3+1 Split and ADM
Leonardo Motta
I'm trying to write a code where I can perform a 3+1 split in a way
that I can use a 3-metric with Einstein convention. I can't see
anything wrong with what I have so far, but I'm obtaining an incorrect
result for the 4d Ricci scalar.
I attached a test notebook.
In a nutshell, here is what is done:
1. I define three manifolds:
* M3, 3 dimensional,
* time, 1 dimensional
* M4 = M3xtime
2. metrich for M3 and metricg for M4.
3. SplitDummySpaceTime is a command that transforms 4-d tensor
expressions into sums over M3 and time indices.
4. I define the usual ADM parameters, except that the vector timevec
is not the time vector of ADM, it is just a quantity that contains
only the zero component equal to one, and its up indices and down
indices are no different. This should be clear from the definition of
the ADM metric replacement rules.
5. ADMExpand is a command that takes metricg and replaces it with ADM
parameters. It only does that after metricg indices have been replaced
with M3+time indices.
6. The test section. The first term in that Ricci scalar is wrong
(dimensionally you can tell, R has dimension of H^2).
Following the notebook, the problem is that each term of R contains at
least two derivatives of the metric. So I'd guess
RicciScalarToGradMetric is correct. I made tests with the ADMExpand,
and each individual rule is doing what it is supposed to. I can't see
the full expression after ADMExpand, but if I apply to RicciInADM the
replacement
shift[-i_] :> 0
I can see immediately the offending term, which is a single time
derivative of the metric.
I thought of the following possibilities:
1. The metrich is independent of time at defintion, so after splitting
the dummy indices into time and M3, perhaps some derivatives of
metrich where lost after ADMExpand. But if this were true, than the
same should have applied to the first and second time derivatives that
appear in the final expression...
2. Metric contractions with metricg after ADMExpand. But I do not
request any metric contraction nor call ToCanonical...
Am I going through this in just the wrong way?
Cheers,
Leonardo
JMM
I've had a look at your code and I get confused at your timevecders1
rules. If I understand what you said in your email then the single
component of this vector is just 1, and hence its partial derivative
should be a zero, shouldn't it?
I would recommend to use `pmQ instead of several `Q when you want to
construct rules for indices in several positions. I would write for
example
timevecders3 = { PD[ _?TangentM3`pmQ ] @ timevec[ _?Tangenttime`pmQ ] -
> 0 }
A trivial additional comment is that if you are going to declare
automatic rules to remove first derivatives of an object (your
timevecders1), then you don't need to declare rules for the second
derivatives (your timevecders2) because these will never appear. Note
also that rules for second order derivatives are too deep for
upvalues, and that is why AutomaticRules stores them as "generic
Rules", meaning that they are not automatic, but stored in the global
variable $Rules, that you need to use by hand when needed.
Summarizing, I think all your timevecders rules can be written as a
single rule:
PD[ _ ] @ timevec[ _ ] -> 0
that you can make automatic (no need to use AutomaticRules) with
timevec /: PD[ _ ] @ timevec[ _ ] = 0
I haven't read the rest of the notebook.
Cheers,
Jose.
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侯紹齊
Have you solved your problem?
Best wishes
Shaoqi Hou