Doubt about Energy Condition in Wormhole: Integral Along Null Geodesic

[wLw]

I am now reading this paperhttps://arxiv.org/pdf/gr-qc/0405103.pdf, which is related to the energy condition in wormhole. Nevertheless, I got a problem in Eq.(6), which derives from so-called ANEC in Eq.(2): $$\int^{\lambda2}_{\lambda1}T_{ij}k^{i}k^{j}d\lambda$$
And I apply the worm hole space time Eq.(3) to calculate as following:

Firstly, I construct an orthonormal null vector: $$k^{i}=Z^{i}-u^{i}$$, where $$Z^{i}=e^{-\phi}(\frac{\partial}{\partial t})^{i}; u^{i}$$ are normed vectors. Then I have $$T_{ij}k^{i}k^{j}=T_{00}+T_{ij}-(T_{0i}-T_{0j})$$ , and the author takes radius component(i=j=r) , so $$T_{ij}k^{i}k^{j}=\rho +p_{r}$$, which is the integral function in Eq.(2), but in this paper, the correct integral has a factor :exp{-2$\phi$} as shown in this paper Eq.(6). In addition, the integral Eq.(6) becomes a closed curve integral. I wonder what is wrong with my calculation and why the integral path becomes closed $\oint$, does that mean the null geodesic is closed? could you help me?

 

[PeterDonis]

I construct an orthonormal null vector

What is $u^i$?

 

[wLw]

What is $u^i$?

the normed spatial vector, in this case , $$u^{i}=\sqrt{1-\frac{b}{r}} (\frac{\partial}{\partial r})^{i}$$

 

[PeterDonis]

the integral Eq.(6) becomes a closed curve integral

I'm not sure that's actually the intent, although the notation certainly suggests it. If you look at the explicit examples in section III none of them appear to be over closed curves.

That said, in a wormhole spacetime closed null geodesics could be possible if the two exterior regions (one on either side of the wormhole) are connected appropriately.

 

[wLw]

That said, in a wormhole spacetime closed null geodesics could be possible if the two exterior regions (one on either side of the wormhole) are connected appropriately.

May be there is a closed null geodesic near wormhole, but I wonder how the Eq(6) derive from Eq(2). My calculation as shown above but lack the factor exp{-2\phi}, do you know what mistake I have made in my calculation?

 

[PeterDonis]

I wonder how the Eq(6) derive from Eq(2).

By substitution from Eq. (4) and implicitly from the null geodesic components derived from Eq. (3). Note, however, that the Eq. (4) components $\rho$ and $p_r$ are in an orthonormal basis, not the coordinate basis, so you have to insert appropriate factors from the metric coefficients.

do you know what mistake I have made in my calculation?

This...

$$
T_{ij}k^{i}k^{j}=T_{00}+T_{ij}-(T_{0i}-T_{0j})
$$

...does not look right.

 

[wLw]

does not look right.

Here is my details : Z,u are orthonormal vectors as above . $$T_{ij}k^{i}k^{j}=T_{ij}(Z^{i}-u^{i})(Z^{j}-u^{j})=T_{ij}(Z^{i}Z^{j}-Z^{i}u^{j}-u^{i}Z^{j}+u^{i}u^{j})=T_{00}+T_{ij}-T_{0j}-T_{0i} $$Ok I miss a sign here, but now it is not still the same as that in Eq(6)

 

[PeterDonis]

Here is my details

They still don't look right. You wrote down the components of $Z^i$ and $u^i$, but you're not using them; you're just treating them as $1$.

 

[wLw]

treating them as 111.

Because they are normed vectors , so the norm is 1

 

[PeterDonis]

Because they are normed vectors , so the norm is 1

The norm is not what appears in the equations you wrote. The components are. The components are not $1$. You wrote them down.

 

[wLw]

The norm is not what appears in the equations you wrote. The components are. The components are not 111. You wrote them down.

$$(Z^{i}-u^{j})(Z_{i}-u_{j})=0$$, so it is a null vector, sorry i could not understand what you mean?

 

[PeterDonis]

so it is a null vector

I am not disputing that it is a null vector. That's not the problem.

sorry i could not understand what you mean?

Do you know the difference between the norm of a vector and its components?

 

[wLw]

Do you know the difference between the norm of a vector and its components?

$$|v^{a}|^{2}=|g_{ab}v^{a}v^{b}|$$

 

[wLw]

Do you know the difference between the norm of a vector and its components?

could you help me write the correct process, so i could study and if i have some questions then i will ask

 

[PeterDonis]

could you help me write the correct process

I have already told you the correct process:

By substitution from Eq. (4) and implicitly from the null geodesic components derived from Eq. (3).

