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Markus Hanke last won the day on January 17
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A different way of looking at the trampoline analogy
Markus Hanke replied to geordief's topic in Relativity
I’m probably forgetting something obvious but…what are you referring to by “types”? Do you mean the Weyl tensor, Ricci tensor, and Ricci scalar? -
Not the AT specifically, but I’ve done a few of the long-distance trails around Europe. Meant to do the Continental Divide Trail a few years back, but had to cancel due to Covid. Agree with everything @TheVat said, especially the bits about water. One can’t be careful enough, there’s nothing more miserable than a stomach bug on trail. Personally I’m partial to my good old Sawyer Squeeze filter, has served me well for many a trail!
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What is gravity?
Markus Hanke replied to Muhammad Owais Isaac's topic in Modern and Theoretical Physics
This would be contributions from distant sources, as opposed to local ones, as well as contributions from any non-zero cosmological constant. Basically anything that stops spacetime from being completely flat before you account for any local energy-momentum. I agree, in this type of scenario you have clear causation in an operational sense. However, I was really thinking more of an isolated system where all parts remain in free fall at all times. Energy-momentum is locally conserved (the divergence of the tensor vanishes) - but then so is curvature (Einstein tensor). You cannot locally create nor destroy Einstein curvature, any more than you can create or destroy energy-momentum. You can only shift these around, and have them change form - so which ‘causes’ which? Not directly, but it contains energy density. -
What is gravity?
Markus Hanke replied to Muhammad Owais Isaac's topic in Modern and Theoretical Physics
While these comments are certainly true, I think the relationship isn’t as trivial as it might appear; after all, a vanishing Einstein tensor doesn’t necessarily imply a flat spacetime, so these equations form only a local constraint on geometry, but they don’t uniquely determine it. Any background geometry is as much considered to be a ‘source’ as is local energy-momentum, when it comes to working out the particular form of a metric at a certain event. Furthermore you have the non-linearities of the constraint itself, which, in some sense, might also be considered a ‘source’. But those contributions of course don’t explicitly appear. Personally, I just think of spacetime as pure geometry - the only difference between vacuum and non-vacuum is how the Riemann tensor decomposes (Weyl and Ricci curvature), so I envision it purely geometrically all the way. In that way of thinking, no question of causation arises, you just have ever-shifting geometries. But maybe that’s just weird old me -
What is gravity?
Markus Hanke replied to Muhammad Owais Isaac's topic in Modern and Theoretical Physics
That’s because mass never appears in the GR field equations - what is generally called the source term here is the energy-momentum tensor. One must also remember what these equations actually say - they state a local equivalence between a certain combination of components of the Riemann tensor (the Einstein tensor) and the energy-momentum tensor. Nowhere does it claim a ‘causative relationship’, but instead it says that these two are the same thing (up to a constant of course); neither one comes first and ‘causes’ the other. -
What is gravity?
Markus Hanke replied to Muhammad Owais Isaac's topic in Modern and Theoretical Physics
Indeed. Just wanted to mention this again quickly in case other readers find it helpful to have some geometric intuition what the various aspects of curvature - Riemann, Weyl, Ricci - actually mean. -
What is gravity?
Markus Hanke replied to Muhammad Owais Isaac's topic in Modern and Theoretical Physics
Another way to say this is that, in this kind of spacetime and under geodesic motion, shapes (ie angles) are preserved, whereas volumes and surfaces are not. -
What is gravity?
Markus Hanke replied to Muhammad Owais Isaac's topic in Modern and Theoretical Physics
This short document might help, particularly chapter 3: https://www.sas.rochester.edu/pas/assets/pdf/undergraduate/first_order_approximations_in_general_relativity.pdf -
What is gravity?
Markus Hanke replied to Muhammad Owais Isaac's topic in Modern and Theoretical Physics
Sorry, I have not been able to keep up with these discussions over the last few days, as I’m busy with a large RL project. What was the question here? deSitter spacetime has non-zero Riemann tensor, so it’s not flat. -
What is gravity?
