
I just wrote down is O bar
moves with v along spatial axis
1 with speed v as seen
by O. And of course gamma
is 1 divided by square
root of 1 minus v squared.
Remember, speed of light is 1.
We don't want to be writing
this crap out every time we have
to transform different
representations,
so we're going to introduce
more compact notation for this.
So we're going to
say delta x mu bar--
this is what I get
when I sum over
index nu from 0 to 3 of lambda.
Mu bar nu, delta x nu.
So that's defining a
matrix multiplication, ,
and you can read out the
components of this lambda
matrix from what
I've got over there.
And even better
yet, delta x mu bar
is lambda mu bar nu delta x nu.
So in this last line, if you
haven't seen this before,
I am using Einstein's
summation convention.
If I have an index that
appears in one geometric
object in the
downstairs position
and an adjacent
geometric objects
in the upstairs
position and it's
repeated-- so repeated indices
in the upstairs and downstairs
position are assumed to be
summed over their full range,
from 0 to 3.
We're going to talk about
this a little bit more,
what's going on with this, after
I've built up a little bit more
of the mathematical structure.
In particular, what
is the distinction
between through the upstairs
and downstairs positions.
Some of you might
be saying, well,
isn't one way of writing it what
we call a covariant component
and wasn't one a
contravariant component?
If you know those
terms, mazel tov.
They're not actually
really helpful,
and so I kind of
deliberately like
to use this more primitive
wording of calling it
just upstairs and downstairs.
Because what we're going to
find, the goal of physics,
is to understand the
universe in a way that
allows us to connect this
understanding to measurements.
And measurements don't
care about contravariant
versus covariant, and all
these things are essentially
just ways of representing
objects with our mathematics
that is sort of a go between
from some of our physical ideas
to what can eventually
be measured.
So covariant,
contravariant, eh, whatever.
At the end of the
day, we're going
to see as we put these
sort of things together
it's how these terms connect
to one another that matters.
The name is not that important.
I do want to make one
little point about this
as I move forward.
It is sort of worth
noting that if I
think of how I relate the
displacement components
according to my barred observer
relative to those as measured
by my unbarred observer, I
can think of this Lorentz
transformation matrix as what
I get when I differentiate
one representation's
coordinates with respect
to the other
representation's coordinates.
Kind of trivial in
this case, and when
we are doing special
relativity there
is a particular form of
the Lorentz transformation
that we tend to
use, but I just want
to highlight this because
this relationship between two
different representations
of a reference frame
is going to come up over
and over and over again.
This is a more general
form, this idea
that you are essentially
taking the derivative.
You're looking at how one
representation varies according
to the other representation.
So using that to think
about how to move
between one inertial
reference frame to another
is going to be very
important to us.
This is a more general form.
I want to make
one further point,
and then I will introduce a
more careful definition of what
is meant by a vector
and I think that will
be a good place for us to stop.
So when I look at
this particular form--
so let's look at the
last thing I wrote there,
where I use the Einstein
summation convention.
This is just a chance for me
to introduce a little bit more
terminology in notation.
So when I wrote down
the relationship delta
x mu bar with lambda
mu bar nu, delta x nu,
how I labeled the index
that I was summing over
is kind of irrelevant.
This is exactly the same as
lambda mu bar alpha delta
x alpha.
I can switch to
something else if you
have enough fonts available.
You can use smiley faces
as your index, whatever.
The key bit is that as long
as you are summing over it,
it's kind of irrelevant
how you label it.
I think that looks silly, so
I'm going to go back to alpha.
When I have an index like that
that is being summed over,
it's going to sort of disappear
at the end of my analysis.
Its only role is to
serve as a place holder.
It allows me to keep
things lined up properly
so that I can do a particular
mathematical operation.
So in this equation, nu or
alpha is called a dummy index.
Now, when I'm doing
it with something
like this, where I'm just
relating one set of one
indexed objects to another
set of one indexed objects,
it's kind of trivial to
move these things around.
We're going to make some much
more complicated equations
later in this class, and
we'll find in those cases
that sometimes it's
actually really useful
to have the freedom to
relabel our dummy indices.
It allows us to sort
of pick out patterns
that might exist
among things and see
how to simplify a relationship
in a really useful way.
On the other hand, I
do not have the freedom
to change that mu bar
that appears there, right.
I have to have
that being the same
on both sides of the equation.
Because I'm free to mix
around things where it's just
going to be summed over
and not play a role,
but in this my mu bar--
pardon me for a second--
mu bar is not a dummy index, and
so I do not have that freedom.
We sometimes call
this the free index.
As I write that down, it seems
like a bit of a strange name,
actually, and it's
really not free.
You're actually constrained
in what it can be.
What can I say, probably
there's some history there
that I don't know about.
So let's do our last
concept for the day.
So let's carefully define
a spacetime vector.
So a spacetime vector is going
to be any quartet of numbers,
those numbers we will call
components, which transforms
between inertial
reference frames
like the displacement vector.
So if I represent some
spacetime vector A
as some collection of
numbers that as observed by o
has components A0, A1, A2, A3.
If a second inertial observer
relates their components
to these by--
if that describes the components
for the observer o bar,
then it's a vector.
If you have taken
any mathematics,
they carefully
defined vector spaces,
this should be familiar to you.
It's a very similar
operation to what
is done in a lot of
other kinds of analysis.
The key to making this notion
of a vector a sensible one
is this transformation law.
So if I had a quartet of
numbers, which is say,
the number of batteries
in my pocket, the number
of times my dog sneezed this
morning, how many toes I
have on my left foot, and--
I'm sick of counting
so let's just say
0 for the fourth component--
that does not transform
between reference frames
by a Lorentz transformation.
It's just a collection
of random numbers.
So not any old
quartet of numbers
will constitute the
components of a vector.
It has to be things that have
a physical meaning that you
connect to what you measure
by a Lorentz transformation.
For it to be a good vector,
it also has to-- that A
has to obey the various
linearity laws that
define a vector space.
So if I had two vectors
and I add them together,
then their sum is a vector.
If I have a vector and I
multiply it by some scalar--
by the way, you have to
be a little bit careful
when I say scalar here,
because you might think
to yourself something
like, you know,
the mass of my shoe,
that's a scalar.
But you be careful when
you're talking about things
in relativity, whether the
scale you're dealing with
is actually a quantity
that is Lorentz in variant.
With quantity like mass, if
you're talking about rest
mass-- we'll get into the
distinction among these things
a bit later--
OK, you're good.
You just have be
careful to pick out
something that actually
is the same according
to all observers.
So when I say scalar, this
means same to all observers.
If this is the case,
then I can define
D to be that scalar times a
vector and it is also a vector.
Question?
Oh, stretching, OK.
I think, actually, I'm
going to stop there.
There is one topic
that I just don't
feel like I can get enough of it
for it to be useful right now,
so I think yes.
I'm going to stop
there for today.
When we pick it up
on Thursday we're
going to wrap up this
discussion of vectors--
again, I want to
kind of emphasize,
I can already see in some places
you're getting the 1,000 yard
stare.
There's no question we're
being excessively careful
with some of these definitions.
They are very
straightforward, there's
nothing challenging here.
But there will be a payoff
when we do get to times where
the analyses, the
geometries we're looking at,
get pretty messed up.
Having this formal foundation
very carefully laid
will help us significantly.
So all right, I'm going
to stop there for today,
and we will pick
it up on Thursday.
