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So following Luhnow and Lipchitz we have introduced tanzer energy energy momentum,

tanzer for gravity t mu nu.

To understand deeper its physical meaning let us consider

gravitational field in weak field approximation.

So it means that we consider the following situation,

that the metric tanzer is approximately mean

cost in tanzer plus small protrubation on top of it.

Small means that all the contributions to this guy are much less than one.

In principle, what are we going to say can be done on top of any background metric.

But, for simplicity, and for illustrative reasons, we consider,

in this lecture, we consider perturbations over the flat background.

So, we choose as the background Minkowski Metric.

So, first we going to restrict ourselves to the linear order in this quantity.

At linear order in this quantity, we want to find the inverse metric tanzer.

To find the inverse metric tanzer, we have solve the following equation.

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G alpha nu = delta nu mu, and at linear order,

we have to solve it for this.

At linear order, this quantity, g mu nu,

is approximately equivalent to eta mu nu minus h mu nu.

Through all this, the rest of the lecture, we assume that the highering and

lowering of the indices is done with the use of the background metric.

So it means that h mu nu is, by definition,

is eta Mu alpha h alpha nu.

So we higher and lower index with the Minkowski and background metric.

So with the same precision, then,

modulus of the determinant of the metric

is approximately equivalent to 1 plus h,

where h is, by definition,

is the trace of this tensor mu nu.

So as follows from the generic infinitesimal transformation,

the generic infinitesimal coordinate transformation is as follows.

As we remember, it is- Defined

with the covariant derivative like this.

So, and in the linearized approximation

one can see that from this and this h mu nu bar(x)

is approximately h mu nu(x) plus short

derivative in the linearized approximation.

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So, in the same approximation, Christoffel symbols,

as follows, defined like this, using all of this.

Gamma mu nu alpha,

it's approximately

one half at mu beta d mu

h alpha beta plus d alpha h

Beta nu minus d beta h nu alpha.

So we see that this linear in h gamma is linear in h.

As a result we remember that in remen tanzer, or in reacher tanzer,

the rad contributions which are d gamma and gamma squared schematically.

So these terms are of the second order in h so

at linear order we drop them off, drop them off, and

as a result Riemann tensor in the linear approximation,

Riemann tensor in the linear approximation.

Is as follows, it's mu nu alpha

beta while it's at mu gamma d alpha

gamma, gamma nu beta minus.

So we keep only these terms that's what we write.

Minus d beta gamma gamma, nu, alpha,

so it's approximate relation.

No, the first approximation.

So it's a definition of the first linear approximation for the Riemann tanzer.

So this is equivalent

to one half d mu d

alpha h mu beta + d mu d

beta h mu alpha-

d mu d alpha H mu

beta minus d nu d

beta h mu alpha.

So we are working in the linearized approximation and

consider presevation over the flat background metric.

And we have written in the linearized approximation expression for

the Reimann tanzer.

Now the expression that follows for

Reacher tanzer, which follows from that expression for

the Reimann tanzer, is as follows,

so it's alpha mu beta nu and

this is approximately equal to-

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In fact, well, this is by definition,

in fact, under such a transformation,

this quantity transforms like this, d mu,

well let me write with upper index.

D mu side bar mu nu is equal to d

mu well by definition it's just h

bar mu nu minus g mu one half g mu

h bar and this is just d mu si.

Mu nu + box epsilon nu,

this epsilon nu.

So we can always adjust this quantity such

that it will cancel this and to put this to 0.

This can be done up to homogeneous Up to such

epsilon which solves the homogenous equation.

After this equation we can always and the remaining

freedom after fixing this gage is the same transformation

like this with epsilon 0 which solves this equation So

in this gauge, it is not hard to see that in this gauge,

R mu nu, well if we use this gauge here,

it's a straightforward calculation.

R mu nu is just one half box h mu nu, very simple.

As a result Einstein equations acquire the following format.

Minus box psi menu, this psi is equal to 16 pi

kapa T menu plus Terms which are of the second order in h so

at linear order we have this equation and

this come from the second order.

To find these terms we have to redo all these calculations to the second order.

Well let's see what happens.

Well the second order, the second order and

we have the following situation that if

we use this and then the tanzer was up

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of the second order it's 8 mu nu minus h mu nu.

plus h mu alpha,

h alpha nu.

Also modulus of g at this order

is approximately 1+h

Plus one half H squared minus

one half H nu nu, H nu mu.

All in all, we can do, well,

this is the second order Now one can find also g,

the Ricci scalar, or Ricci tensor,

Ricci tensor in this approximation,

and the second order, to the second order,

looks quite tedious but it is like this,

it's one half H row sigma d mu d nu

h row sigma minus h row sigma d row d mu.

H nu alpha and symmetrized over the indices mu and nu.

Plus one quarter

of d mu h rho sigma d

nu h rho sigma plus

remaining terms also

quite tedious.

And hope we will fit into the board.

So, it's a d

sigma h rho mu

times d rho h g

sigma h rho mu.

And to symmetries over

this synthesis.Plus

one-half d sigma

(h rho sigma d rho h mu

nu)- one quarter

d rho H mu nu gyro of H,

and finally minus

D sigma H ro Sigma

minus one half zero

of h times d nu h nu

row zenith rise.

So, this is the Rehman Ritchie Scaler At the second

order to find these terms one has to day mind that

these terms are approximately the falling.

So, reach it turns to the second order

minus one half at the menu times r to

the second minus One half h mu nu times

a scaler to the first order.

After straight forward calculation one can.

Find that this quantity coincides

with models with 16 pi kappa times t mu nu.

As calculated from the expression that we have written for this for

quantity before in this lecture we have the next expression for this quantity.

If we substitute into that expression, this and

do the calculation of the second order, this will conside with this expression.

With the second order.

Here with the second order in h squared here.

So the meaning of this quantity is actually that it is nonlinear part,

if we consider linearized perturbations and

expand in powers of h, this is nonlinear part in h, so

it can be attributed to the right hand side Of the Einstein equations.

So now, one can easily understand the reason

why this quantity cannot be made tensorial.

You see, in the gravity, to specify what means energy flux, so

the flux through some distance surface we have to, first fix this surface.

To fix this surface we have to fix a background, so

we have to have a background and consider perturbations over this

background which carry the flux through that surface.

So we have to consider background and perturbations But

this separation on what means background and

what means preservation changes.

If we have caught in a transformation, this terms mix on the arbitrary.

Non infant decimal caught in it not necessary, non linearized.

Not such transformation, but Genetic.

We make an observation.

Then we have to separate again what means background and what means perturbations.

And this separation is ambiguous and

that's the reason this quantity it doesn't have tensorial properties.

And finally, one can see that under these transformations Under this transformations

this quantity transforms as a tanser under such linearized transformations.

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