Recent content by wLw

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...

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

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

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...

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

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

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

Because they are normed vectors , so the norm is 1

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)

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?

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

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...

yeah but i wonder how do Eq.(38) derive?

I am reading this article now :http://www.cmp.caltech.edu/refael/league/thorne-morris.pdf. And i am a little bit confused about the Eq.(38a)and Eq.(38b), which means the time measured by traverller and people in the sataion, and I just think the time measured by traveller should be his proper...

https://arxiv.org/pdf/1812.06239.pdf In this paper,the authors use ricci flow to construct Lifshitz spaces. But it is known that ricci flow is limited by Riemannian manifold, which has a positive metric. but in this paper the author use ricci flow in a lorentz manifold, whose signature...