hello and thanks for joining me today on what will be a bit of an adventure today
we're going to try three different tensors at three vastly different price
points in an effort to widen the scope of our palate in physics and math I will
warn you though that this video will go from zero to 100 so you might want to
buckle up because it'll be a bit of a wild ride I'm joined in this video by
Andrew Dotson who will take us through today's delicacy item the
energy-momentum tensor but first I've scraped together some loose change to
purchase today's budget items so let's have a look at what's in here first of
all I have my order number now since this is just a simple number or a scalar
with no direction needed to describe this single component it would be a
tensor of rank zero it wouldn't change if I change my coordinate system this
would read 142 no matter how I look at it so let's see what else I have
in here well I also have some greasy vectors now if I wanted to describe the
length and orientation of this chip I could do so using a vector that has a
magnitude and direction since I'm in 3d space I could set up a
XYZ axis and describe it using three components because each of those three
components could be described using one direction this would be a tensor of Rank
1 and if I changed my coordinates it would change how I write the vector but
it wouldn't change the vector itself for our mid-level price point we could
consider a slab of jelly which will have various stresses inside which are
internal forces between neighboring parts of the material these forces keep
the material in place and if I was to cut it in half then the two pieces would
deform slightly there are tangential shear forces as well as perpendicular
forces to consider at each point if we consider the point right at the centre
of each of our jellies call it point O we could describe all of the forces on
O as a tensor if we were to make a cut in the Y plane it would expose the face
which is perpendicular to the X direction
and we would see the part where it would have been experiencing forces from the
front half of the jelly call these forces Pxx Pxy and Pxz if we could repeat
this for a cut which exposes the face of the jelly which is perpendicular to the
y direction and again for a face that will be perpendicular to the z direction
considering the forces on each of these faces we would end up with a matrix with
nine components the first index of each component would refer to the direction
that each face is perpendicular to and the second index would refer to the
direction of the force on that face because two directions are used to
describe each component this would be a tensor of Rank 2 this tensor is also
invariant under a coordinate transform and in fact all of these tensors are
invariant because when operated on by a coordinate transform the magnitude of
the components of the tensor change according to that transform whilst the
basis vectors or the directions of the components change in an inverse manner
allowing the entire tensor to be invariant Andrew is now going to tell us
about a rank two tensor in four dimensions and this is where things get
a little bit fancy if you struggle to keep up Andrew does have a lot of other
videos about tensors on his channel but I at least hope that even if you don't
totally know what's going on you can still appreciate the elegance of
the dish. What got me in the tensors well definitely my dear dear friend John
Mayer hey John do you have something to say about that? I met Andrew
back in 2006 it was after I wrote the song gravity and at the time people
were fascinated with understanding the meaning of the song and who the
inspiration for it was rather which was so so disappointing because to me the
song wasn't about a person it was I thought clearly about the
energy-momentum tensor and Andrew knew that he reached out to me on MySpace and
we got to talking about its properties and where it pops up in physics which i
think is what he's gonna be talking about here shortly and we just been
rocking ever since but once you understand how the tensor fits into the
song you see how it just writes itself here let me play the entire thing.
No that's alright thanks John Mayer what's going on smart people level 12 tensor
boi here to talk about the energy momentum tensor without a doubt one of
my favorite tensors I just think it's the coolest I'll get to why a little bit
later but for now let's just talk about what it is so from the name you've
probably gathered that it's a tensor it is a second rank tensor which means that
if we can calculate its components in one basis let's call that T Alpha
Lambda so as long as my big head isn't in the way which I'm sure it is T alpha
lambda and someone else comes it calculates it in another basis say in
the prime frame T mu nu please don't make fun of my mus I'm super
self-conscious about them these two are related by two transformation
coefficients so it's gonna be a D X prime mu D X alpha D X prime nu D X lambda this
is the transformation rule for a second-rank contravariant tensor now
whenever someone says something transforms as a tensor your question
should always be under what transformations are we considering and
typically if you see something indexed with Greek indices that means we're
probably considering Lorentz transformations that's something that
you can infer so for fields like quantum field theory or special
relativity that's what we're going to be considering if you go to more something
like general relativity you're widening your scope to more general coordinate
transformations called diffeomorphisms we're not going to be getting that
technical in this video we will talk a little bit about general relativity but
just throwing that word at you that's something to consider
but you should always know what transformations you're considering now
if we're indexing it with the Greek letters that typically means the the
letters are going from zero to three zero being the time index and one
through three being spatial bits now if we want to know what these components
are it's very common to represent the tensor components as matrix elements all
second-rank tensors can be represented as square matrices but not all square
matrices are tensors that is important and I've made a video showing a counter
example of that but let's go ahead and throw all of these components into a
square matrices into a square matrix we have going from 0 to 3 here 0 to 3 here
so that's going to be a 4 by 4 matrix great we threw it all in a matrix let's
talk a little bit about the physics that it represents so formally it's defined
