KJW

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.

Or as Rowan Atkinson has said: The freedom to be inoffensive is no freedom at all.

I don't agree with this because one also has to consider that the laws of physics determine the intrinsic size of the units that we define. The constancy of the fundamental constants is implied by the constancy of the laws of physics. For example, suppose you measure the length of some object. You use a steel ruler to measure the object in metres. But you could count the number of iron atoms along the edge of the ruler. Thus, instead of measuring the length of the object in metres, you have measured the object in terms of iron-to-iron interatomic distances. Any change in the laws of physics that alters the iron-to-iron interatomic distance would also alter the length of the object by the same amount, and therefore the length of the object in terms of iron-to-iron interatomic distances would be unchanged. But this invariance implies that the changes of the laws of physics cannot be measured, which justifies the assumption of the constancy of the laws of physics. When you express the fundamental constants in terms of their dimensions, the result is a system of equations. When this system of equations is inverted, you obtain a definition of the Planck units in terms of the fundamental constants. In principle, you could measure everything in terms of the Planck units. The laws of physics govern the intrinsic size of the Planck units, but you can't actually measure the Planck units because everything is measured relative to the Planck units, and therefore the laws of physics cannot be anything but constant. Also, because the Planck units are expressed in terms of the fundamental constants, the fundamental constants cannot be anything but constant. As I see it, the use of natural units is about making all the ostensibly different dimensions of measurement the same. So, whereas time and length appear to be different, multiplying time by c rescales time so that it is the same as length. And when this is done, c becomes 1 and dimensionless (but only because time and length now have the same dimensions).

@Killtech, I think you are on the verge of realising that c (and the other fundamental constants) must be constant because when we measure something, it is relative to the definition of the units that have been used, and therefore, in order to obtain a definite value for a measurement, the units of measurement have to be assumed to be intrinsically constant.

When I first saw this a few days ago, I thought it might be a joke. It reminds me of the Scunthorpe problem.

The deflection of light by the Sun is twice that predicted by Newtonian gravity. Half of this value (equal to the value predicted by Newtonian gravity) satisfies the equivalence principle and is the result of the gravitational time dilation. The remaining half is due to the curvature of the three-dimensional space. I believe that these two halves are equal because the spacetime trajectory is lightlike.

Perhaps it would help if you read the Wikipedia article on the history of atomic theory: https://en.wikipedia.org/wiki/History_of_atomic_theory Knowing the history of atomic theory provides an understanding of how the knowledge of atoms developed over time.

Another example is a chemical synthesis, the product of a particular chemical reaction on a starting material whose structure is known by whatever means. It may be that the substance produced has never been produced before. In this case, there is no known sample with which to compare our substance produced. But the substance produced is not entirely unknown, either. It is likely to be the substance that was intended to be produced on the basis of what is known about the chemical reaction. And if it is not the substance that was intended to be produced, then it is likely to be in some way related to the starting material or the substance that was intended to be produced. In either case, it becomes much easier to analyse the spectra of the substance than if the substance is truly unknown. Proton nuclear magnetic resonance spectroscopy is especially useful in this regard.