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Spatial slice of Schwarzschild metric in isotropic coordinates. More...
Public Member Functions | |
| def | __init__ (self, m, kw) |
| Create a Schwarzschild slice metric of a given mass. More... | |
| def | horizon_area (self) |
| Area of the Schwarzschild horizon. More... | |
| def | horizon_coord_radius (self) |
| Schwarzschild radius w.r.t. isotropic coordinates. More... | |
| def | psi (self, point, derivatives=False, max_deriv=1) |
| Public Member Functions inherited from _ConformallyFlatMetric | |
| def | __init__ (self, force_fd=False, kw) |
| Base class init for conformally flat metrics. More... | |
| def | conformal_factor (self, point) |
| Compute the conformal factor at the given point. More... | |
| def | diff (self, point, inverse=False, diff=1, kw) |
| Analytically compute derivatives w.r.t. x, y, z. More... | |
| def | diff_lnsqrtg (self, point, kw) |
| Return x,y,z derivatives of ln(sqrt(det(g))). More... | |
| def | psi (self, point, derivatives=False, max_deriv=1) |
| Compute the value of psi at a given point. More... | |
| Public Member Functions inherited from _ThreeMetric | |
| def | __init__ (self, dx=1e-8, fd_order=3) |
| Base class init for metrics. More... | |
| def | at (self, point) |
| Return the metric tensor at a given point. More... | |
| def | diff (self, point, inverse=False, diff=1, kw) |
| Return derivatives in x,y,z directions at the given point. More... | |
| def | diff_lnsqrtg (self, point, kw) |
| Return x,y,z derivatives of ln(sqrt(det(g))). More... | |
| def | christoffel (self, point) |
| Compute the Christoffel symbols of this 3-metric. More... | |
| def | christoffel_deriv (self, point) |
| Compute the derivatives of the Christoffel symbols. More... | |
| def | ricci_tensor (self, point) |
| Compute the components of the Ricci tensor. More... | |
Public Attributes | |
| m | |
| Public Attributes inherited from _ConformallyFlatMetric | |
| force_fd | |
| Public Attributes inherited from _ThreeMetric | |
| dx | |
| Step length for numerical differentiation. More... | |
| fd_order | |
| Finite difference approximation order for numerical differentiation. More... | |
Spatial slice of Schwarzschild metric in isotropic coordinates.
In isotropic coordinates, the Schwarzschild spacetime metric becomes
where is the flat 3-metric and the conformal factor is
See e.g. Equation (10.89) in [1] or Equation (3.16) in [2].
For a spatial slice, we just take the spatial part of this metric for constant .
Notes
Note that this is not the form the metric is usually written and as such, the physical quantities expressed in values of the "radial" parameter have different expressions. Specifically, the value of this parameter at which the horizon lies is given by (cf. [1]) . This is not to be confused with the Schwarzschild radius , which has a geometric meaning related to the surface area of the horizon and at which the horizon lies in the traditional Schwarzschild coordinates.
Derivatives of the conformal factor are computed in psi() using
References
[1] Schutz, Bernard. A first course in general relativity. Cambridge university press, 2009.
[2] Straumann, Norbert. General relativity. Springer Science & Business Media, 2004.
| def __init__ | ( | self, | |
| m, | |||
| kw | |||
| ) |
Create a Schwarzschild slice metric of a given mass.
| m | (float) ADM Mass of the metric. |
| **kw | Optional arguments for the parent class (e.g. dx ). |
| def horizon_area | ( | self | ) |
Area of the Schwarzschild horizon.
The area is given by , where is the Schwarzschild radius. Note that in isotropic coordinates, the value of this radius cannot be taken as being a meaningful value of our r
coordinate.
| def horizon_coord_radius | ( | self | ) |
Schwarzschild radius w.r.t. isotropic coordinates.
In the typical Schwarzschild coordinates, the horizon is at a "radial" coordinate 2m
. This class implements the slice in isotropic coordinates, where the "radius" parameter value at which the horizon is located is m/2
instead.