Markus Hanke replied to Muhammad Owais Isaac's topic in Modern and Theoretical Physics
In fairness, I think this misrepresents what KJW is trying to say. How does a freely-falling test particle under the influence of gravity move? It follows a geodesic in spacetime, which is a particular solution to the geodesic equation. This equation is itself a particular form of the principle of extremal ageing, ie the tendency will be for the test particle to move such that a comoving clock will record an extremum of proper time between any given pair of events along the trajectory. When you actually perform this variational problem, of course all components of the metric are technically involved. However, when you are dealing with situations that are in some sense close to being Newtonian / not too relativistic, such as Earth for example, the tt-component of the metric will be much larger than the rest of the metric, by a factor of ~c^2. It will dominate the calculation - meaning time dilation plays a much larger role than tidal effects. In that sense, it is indeed almost exclusively time dilation that gives rise to our daily experience of “downward gravity” here on Earth. Of course, this would not be true in other situations, like near the EH of a solar mass BH, where tidal gravity plays a major role. This doesn’t mean that the source of gravity isn’t energy-momentum and curvature, it just means that under certain circumstances the tidal components don’t play a major role, leaving mostly just time dilation as the dominating effect. -
Do inspiral charged black hole pairs radiate light?
Markus Hanke replied to md65536's topic in Relativity
The kind of wavelengths you get would depend on the specifics of the setup - it’s conceivably possible to get visible light too. For a stationary charge supported in a gravitational field, the result I am familiar with from the literature (see link further up in the thread) would indicate that a comoving detector would not detect any radiation, but another detector freely falling past the charge, would. Good point. But I think given enough charge and enough acceleration, it should be detectable. I must admit I’m not sure what the actual numbers are like, I never looked at this in that much detail. -
Do inspiral charged black hole pairs radiate light?
Markus Hanke replied to md65536's topic in Relativity
AFAIK (and can remember) it comes basically from the general definition of the Hamiltonian, with the potential field \(A_{\mu}\) plugged in. I’m a bit pressed for time these days (involved in a big project here at the monastery), so I wasn’t able to immediately find a proper textbook reference; but the second-to-last answer on this PSE thread outlines what I mean: https://physics.stackexchange.com/questions/283519/derivation-of-electromagnetic-stress-energy-tensor-in-curved-spacetime I’ll have to dig through my notebooks when I get a chance, I know I’ve got a proper reference on this somewhere. -
Do inspiral charged black hole pairs radiate light?
Markus Hanke replied to md65536's topic in Relativity
Sure - isn’t that already a suitable model for the situation at hand? The charge is seen to radiate in some frames but not in others. Ok - this doesn’t sound like too hard of a test to perform, I wonder if this has been done? I remember having seen this done, so I’m aware of the concept. What problem do you see with this? -
Do inspiral charged black hole pairs radiate light?
Markus Hanke replied to md65536's topic in Relativity
But we’re not giving up covariance, are we? We’re simply considering how the EM field - a covariant object - decomposes in a particular frame. Is this not a form of non-locality? Basically you’re saying that whether or not a charge radiates in some local region depends on the existence of potentially distant sources (=external field). I need to think about this one first -
Do inspiral charged black hole pairs radiate light?
Markus Hanke replied to md65536's topic in Relativity
I guess what you mean is that the radiation field (and the EM field in general) will always be much larger than the local free-fall frame. There’s also the issue of the field “back-reacting” with the charge, which would make true free-fall impossible in the first place. These are good points, and I’m not sure how they influence the analysis of this situation. I’m struggling to understand this - why would the absence of a magnetic field contradict the charge not radiating? I completely agree, and this insight should be all that’s needed to understand why some observers see radiation and others don’t. That’s fair enough - how would you yourself evaluate and understand this situation?