KJW

Yes and no. Actually, the gravity with which we are familiar is caused by time dilation, and the curvature of space plays no part in this. But the space surrounding the earth is curved. The Schwarzschild solution, which models gravity around an ideal non-rotating spherical object, requires the surrounding space to be curved for it to describe pure gravitation, even though it is only the time dilation which provides the familiar gravity. It is worth noting that the deflection of starlight by the sun is double that which can be accounted for by time dilation alone. It is the curvature of space which provides the additional half of the deflection. In general relativity, pure gravitation is described by the Weyl conformal tensor field, which has 10 independent components in four-dimensional spacetime. Thus, it is more complicated than the gravity with which we are familiar. Bear in mind that spacetime is the unification of space and time, and the notion of different frames of reference implies that one can't separate space and time. The mathematics of general relativity places space and time on equal footing, with the same formulae applying to both space and time equally.

Should be fixed now. I'll need to tweak it, since it's an old version of MathJax, but it should behave the same as the old system now. It was fixed but now it's gone.

Agreed. I thought this might be a preference option but couldn't find any way to change it.

There is no LaTex!

Hydrogen is always going to be released at the cathode (unless mercury is used as the cathode, in which case sodium is produced dissolved in the mercury). As for what's produced at the anode, whether oxygen or chlorine, I am also curious. I recall that in another thread, @exchemist said that both are produced, depending on the chloride concentration.

As I see it, there are two distinct ways of taking sides. You appear to be taking sides in one way, I am taking sides in the other way. One way of taking sides is with regards to whether or not you support Trump as President, whether or not you would vote for him at an election, whether or not you agree with his policies, etc. This is the way of taking sides that I assume you are referring to. I believe I have made myself clear on how I view Trump. However, I should point out that as an Australian, I am more distant from what is happening than are many other people. The side I am taking is with regards to what the US Constitution says about whether or not Trump can legally serve a third term as President. I believe he can. I am not in any way suggesting that is a good thing, only that it does seem to be legal. But I am only taking my cue from what I have read. I am not a legal expert, certainly not in regard to US law. Indeed, it is not so much that I believe Trump can legally serve a third term as President, but rather that what I have read seems to indicate that Trump can legally serve a third term as President. Why wouldn't it be the common view? It seems to be generally accepted that US Presidents serve two terms and no more. Anyone who knows anything about US politics (not just Americans) would know about the two-term limit.

Excerpt from the second link you shared, see this interpretation is disturbing to me because it negates the idea of a living document and flies in the face of article V of the original Constitution establishing amendment power, where it makes clear the only limits it place on said amendment power was a temporary prohibition on changes to clauses 1 and four of section 9 of article I. This was just one consideration. Further down the article, the authors do consider the meaning of "eligibility" under the Twelfth Amendment after the adoption of the Twenty-Second Amendment. Their view seems to be that because the Twenty-Second Amendment doesn't prohibit a twice-elected president from becoming president by means other than by re-election, it cannot be said that the Twelfth Amendment prohibits a twice-elected President from serving as Vice President. Then we are in agreement. I don't see this as a matter of choosing sides. I see this as a possibility that is available to Trump at the end of his second term. I chose this side of the debate only as a counterbalance to the commonly stated view that Trump is ineligible for a third term as President. Bear in mind that the article was written in 1999 during the Bill Clinton presidency, so the idea that a twice-elected President can serve a third term as President is hardly new. So why didn't Bill Clinton, George W. Bush, or Barack Obama try for a third term as President? It seems rather obvious to me... they weren't megalomaniacs. And let's face it, although it might be legal for a twice-elected President to serve a third term as President by becoming Vice President, it's hardly a respectable way of becoming President. Instead, it seems rather sleazy. But then, respectability is not really a feature of the Trump presidency. One further point to consider: Is anyone who has just been elected President really going to voluntarily give up that position to someone who has already served two terms as President and is trying to use a loophole to serve a third term?

