The Pullback of Volume Forms

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  • เผยแพร่เมื่อ 24 ธ.ค. 2024

ความคิดเห็น • 11

  • @dededededededewedq
    @dededededededewedq 2 หลายเดือนก่อน +2

    Thank you for your help ! Just started second year of university and we are studying this. I understand it better now

    • @TensorCalculusRobertDavie
      @TensorCalculusRobertDavie  2 หลายเดือนก่อน +1

      Thank you for saying that. Best wishes for your studies. Cheers!

  • @johnwarren8032
    @johnwarren8032 11 วันที่ผ่านมา

    Thank you again. Your patient explanations of all these details adds a lot of value.
    I have a critique of the illustration at 1:30 that may be useful (or maybe not). I think the notation
    f* \omega _ f(p) on the purple arrow could be misleading. The arrow refers to the pullback as a mapping between spaces, so it would make sense to me to denote it simply by f*. f* \omega _ f(p) itself is the image of the specific form \omega_f(p) under this pullback map so that it would be a (co)vector within the cotangent space of M. Likewise, I would prefer to see that image form, currently denoted by \omega_p in the cotangent space of M, written as f* \omega_f(p).
    Please correct me if I'm wrong. Thanks again.

  • @pushpitgahtori5282
    @pushpitgahtori5282 2 หลายเดือนก่อน +2

    Hey
    I am very interested in your works and content
    on general relativity. I have some questions i
    am very curious to ask. They are slightly long
    so can i please get your email of something so
    we can communicate

    • @TensorCalculusRobertDavie
      @TensorCalculusRobertDavie  2 หลายเดือนก่อน +1

      Please feel free to ask your questions on this forum.

    • @pushpitgahtori5282
      @pushpitgahtori5282 2 หลายเดือนก่อน +3

      @@TensorCalculusRobertDavie i want to test a hypothesis i had that- the frame drag of the entire galaxy spins the spacetime curvature of the galaxy as a whole so this gives extra velocity to the stars orbiting on the edge of the galaxy as they reside on that spinning spacetime curvature. The observed extra velocity is said to be due to dark matter but i wanna test this hypothesis but I don’t have the understanding required to do the calculations so i ask of you to either prove or disprove this hypothesis. Thank you and please let me know what are your thoughts on this

    • @TensorCalculusRobertDavie
      @TensorCalculusRobertDavie  2 หลายเดือนก่อน +1

      @@pushpitgahtori5282 It's a very interesting hypothesis, and I can see why you're considering the effects of frame-dragging on galactic scales. Let's break down the idea and address its feasibility with some key points.
      Understanding Frame-Dragging:
      Frame-dragging is an effect predicted by General Relativity where a rotating massive object (like a planet, star, or black hole) "drags" spacetime around it. This effect is significant near extremely massive, compact objects (such as black holes) and causes nearby objects to move differently than they would if the central object were not rotating. The effect was famously confirmed around Earth by the Gravity Probe B experiment.
      Applying Frame-Dragging to a Galaxy:
      Scale of Frame-Dragging: Frame-dragging effects are localized and typically strong only very close to the massive, rotating body causing them. Around something like a black hole, these effects drop off quickly with distance. For frame-dragging to affect stars across an entire galaxy, the rotating source would need to be of immense mass and extend its effects over tens of thousands of light-years.
      Rotation of the Galaxy: Galaxies do rotate, and the stars in a galaxy generally move in circular orbits around the galactic center. However, the frame-dragging effect from a central black hole (or even the supermassive black hole at the center of the galaxy) does not extend far enough to influence the motion of stars across the whole galaxy. The typical frame-dragging region near a supermassive black hole is only a few light-hours across, not tens of thousands of light-years.
      Dark Matter Explanation: The "extra" velocity of stars at the edge of galaxies-meaning the stars move faster than expected from the visible mass alone-is a well-observed phenomenon. The hypothesis of dark matter was developed because these velocities cannot be explained by the gravitational pull from the stars, gas, and other visible matter within the galaxy. The explanation relies on the idea that there is additional unseen mass (dark matter) providing extra gravitational pull, allowing stars to maintain their observed velocities.
      Galactic Frame-Dragging Hypothesis: For your hypothesis to be correct, it would require the entire galactic rotation to induce a significant, coherent frame-dragging effect over the whole galaxy. However, current theories and models of general relativity suggest that frame-dragging doesn't work this way on such a large scale. Even if you consider the cumulative effect of all the rotating matter in the galaxy, the resulting spacetime curvature would not match the observed "flat" rotation curves of galaxies. Instead, the effects would be much weaker and localized around rotating objects.
      Testing the Hypothesis:
      To definitively prove or disprove the hypothesis, you would need to set up a model using the equations of General Relativity, particularly focusing on the Kerr metric for rotating bodies and trying to generalize this to the whole galaxy. The challenge is that no existing models of galaxy-scale frame-dragging predict effects strong enough to account for the observed extra velocities of stars on the edges of galaxies.
      In summary:
      Frame-dragging is a localized effect that is significant near massive, rotating bodies (like black holes) but not over galactic scales.
      The observed velocity curves in galaxies are consistent with the presence of dark matter, which is why dark matter remains the leading explanation.
      Testing your hypothesis would require solving Einstein's field equations for a model representing the entire galaxy’s rotation, but it's unlikely that frame-dragging alone can explain the observed rotational curves.
      I appreciate your creativity in thinking about this, and it’s great to question and explore these ideas! If you want to dive deeper, I'd recommend looking into the Kerr metric and the mathematical formulation of frame-dragging, as well as reviewing studies on galactic rotation curves and dark matter distribution models.

