Recent content by James1238765

1. The Riemann curvature in terms of Christoffel coefficients are: 2. The Riemann curvature in terms of ##\{a,b,c,d,e,f,g,h,i,j\}##, ##\{a',b',c',d',e',f',g',h',i',j'\}## are: 3. The Ricci curvature in terms of ##\{a,b,c,d,e,f,g,h,i,j\}##, ##\{a',b',c',d',e',f',g',h',i',j'\}##...

1. For the boundary value problem in 10 variables ##\vec r = [a,b,c,d,e,f,g,h,i,j]## : $$\frac{d^2u}{da^2} + \frac{d^2u}{db^2} + \frac{d^2u}{dc^2} + \frac{d^2u}{dd^2} + \frac{d^2u}{de^2} + \frac{d^2u}{df^2} + \frac{d^2u}{dg^2} + \frac{d^2u}{dh^2} + \frac{d^2u}{di^2} + \frac{d^2u}{dj^2} = f(\vec...

1. Random numerical ##g_{ij}## will generate valid ##T_{ij}##, with matter everywhere: Thus, the difficulty is in generating the black vacuum (zeros), instead of the (white) matter. 2. Combinatorially, we consider a 2x2x2 universe grid, with only 2 matter states (matter ON, and matter OF)...

@romsofia thank you, the two books seem nicely computational! I have looked at a few books/articles on 3+1 method, a lot of new structures (folliation, ##\gamma##) seem needed for ADM & harmonic differential equations solving... I would like to try and exhaust simpler methods while reading up on...

thank you. I think I roughly understand the general analytic strategy now, ie. try to guess the functional form of the independent ##g_{ij}## components and their dependencies on ##\{t, r, \theta, \phi \}##, such that as many of ##\frac{dg_{ij}}{d?}## becomes zero, which causes many ##\Gamma##...

I think I see the differential equations now. If every step from ##g_{ij} \rightarrow R## is expanded into single equations form, we would get 16 equations with 10 variables (the ones in the original metric ##g_{ij}##), such as from [here]. So what are the standard methods to solve (in 1 go)...

Which Einstein field equation? The geodesic ones will just calculate trajectories, don't they? What field differential equations precisely will arise from $$ R_{ij} + Rg_{ij} = T_{ij}$$ ? I would greatly appreciate it if someone could link to a straightforward worked example in numerical...

Starting with ##T_{ij}## we have 16 numbers for each 3D gridpoint, with particular places we want to have all 0 values. The problem seems to be trying to reverse engineer what R and ##R_{ij}## will give rise to this result, then reverse engineer the ##\Gamma^i_{jk}## and all the way to...

Ok. will try it backward from ##T_{ij}## next. Are all the reverse algorithms ##R \rightarrow R_{ij} \rightarrow R^i_{jk} \rightarrow \Gamma^i_{jk} \rightarrow g^{ij} \rightarrow g_{ij}## well defined and calculable for all the intermediate quantities?

all that really long chain beginning with ##g_{ij} \rightarrow g^{ij} \rightarrow \Gamma^i_{jk} \rightarrow R^i_{jkl} \rightarrow R_{ij} \rightarrow R##

The heavyweight computations used original Schwarzchild coordinates, and I was wary if changing to Cartesian suddenly might break something. Yes otffset being time dependant would mean moving bodies, though the solution as currently obtained gives nonsensical ##T_{ij}## anyways, so it's still...

Sorry the coordinate notation was abstract. Here's a real example, the point (1, 1, 1) in spherical coordinate is first transformed into cartesian via: $$x = r \sin \theta \ cos \phi$$ $$y = r \sin \theta \ sin \phi$$ $$z = r \ cos \theta$$ we obtain the new cartesian coordinates: Next we...

Adding 2 sources for ##g_{ij}## could be done pointwise as follows: $$ g_{total} (r, \theta, \phi) = g_{ij} (r, \theta, \phi) + $$ $$g_{ij}(Sph_r(Cart_x(r,\theta,\phi)+dist_x, Cart_y(r,\theta,\phi)+dist_y, Cart_z(r,\theta,\phi)+dist_z))$$ Single source metric is a vacuum solution: Two...

@Ibix from differential equations perspective, could we recover completely ##g_{ij}## from the full set of descendant ##\Gamma^i_{jk}## (post 33)? As to why I insist on trying this, even if the solution is wrong, the practice is often invaluable. We can see in a single picture if the ##T_{ij}##...

Is it possible to recover ##g_{ij}## completely from all the descendant ##\Gamma^i_{jk}##? Adding ##\Gamma^i_{jk}## can easily be done gridpoint-wise. The main problem is the combined metric ##g^{ij}## is needed to calculate R. This combined metric is probably ugly looking, but can it be...