in terms of a flux of 4 momentum P mu for momenta because it has four
components transforms as a four vector so with the flux of four momentum P mu
across some surface at constant X nu that's helpful right no okay let's try
to make that a little bit more tangible because the first that didn't really
mean anything to me either so really the energy-momentum tensor at
least in the context of general relativity is kind of like a
generalization of the matter density that you use in say Newtonian physics
that rho on the side of poisson's equation so with that it kind of follows
that this term here T zero zero at least if we're looking at say some fluid some
infinitesimal patch of fluid in its rest frame this is going to
represent its energy density okay energy density and once we want totals we can
just integrate over all space similarly in that same frame we can take a look
at these terms here and these are going to be momentum densities
okay that's also kind of like an energy flux now this bottom right block
probably deserves a bit more of an explanation I'm also out of different
colors so I'm just going to use indexing so all the ones twos and threes and
combinations not the zeros I'm just gonna call that T IJ so I J or either
one two or three that has a name and it's called momentum flux
and for the momentum flux it split into two different categories the diagonal
terms which are pressure and the off diagonals which are shear stress so for
I equal to J that's pressure it's not too hard to see how that's related to
momentum and energy pressure is force divided by area it's a force being
applied perpendicular to a surface like punching your hand and force is just the
time derivative of the momentum now for the energy it's kind of like if you
compress a piston full of gas you increase the internal energy now for the
shear stress it's a little bit more subtle so for I not equal to J so that's
these terms shear stress if the pressure is kind of like punching the surface of
your hand and the shear stress is like rubbing them together it's like the
force being applied parallel to the surface before all the engineers comment
in the comment section it doesn't have to be it can also be perpendicular
that's just not the main takeaway as far as the energy goes with the piston
analogy for pressure for the shear stress it's like twisting an object so
in the context of general relativity the shear stress terms could pop up if
you're looking say at a fluid that has nonzero viscosity so if you have a
perfect fluid these off diagonal terms would just be zero cool part of the
reason I like the energy momentum tensor so much is because of how important it
is in general relativity general relativity is governed by the Einstein
field equations and the energy momentum tensor is literally the other half of
the equation this is the covariant version but it amounts to the same thing
now on the left hand side we have the metric tensor and much like how in
differential equations your solution usually isn't just a
number that would be boring it's a function in the same sense the field equations
solution is the metric tensor now I say equations because this isn't just one
equation each of these objects is a second-rank tensor and as we saw those
had 16 components so this is actually 16 equations but because of symmetry it
reduces down to you only having to solve 10 partial differential equations the
metric tensor contains information regarding the geometry of the space-time
from which you can deduce if your space is curved at all by calculating what's
called the Riemann curvature tensor now that's not explicitly shown in field
equations but the contraction of the Riemann curvature tensor gives you this
object which is called the Ricci tensor now taking the trace of the Ricci tensor
gives you the Ricci scalar scalars are nice it's a nice invariant and then over
here we have the cosmological constant because dark energy so the left-hand
side of the equation is all about the geometry of space-time and as we know
now matter and energy is represented in the form of the energy momentum tensor
now the energy momentum tensor is to find T mu nu is equal to minus 2 1 over
root minus g and then it's a functional derivative of the action with respect to
the metric where the g without the index corresponds to the determinant of the
metric okay so this is how you actually define them the energy momentum tensor
and now that you know all of this you can finally understand the famous
expression that says matter tells space-time how to curve and curved
space-time tells matter how to move now the fact that the energy momentum tensor
is defined in terms of the metric actually leads to a very important
result it stems from the fact that the metric tensor is symmetric which means
that if I swap mu with nu nothing should change there are theories where
that doesn't hold they're said to have torsion that does not apply to us here
and it can't for the sake of consistency for logical consistency because this not
being symmetric say we have some point here and two paths connecting it A and B
the metric tensor not being a symmetric tensor would be like
saying the angle between A and B is different from the angle between B and A
that just doesn't make sense the metric tensor has to be a symmetric tensor and
because the energy-momentum tensor is defined in terms of it the way that it
is it guarantees that the energy-momentum tensor is also symmetric
now the energy-momentum tensor isn't only used in general relativity it's
also used in quantum field theory so it's defined from a different way it's
derived in a different way usually from noether's theorem and from noether's
theorem you derived the energy momentum tensor of the form T mu nu I'm using
contravariant indices it's equal to the sum over n dl d d mu phi sub n d nu phi sub
n minus g mu nu l now this part's all fine because the metric is symmetric but
this part is not manifestly symmetric and because of this typically the
definition that you see in the quantum field theory textbooks of the energy
momentum tensor is kind of just for academic purposes because it doesn't
really help you unless it's a symmetric tensor so people have gone out of their
way to find a way to modify the energy momentum tensor such that it retains its
symmetry properties it's called the Bellefonte tensor but anyways it's also
used in quantum field theory it's a bit more subtle but I've taken enough time
so thank you Toby for letting me talk about the tensor it's definitely one of
my favorites be sure to let us know in the comments section what yours is and
I'll see you guys there