Undergrad Covariant Derivative: Limits on Making a Tensor?

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SUMMARY

The discussion centers on the limitations of covariant derivatives in relation to non-invariant quantities and tensors. It is established that covariant derivatives can only be applied to tensors, not to individual components. Therefore, one cannot derive an invariant tensor quantity from non-invariant components using covariant derivatives. This highlights the fundamental nature of tensors in differential geometry.

PREREQUISITES
  • Tensor calculus
  • Understanding of covariant derivatives
  • Knowledge of invariant quantities in differential geometry
  • Familiarity with the properties of tensors
NEXT STEPS
  • Study the properties of covariant derivatives in detail
  • Explore the relationship between tensors and invariant quantities
  • Learn about the applications of tensors in physics
  • Investigate the implications of tensor calculus in general relativity
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics and physics, particularly those focusing on differential geometry and tensor analysis.

dsaun777
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Can you take any non invariant quantity like components and take the covariant derivative of them and arrive at an invariant tensor quantity? Or are there limits on what you can make a tensor?
 
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dsaun777 said:
Can you take any non invariant quantity like components and take the covariant derivative of them and arrive at an invariant tensor quantity?

You can't take covariant derivatives of components. Covariant derivatives operate on tensors.
 
PeterDonis said:
You can't take covariant derivatives of components. Covariant derivatives operate on tensors.
Sorry my brain is not working long day...
 

Thread 'EPR revisited'

In an inertial frame of reference (IFR), there are two fixed points, A and B, which share an entangled state $$ \frac{1}{\sqrt{2}}(|0>_A|1>_B+|1>_A|0>_B) $$ At point A, a measurement is made. The state then collapses to $$ |a>_A|b>_B, \{a,b\}=\{0,1\} $$ We assume that A has the state ##|a>_A## and B has ##|b>_B## simultaneously, i.e., when their synchronized clocks both read time T However, in other inertial frames, due to the relativity of simultaneity, the moment when B has ##|b>_B##...
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