Relativity 109c: Gravitational Waves – Wave Derivation (The Lorenz Gauge) – YouTube

Relativity 109c: Gravitational Waves - Wave Derivation (The Lorenz Gauge)



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  • Video Views: 8615
  • Published On: 2022-02-08 04:00:06
  • Video Published/Author: eigenchris
  • Video Duration: 00:23:55
  • Source: Watch on YouTube


Full relativity playlist: https://www.youtube.com/playlist?list=PLJHszsWbB6hqlw73QjgZcFh4DrkQLSCQa

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Some helpful PDFs on the Lorenz Gauge:
https://pages.pomona.edu/~tmoore/LesHouches/les-houches-4.pdf
https://pages.pomona.edu/~tmoore/LesHouches/les-houches-5.pdf

Other Sources on Gravitational Waves:
Sean Carroll: https://preposterousuniverse.com/wp-content/uploads/grnotes-six.pdf
Wikipedia: https://en.wikipedia.org/wiki/Linearized_gravity
YouTube: https://www.youtube.com/watch?v=kIBdtbCszB0
PDF 1: http://star-www.st-and.ac.uk/~hz4/gr/hendry_GRwaves.pdf
PDF 2: https://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/NS.BH.GW_files/GW_Physics.pdf

0:00 Introduction to Lorenz Gauge
2:10 Proving 3 terms in G go to zero
5:09 Displacement Field
8:03 Metric Tensor transformation
10:30 Riemann Tensor transformation
12:05 h-bar transformation
14:43 Lorenz gauge gives the Wave Equation
17:57 Gauge Transformations in Electricity and Magnetism
22:12 Gauge Transformations in Linearized Gravity

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25 comments
  1. Dear EigenChris, very good.. thanks! Am finally starting to understand 4D-SpaceyTime Tensor/Up_Down_Index notation! And thanks for clarifying the Lorenz/Lorentz confusion, which may be a kind of cosmic joke.

  2. I think this is the first time in your videos that you explicitly deal with a displacement as a coordinate transformation. I always understood the coordinate transformations you used to basically be linear maps.
    And I actually made the assumption that tensors are only invariant under those, and not those that change the origin, but I see why that would be wrong.
    But is there something else we have to pay attention to when we use something other than a linear map as a transformation? Or is that just trivial?

  3. Thank you Cris, for another great video and explanation! I have a question about the gauge invarience in GR. The gauge invarience in the EM filed results in charge conservation. What is the quantity conserved by the gauge invarience in ( curvature?) the Riemann Tensor?

  4. This was great! I have a question regarding the approximations made to get to this point. So first, you walked us through a derivation of the linearized Einstein equations, which is a perfectly sensible thing to do and very common. But I'm confused about the derivation of the Lorenz gauge in linearized gravity, since it seems like that's only an approximate gauge, at least as you walked through here, since you made assumptions about terms being small. This confused me since it seems like even in linearized gravity, the Lorenz gauge is an approximation, and possibly not satisfied exactly by any real transformation. Is that true?

  5. Any insight how we might interprete Lorenz gauge cond.? Its a "continuity equation", right. For the EM 4 potential it can be interpreted as choosing coordinates where the "scalar potential (the zeroth component) is conserved". This can be seen by integrating the divergence equation 21:55 and using divergence theorem.

    So, could this metric Lorenz gauge be interpreted as choosing a frame where "the metric is conserved" in the sense that the contractions cannot just vanish or appear out of nowhere? Metric must propagate if it wants to change. This would make sense if we want to see Gwaves?

  6. Is there a concrete simple example of a Lorenz Gauge frame? I understand that it's just a simple way of writing the field equations in a way that makes it apparent that it is a wave equation, but I would like to picture what kind frames satisfy the Lorenz Gauge.

  7. So we chose a coordinate system that obeys a wave equation to get a metric that obeys a wave equation. How can we be sure that the waves this predicts are really a feature of the metric and not a coordinate artifact?
    Isn't that why Einstein changed his mind about gravitational waves like 15 times?

  8. phenomenal video as always, and incidentally perfect timing with the little explanation on gauges and gauge transformations. i just recently i decided to start reading papers on Gauge Theory Gravity, a flat spacetime formulation of General Relativity using Clifford Algebra based Geometric Algebra/Calculus and new gauge-related axioms to constrain the field equations. Up until now, i was having a hard time understanding why transforming from flat spacetime to another flat tangent spacetime at some point would matter at all, but i think you tipped me off into the right direction. again, thank you for all the fantastic videos.

  9. I only understood about 10% of this, but even so I think I got the punch line.

    By choosing the right gauge the equations simplify so it is much easier to understand, and although the answer is not perfect, it is close enough for government work.

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