The null geodesic components are the $k^i = Z^i - u^i$; you have already written what $Z^i$ and $u^i$ are, and since $Z^i$ only has a $t$ component and $u^i$ only has an $r$ component, the vector $k^i$ is easy to write down: it's just $k^0 = Z^0$ and $k^1 = u^1$.

The only part you need to figure out is what the components $T_{00}$ and $T_{11}$ are in the coordinate basis (you know they are $\rho$ and $p_r$ in an orthonormal basis, but the coordinate basis is not orthonormal since the coordinate basis vectors do not have norm $1$).

 

[wLw]

in the coordinate basis (you know they are ρρ\rho and prprp_r in an orthonormal basis

BUT the $\rho ,p_{r}$ are calculated in normed basis not in the coordinates , see page3:'Using the Einstein field equations, the components of the diagonal energy–momentum tensor in an orthonormal basis turn out to be (units — G = c = 1) [2] ', that is why i did not use coordinate but use normed basiss

 

[PeterDonis]

the $\rho ,p_{r}$ are calculated in normed basis not in the coordinates

Yes, I know, but the integral equation, Eq. (2) in the paper, is written in a coordinate basis, so you have to figure out what the stress-energy tensor components are in that coordinate basis.

 

[wLw]

but the integral equation, Eq. (2) in the paper, is written in a coordinate basis,

oh really ? But I can not find the information about calculation in coordinateof Eq(2) by the author.

 

[PeterDonis]

I can not find the information about calculation in coordinateof Eq(2) by the author

That's because the paper you linked to is intended for advanced readers, who already know how to perform such calculations and don't need them laid out step by step. You marked this thread as "A" level, which means you are expected to have that background knowledge.

 

[wLw]

, Eq. (2) in the paper, is written in a coordinate basis, so you have to figure out what the stress-energy tensor components are in that coordinate basis.

you mean Eq.(2) is based on coordinate basis, $k^{i}=((\frac{\partial}{\partial t})^{i}-(\frac{\partial}{\partial r})^{i})$ here r, t are coordinate ? and one could get

T00+Tij−T0j−T0i​

 

[PeterDonis]

you mean Eq.(2) is based on coordinate basis

How else would you fill in the integrand $T_{ij} k^i k^j$? Those are components of $T$ and $k$ in the chosen coordinate chart.

$k^{i}=((\frac{\partial}{\partial t})^{i}-(\frac{\partial}{\partial r})^{i})$

No, that's not a null vector. Your vector $Z^i - u^i$ is a null vector; can you write down its components?

You marked this thread as "A" level but you don't appear to have an "A" level knowledge of the subject. What background in General Relativity do you have?

 

[wLw]

How else would you fill in the integrand $T_{ij} k^i k^j$? Those are components of $T$ and $k$ in the chosen coordinate chart.
No, that's not a null vector. Your vector $Z^i - u^i$ is a null vector; can you write down its components?

You marked this thread as "A" level but you don't appear to have an "A" level knowledge of the subject. What background in General Relativity do you have?

Yes I know that is not null vector , I just follow your information that Eq(2)should be calculated in coordinate. But as you seen , only use coordinates could not construct a null vector . And in my previous thread I mentioned u and Z are normed , so $T_{ij}=diag(\rho,p,p,p)$under these frames my question is how does Eq(6)derive from Eq(2), why there is a factor exp(-2phi), could you write the process directly, so that I could study that .

 

[PeterDonis]

I know that is not null vector , I just follow your information that Eq(2)should be calculated in coordinate.

No, you're not. See below.

as you seen , only use coordinates could not construct a null vector

You are very confused. Calculating using coordinates does not mean the only vectors you can use are the coordinate basis vectors themselves. You already know this, because you wrote the vectors $Z^i$ and $u^i$ in terms of the coordinate basis vectors:

$$
Z^{i}=e^{-\phi}(\frac{\partial}{\partial t})^{i}
$$

$$
u^{i}=\sqrt{1-\frac{b}{r}} (\frac{\partial}{\partial r})^{i}
$$

In the coordinates used in the paper, $(\frac{\partial}{\partial t})^{i} = (1, 0, 0, 0)$ and $(\frac{\partial}{\partial r})^{i} = (0, 1, 0, 0)$, so the vector $Z^i - u^i$ has components $(e^{-\phi}, \sqrt{1-\frac{b}{r}}, 0, 0)$. It is easily verified that this is indeed a null vector. So there you are: you can "construct a null vector" by "only using coordinates".

The above is basic tensor algebra; if you are not already familiar with it you do not have the background knowledge to read the paper you linked to in the OP. You need to get that background knowledge first.

could you write the process directly

No. You are basically asking for a course in tensor algebra. That's way beyond the scope of a PF discussion. You need to take the time to learn how to do these things from a textbook.

This thread is closed.