    • @pushpitgahtori5282
      @pushpitgahtori5282 2 หลายเดือนก่อน +2

      @@TensorCalculusRobertDavie thanks for your explanation. I understand what you meant but i have few remaining questions
      Isn’t the 2/3 of the galaxy massive enough to influence the remaining 1/3 on the edge of the galaxy to move just 150 km/s faster than they should because thats all thats required to account for their extra velocity?
      Is solving this problem with general relativity and kerr matric a messy or lengthy task that you may have a trouble solving?
      I appreciate your attention

    • @TensorCalculusRobertDavie
      @TensorCalculusRobertDavie  2 หลายเดือนก่อน +1

      @@pushpitgahtori5282 It's great to see your continued interest in exploring this idea, and you've raised a valid question. Let me address each part of your follow-up to clarify the situation.
      1. Could 2/3 of the Galaxy Influence the Remaining 1/3 on the Edge?
      The idea that the inner part of the galaxy could influence the motion of stars at the edge is reasonable in a general gravitational sense. Indeed, the gravitational pull from the central regions of the galaxy does contribute to the motion of stars in the outer regions. However, the discrepancy we're trying to account for - the "extra" velocity of 150 km/s - requires more than just the usual gravitational effects.
      In Newtonian mechanics, the gravitational influence from the inner part of the galaxy would decrease with distance from the center, leading to lower orbital speeds at the edges (similar to how planets orbit the Sun more slowly as they get farther away). Observationally, however, we see that the speeds remain relatively constant, which is the famous "flat rotation curve" problem.
      For frame-dragging to be responsible, the entire rotating mass of the galaxy would need to exert a coherent dragging effect strong enough to affect stars tens of thousands of light-years away. Current understanding of General Relativity indicates that frame-dragging doesn’t have this kind of reach on galactic scales. Even if the inner two-thirds of the galaxy were very massive, the frame-dragging effect would be localized and not likely to uniformly boost the orbital velocities of stars on the outskirts by 150 km/s.
      2. Would Solving This Problem with General Relativity Be Complicated?
      Yes, it would be a complex problem, but not impossible to approach with the right mathematical tools. Solving this problem rigorously would involve setting up and solving Einstein's field equations for a rotating galaxy, which would require a detailed understanding of the distribution of mass and rotation within the galaxy. Here's why it gets messy:
      Kerr Metric Limitations: The Kerr metric, which describes the spacetime around a rotating black hole, is a solution for an idealized rotating, spherically symmetric object. Applying this directly to a galaxy, which is a vast, disk-like structure with complex mass distribution, would not give an accurate result without substantial modifications.
      General Relativity vs. Newtonian Approximation: On galactic scales, the effects of general relativity are usually very small compared to the standard gravitational influences described by Newtonian mechanics. To see a significant relativistic effect (like frame-dragging), you need extremely compact and massive objects, like black holes. A galaxy doesn’t behave in the same way because it’s not compact and dense enough overall, despite its total mass.
      3. Why Is the Flat Rotation Curve Still a Mystery?
      The reason the flat rotation curves are still a major issue is that general relativity doesn’t predict a significant frame-dragging effect on this scale. That’s why astrophysicists have proposed dark matter: it’s a way to explain the observations without requiring an improbable relativistic effect. Dark matter models can predict the correct velocities across galaxies by adding extra mass that does not interact electromagnetically (so we don’t see it) but does interact gravitationally.
      Conclusion:
      In theory, the gravitational influence from the inner parts of the galaxy does play a role, but it's not enough to account for the flat rotation curves. For frame-dragging to account for the extra velocity of stars at the edge of the galaxy, the effect would have to be significantly stronger than what we expect based on current models of general relativity, and galaxies just aren’t dense enough to create that kind of effect.
      While the idea is fascinating, current understanding suggests that this extra velocity is more likely explained by the presence of dark matter rather than frame-dragging on a galactic scale. Solving this with a full relativistic treatment would indeed be a lengthy and complex task, and the results would likely reinforce that dark matter is a better explanation.
      I hope this helps clarify things! If you’re interested in more technical details, I’d be happy to explain further, or suggest some reading on dark matter, frame-dragging, and general relativity.