Fdtxjr

Estimate capacitor parameters

grandmas drink with lemon juice

Active filter with series inductor and resistor - do these exist?

Statistical model of ligand substitution

Can a non-EU citizen traveling with me come with me through the EU passport line?

Stopping real property loss from eroding embankment

Stop battery usage [Ubuntu 18]

Is above average number of years spent on PhD considered a red flag in future academia or industry positions?

What was the last x86 CPU that did not have the x87 floating-point unit built in?

What do you call the holes in a flute?

How to politely respond to generic emails requesting a PhD/job in my lab? Without wasting too much time

How can I protect witches in combat who wear limited clothing?

Training a classifier when some of the features are unknown

How is simplicity better than precision and clarity in prose?

Why don't the Weasley twins use magic outside of school if the Trace can only find the location of spells cast?

Is there a documented rationale why the House Ways and Means chairman can demand tax info?

Is there a service that would inform me whenever a new direct route is scheduled from a given airport?

Strange behaviour of Check

How do I keep my slimes from escaping their pens?

What LEGO pieces have "real-world" functionality?

Can a zero nonce be safely used with AES-GCM if the key is random and never used again?

When is phishing education going too far?

Unable to start mainnet node docker container

Direct Experience of Meditation

Why if a tensor vanishes in some coordinates system it vanishes in all coordinates systems?

0

$begingroup$

If we take how the components of a tensor on a pseudo riemannian manifold transform

$$T'^munu(x')=fracpartial x'^mupartial x^alphafracpartial x'^nupartial x^betaT^alphabeta(x),$$

if $T^alphabeta(x)=0$ in a certain coordinates system, we get that the tensor vanishes only in the intersection of the charts, but this nothing tells me about what the tensor does where $U_xcap U_x'=emptyset$, with $U_x$ and $U_x'$ neighborhoods of the two charts.

So, as far as I know from the above relation, my tensor is null in the intersection but may be non vanishing in the rest of the chart.

Do all the charts intersect somehow and thus going through all the charts I can see it?

differential-geometry tensors

share|cite|improve this question

edited Apr 9 at 8:16

José Carlos Santos

175k24134243

asked Mar 25 at 20:56

WideepWideep

518

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It doesn't. If the components of a tensor vanish in some coordinate system, all you can deduce is that this tensor is zero on the neighborhood over which the coordinate system is defined.
  $endgroup$
  – levap
  Mar 25 at 21:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Because in physics I have always heard "if a tensor vanishes in a coordinates system it vanishes in all coordinates".
  $endgroup$
  – Wideep
  Mar 25 at 21:09
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The precise statement would be "if a tensor vanishes on a collection of coordinate systems which cover the whole manifold then it vanishes in all possible coordinate systems".
  $endgroup$
  – levap
  Mar 25 at 21:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Wonderful, it was as I thought it should be! Thanks.
  $endgroup$
  – Wideep
  Mar 26 at 8:31
  
  

  
  
  
  

add a comment | 

0

$begingroup$

If we take how the components of a tensor on a pseudo riemannian manifold transform

$$T'^munu(x')=fracpartial x'^mupartial x^alphafracpartial x'^nupartial x^betaT^alphabeta(x),$$

if $T^alphabeta(x)=0$ in a certain coordinates system, we get that the tensor vanishes only in the intersection of the charts, but this nothing tells me about what the tensor does where $U_xcap U_x'=emptyset$, with $U_x$ and $U_x'$ neighborhoods of the two charts.

So, as far as I know from the above relation, my tensor is null in the intersection but may be non vanishing in the rest of the chart.

Do all the charts intersect somehow and thus going through all the charts I can see it?

differential-geometry tensors

share|cite|improve this question

edited Apr 9 at 8:16

José Carlos Santos

175k24134243

asked Mar 25 at 20:56

WideepWideep

518

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It doesn't. If the components of a tensor vanish in some coordinate system, all you can deduce is that this tensor is zero on the neighborhood over which the coordinate system is defined.
  $endgroup$
  – levap
  Mar 25 at 21:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Because in physics I have always heard "if a tensor vanishes in a coordinates system it vanishes in all coordinates".
  $endgroup$
  – Wideep
  Mar 25 at 21:09
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The precise statement would be "if a tensor vanishes on a collection of coordinate systems which cover the whole manifold then it vanishes in all possible coordinate systems".
  $endgroup$
  – levap
  Mar 25 at 21:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Wonderful, it was as I thought it should be! Thanks.
  $endgroup$
  – Wideep
  Mar 26 at 8:31
  
  

  
  
  
  

add a comment | 

$begingroup$

If we take how the components of a tensor on a pseudo riemannian manifold transform

$$T'^munu(x')=fracpartial x'^mupartial x^alphafracpartial x'^nupartial x^betaT^alphabeta(x),$$

if $T^alphabeta(x)=0$ in a certain coordinates system, we get that the tensor vanishes only in the intersection of the charts, but this nothing tells me about what the tensor does where $U_xcap U_x'=emptyset$, with $U_x$ and $U_x'$ neighborhoods of the two charts.

So, as far as I know from the above relation, my tensor is null in the intersection but may be non vanishing in the rest of the chart.

Do all the charts intersect somehow and thus going through all the charts I can see it?

differential-geometry tensors

share|cite|improve this question

edited Apr 9 at 8:16

José Carlos Santos

175k24134243

asked Mar 25 at 20:56

WideepWideep

518

$endgroup$

share|cite|improve this question

edited Apr 9 at 8:16

José Carlos Santos

175k24134243

asked Mar 25 at 20:56

WideepWideep

518

share|cite|improve this question

edited Apr 9 at 8:16

José Carlos Santos

175k24134243

asked Mar 25 at 20:56

WideepWideep

518

edited Apr 9 at 8:16

José Carlos Santos

175k24134243

edited Apr 9 at 8:16

José Carlos Santos

175k24134243

asked Mar 25 at 20:56

WideepWideep

518

asked Mar 25 at 20:56

WideepWideep

518