A transformation of the type: [math]g'_{pq} = \phi\ g_{pq}[/math] where [math]\phi[/math] is a scalar function of the coordinates, is called a "conformal transformation". Conformal transformations vary the scale over spacetime location but do not change angles. In spacetime, angles include the speed of light in a vacuum, which is also invariant under conformal transformations. An important property of conformal transformations is the invariance of the Weyl conformal tensor field of the form [math]{C_{pqr}}^{s}[/math] (other forms of the Weyl conformal tensor field transform according to the metric tensor field used to raise or lower indices). The Weyl conformal tensor field is the part of the Riemann curvature tensor field that describes the tidal effect associated with pure gravitation (external to any distribution of energy-momentum). It should be noted that the connection object field transforms such that the covariant derivative of the metric tensor field remains zero (a non-zero covariant derivative of the metric tensor field transforms covariantly): [math]\nabla'_r g'_{pq} = \phi\ \nabla_r g_{pq} = 0[/math] If a conformal transformation is applied to a metric tensor field of flat spacetime, the resulting spacetime is called "conformally flat". A conformally flat spacetime has zero Weyl conformal tensor field, but non-zero Ricci tensor field in general. An example of a conformally flat spacetime is a flat-space Friedmann-Lemaître-Robertson-Walker (FLRW) metric used in cosmology. By coordinate-transforming time to expand the same as space, the conformal flatness of the FLRW spacetime becomes explicit. In this case, lightlike trajectories become simply described as straight lines, simplifying the calculation of cosmological redshift.

It could actually be just an algebraic problem: Given [math]{R_{pqr}}^{u}[/math] (obtained from [math]{\Gamma_{qr}}^{u}[/math]), find [math]g_{us}[/math] such that: [math]{R_{pqr}}^{u} g_{us} + {R_{pqs}}^{u} g_{ur} = 0[/math] [math]{R_{pqu}}^{u} \ne 0[/math] is an obstruction to this, but I'm not sure if it's the full obstruction. (Note: This is NOT the Ricci tensor field) However, if [math]{R_{pqr}}^{u} = 0[/math], then all [math]g_{us}[/math] satisfies the above equation, but not all [math]g_{us}[/math] satisfies [math]{R_{pqr}}^{u} = 0[/math], so a solution to the above equation is not necessarily a metric corresponding to the given connection object field. Nevertheless, [math]{R_{pqu}}^{u} \ne 0[/math], which is derived entirely from the given connection object field, implies that a metric (with zero covariant derivative) does not exist. The significance of this is that the connection object field emerges to transform a non-tensorial partial differential operator to a tensorial covariant differential operator, but this does not suffice to produce the metric tensor field (with zero covariant derivative, a seemingly important property of the metric tensor field).

I had already posted this link to an article from the Cornerstone Law Firm: https://cornerstonelaw.us/22nd-amendment-doesnt-say-think-says/ I also found this article from the University of Minnesota Law School: Peabody, Bruce G. and Gant, Scott E., "The Twice and Future President: Constitutional Interstices and the Twenty-Second Amendment" (1999). Minnesota Law Review. 909. https://scholarship.law.umn.edu/cgi/viewcontent.cgi?article=1908&context=mlr See page 55 / 73 of the PDF (page 618 of the text).

It's hardly "straightforward". Otherwise, there wouldn't be a legal debate on the issue. Actually, the 22nd and 12th Amendments taken together contain a peculiar circularity, rendering them unable to decide whether Trump can legally become Vice President after the conclusion of his second term. That is, if Trump can legally become Vice President, he can legally become President by becoming Vice President, and therefore he can legally become Vice President; whereas if Trump can't legally become Vice President, he can't legally become President by becoming Vice President, and therefore he can't legally become Vice President.

Even if the counterargument comes from the ultra-far-right, it may still be valid based on the precise wording of the Amendments. As I see it, whether or not unable to be elected means ineligible is something that neither you nor I can say with any confidence when it comes to how the courts will decide. It seems to me that unable to be elected does not mean ineligible, but I'm guessing the argument will come down to what the drafters of the Amendments intended (why they were worded the way they were, etc). Also, given that we are in a political environment in which Trump is now president in spite of what happened on January 6, 2021, I wouldn't hold out too much hope that Trump will be legally denied his third term.

Unfortunately, the 22nd Amendment does not say that Trump will be ineligible to the office of President after the completion of his second term, only that he can't be elected to the office of the President. Thus, the 12th Amendment doesn't seem to apply. One thing that the article: https://cornerstonelaw.us/22nd-amendment-doesnt-say-think-says/ points out is that "if the 22nd Amendment’s purpose was to ensure that there was a 10 year maximum on service for anyone regardless of how they became President, it could have said so".