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Usually there are many equivalent ways to write an if
statement.
Here is the original fragment, which uses AND:
if ( !(speed > 2000 && memory > 512) ) System.out.println("Reject this computer"); else System.out.println("Acceptable computer");
Here is an equivalent fragement that uses OR:
if ( speed <= 2000 || memory <= 512 ) System.out.println("Reject this computer"); else System.out.println("Acceptable computer");
Yet another equivalent fragment reverses the order of the true and false branches of the statement.
if ( speed > 2000 && memory > 512 ) System.out.println("Acceptable computer"); else System.out.println("Reject this computer");
The last fragment is probably the best choice because it is the easiest to read. It follows the pattern:
if ( expression that returns "true" for the desired condition ) perform the desired action else perform some other action
Generally, people find statements that involve NOT to be confusing. Avoid using using NOT, if you can. If NOTs are needed, try to apply them to small subexpressions, only.
Rewrite the following natural language statment into an equivalent easily understood statement.
It is not true that robins are not seen when the weather is not fair.


This is a paragraph.
from time import sleep for i in range(7): print(i * 10) print(i + 10) sleep(2)
This is a paragraph.
This is a paragraph.
This is a paragraph.
This is a paragraph.
for i in range(5): print(i)
This is a paragraph.
This is a paragraph.
This is a paragraph.
ąłńżźćęóążź


Intro
Think about CSS... and forget Cascading.
A css is just a list of queries (css selection query) that apply styles to matched DOM elements.
A deep sheets it's all the same :
A list of queries (deep-queries for the moment) that apply something (and somehow) to matched javascripts properties from a particular root object.
You define it through a classical js object, where properties are sheets entries.
trough deep.sheets or deep.nodes(...).sheets(...)
dq.up & dq.bottom
Obviously the two more important sheet's actions ar 'up' and 'bottom'.
Simple example with up...
deep.nodes({ a:{ b:true } }) .up({ _deep_sheet_:true, "dq.up::/a":{ hello:"world" } }) .log()
deep.nodes({ a:{ b:true } }) .query("/a") .up({ hello:"world" }) .query("/!") .log()
Obviously, you could have severals entries in a sheet :
var sheet = { _deep_sheet_:true, "dq.bottom::./!":{ array:["from bottom"] }, "dq.up::./a":{ test:123, other:true } }; deep.nodes({ array:["base entry"], a:{ test:1 } }).up(sheet).log();
dq.transform
deep.nodes({ a:true, b:12 }) .up({ _deep_sheet_:true, "dq.transform::./b":function(node){ return node.value + 20; } }) .log();
The callback if fired with each selected node.
RemarqueIf you need to perfom async stuffs in transformer callback, please read below.
Compilation rules
When you apply (deep.up) a sheet on an object, the sheet is executed.
When you apply a sheet on (deep.up) an sheet, the sheets are merged.
When you apply an object on (deep.up) a sheet, the sheet is ignored.
sheets & backgrounds
deep.nodes({ a:true, _backgrounds:[ { b:{ hello:'world' } }, { _deep_sheet_:true, "dq.up::./b":{ c:123 } } ] }) .flatten() .log();
Sheets foregrounds
You could gives sheets list directly in your objects, at any levels (as backgrounds) that will be applied after backgrounds application. The difference from backgrounds application is that those sheets will be applied 'up'
Sheets list could contains, direct sheets, or any string reference to ressource (with its protocol) e.g. "json::/my/sheet/path.json"
deep.nodes({ a:true, _transformations:[ { _deep_sheet_:true, "dq.up::./!":{ c:123 } } ] }) .flatten() .log();
Async API
When you want to manipulate sheets that do async stuffs, you need to use deep.sheets in place of deep.up to apply sheets on objects.
The main reason is that deep.up and deep.bottom are absolutly sync and does not wait promises returned by sheet's application.
dq.transform
Each selected value is passed as argument in provided function AND the return of this function replaces original value
var sheet = { "dq.transform::./*":function(node){ return deep.delay(100) // async simulation .done(function(){ return "async:"+node.value; }); } }; deep.sheet([1,2,3,4,5], sheet) .log();


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