While the question of uniqueness is a reasonable question, my only interests are whether the solution metric tensor field exists, and if it does not exist, what the obstruction is to its existence. The question is basically asking whether an arbitrary connection object field admits a metric tensor field (with zero covariant derivative). As for whether or not the torsion tensor field is zero, I don't think this actually matters to the question being posed. I believe that [math]\nabla_{\lambda} a_{\mu \nu}[/math] is independent of the torsion tensor field in the sense that the expression for the connection object field in terms of [math]\nabla_{\lambda} a_{\mu \nu}[/math], [math]\partial_{\lambda} a_{\mu \nu}[/math], and [math]a_{\mu \nu}[/math] will contain the torsion tensor field as an arbitrary field (similar to the expression for the connection object field in terms of [math]\partial_{\lambda} g_{\mu \nu}[/math], [math]g_{\mu \nu}[/math], and [math]\nabla_{\lambda} g_{\mu \nu} = 0[/math], where the torsion tensor field is undetermined or asserted to be zero).

One question that interests me, though I have yet to fully explore, is: Given only a nondegenerate symmetric tensor field [math]a_{\mu \nu}[/math] and its covariant derivative [math]\nabla_{\lambda} a_{\mu \nu}[/math], is it possible in general to determine the metric tensor field [math]g_{\mu \nu}[/math] (with zero covariant derivative)?

If one has a given connection object field, then changing the metric tensor field changes the covariant derivative of the metric tensor field. One way to look at this (though I make no claim to its equivalence) is that one can change the metric tensor field in which the covariant differential operator is equal to the partial differential operator. Usually, (for spacetime) this is the Minkowskian metric tensor field, but it need not be. This is a change in definition, but it results in corresponding adjustments that keeps everything correct.

@grzegorzsz830402, what is your take on the delayed choice quantum eraser experiment?

Fair enough. But your proposed idea of applying just lime powder will not be fungicidal. You need the Cu for that which, as I say, is the key ingredient in Bx mixture for its intended application, in vineyards. Oh!! "Lime" as in calcium. I was thinking "lime" as in citrus (usually it's orange blossom in cleaners).

I believe that this violates the principle of general relativity. The principle of general relativity requires that clocks and rulers be allowed to behave naturally as clocks and rulers, whereas you are applying corrections to the clocks and rulers based on gravitation in violation of the principle. That the application of the corrections leads to flat spacetime means that the corrections are destroying information about the spacetime being measured. All the information about the measured spacetime is contained in the applied corrections and not at all in the flat spacetime. So, unless you somehow retain the information contained within the corrections and use that information in the description of the measured spacetime, the flat spacetime will not be a valid description of the measured spacetime. You appear to be constructing the following: [math]g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}[/math] where: [math]g_{\mu\nu}[/math] is the measured metric tensor field using uncorrected clocks and rulers [math]\eta_{\mu\nu}[/math] is the flat spacetime metric tensor field [math]h_{\mu\nu}[/math] is the corrections field One difficulty of the above worth noting is that [math]\eta_{\mu\nu}[/math] and hence [math]h_{\mu\nu}[/math] can be mathematically chosen independently from [math]g_{\mu\nu}[/math], whereas given [math]g_{\mu\nu}[/math], there would seem to be a natural choice of [math]\eta_{\mu\nu}[/math] and hence [math]h_{\mu\nu}[/math]. In obtaining the curvature tensor fields, you would substitute the above expression for [math]g_{\mu\nu}[/math] into the expression of the curvature tensor fields in terms of [math]g_{\mu\nu}[/math] to obtain the expression of the curvature tensor fields in terms of [math]\eta_{\mu\nu}[/math] and [math]h_{\mu\nu}[/math].

Also no. It leverages that fact, but defines the meter in terms of the second and the numerical value of c That's actually an interesting point. Although defining c to have a particular numerical value does seem to be forcing the speed of light in a vacuum to be constant, in fact measuring out a metre of length involves creating the distance travelled by physical light in 1/c of a second. That is, the light will travel at whatever speed it wants to travel and isn't being forced to conform to a constant speed. But the use of physical light to define the metre does rely on the speed of light in a vacuum being able to fulfil the properties of a standard.

This forum is not a chain letter service.

Yes, we do have the right to complain. That is our right to free speech. Trump does have his right to free speech, but he doesn't have the right to break the law.

This fails to distinguish between a change caused by a change of the laws of physics and a change caused by some physical field. The problem with changing the laws of physics is that there needs to be a basis for that change. If the change is physically real, then that implies the existence of some field that gives rise to the measured change. And there also has to be underlying laws of physics that govern the basis for that change. So, the original set of changing laws of physics become replaced by a new set of constant laws of physics. In general relativity, this gives rise to covariance.

How can it not be about legal rights? In any society, it is inevitable that there will be disagreement, and therefore the need for some form of arbitration with the power of enforcement.

No, the religious people do have the right to express that they were offended by the performance, but do not have the right to have the performance banned.