joigus

invisible braces: \[ {\left. \delta^{\mu} \right. }_{\nu}\]

Funny that you think that, as nobody else does. Without going into details, it's quite clear for anybody who's worked with tensor calculus for some time that the most likely thing going on is that you're confusing invariant properties with coordinate-dependent properties, and mixing them all up in one big mess. Some tensor identities can be proved by appealing to some tensor being zero in one particular coordinate system. Then it must be zero in all coordinate systems. Conversely, non-zero in one system <=> Non-zero in all. On the contrary, the Christoffel symbols can always be chosen to be zero in one coordinate system, but non-zero in infinitely many other coordinate systems. Playing with these two properties facilitates some proofs, but you must know what you're doing. Handling index expressions without any care of what is a tensor and what only holds in one particular system is the wrong way to go. It is a real pain to go over every step of a calculation that somebody only too obviously did wrong, because you have proven the theorems forwards and backwards and gone through all the examples. A tensor being diagonal, e.g., is not an invariant property under SO(3) or O(3). On the other hand, tensors like the identity \( {\left. \delta^{\mu} \right. }_{\nu} \) or \( \epsilon_{\alpha\beta\cdots} \) are called isotropic tensors, because they look the same (have the same components) in all coordinates systems. So you cannot safely assume that a diagonal tensor takes part in any tensor equation. "Diagonal matrix" makes sense, meaning "a matrix that looks diagonal in a particular base". "Diagonal tensor" does not.

I would have to be on top of the hill to look down on others here to evaluate them. I'm not in such position. How smug would I be if I did? I stand by my words: A superb explanation --especially considering the limited amount of time and manoeuvre we all have here-- by a person well versed in mathematics who has all my respect. MigL's explanation was also very helpful, although in a very different style and spirit. As to your kind offering of starting another thread, I'm not so interested in judging people as in examining ideas, and trying to understand some of the most difficult ones. But you're free to open that thread if you want. Here's smiling at you

Easy, because as @swansont told you, you're going around in circles. \[ \frac{\sqrt{FE_R} \left( \sqrt[4]{FE_R} \right)^2}{\frac{E_R}{c^2}} = \left( FE_R \right)^{\frac{1}{2}+\frac{2}{4}} \frac{c^2}{E_R} =\] \[ = Fc^2 = \frac{G}{c^2}c^2 = G \] You derive guess an equation from your definition. You substitute your definition, so you get to an identity. Doesn't matter that your definition dimensionally has no relevance. And your \( g_\text{photon} \) has the funny dimensions of (length)3/2(time).1.

\[ \frac{\sqrt{FE_R} \left( \sqrt[4]{FE_R} \right)^2}{\frac{E_R}{c^2}} = \left( FE_R \right)^{\frac{1}{2}+\frac{2}{4}} \frac{1}{c^2} =\] \[ =\frac{FE_R}{\frac{E_R}{c^2}} =Fc^2 = \frac{G}{c^2}c^2 \]

If I may say something... I was aware that @wtf was giving a superb mathematician's exposition of the topic, while @MigL who had had some previous experience with the OP, was quite deliberately trying to dumb it down. It was fun seeing you interact. But your effort is not in vain, wtf. Thank you. I appreciate it.

\( F= \frac{G}{c^2} \) has units of (mass)-1x(length) Energy has units of (mass)x(length)2x(time)-2 So \( \sqrt[4]{F\times E_\text{photon}} \) has dimensions of (length)3/4x(time)-1/2. So that's a non-starter from dimensional analysis alone. Sorry, I honestly thought you were joking in the Physics section. I immediately removed the neg reps.

Happy birthday and many happy returns!

You are serious, then? I couldn't believe you were serious.

As long as it's just a tool. Now that I think of it my metaphor of the root and the seed was not particularly illuminating. LOL Edit: x-posted with iNow.

My French is a bit rusty, but I can tell you as much as this: Fire does not involve the disappearance of matter. The concepts of an infinitely big Mandelbrot set and and infinitely small Mandelbrot set do not make mathematical sense. The Mandelbrot set is invariant under discrete "zoomings". The sentence, "The only way to represent the infinity of a material thing is the circular shape, where the beginning and the end merge" is ambiguous enough so as not to make any mathematical sense.

A part of me wants to believe that religious types who drop by have a part in them who is desperate to be won over by a set of more solid arguments... or perhaps more positive, constructive doubt. It's a bit disappointing, rather than offending, when they turn to calling you names. I notice that frequently people with strong opinions rarely drop them in front of you. We all are hardwired not to lose face. That's probably because our competitive primate nature has grown ramifications into language itself. Ideas are more like the proverbial seed that our present interlocutor has mentioned, rather than roots that try to break through the ground. They normally sprout when you're not looking.

Oh, we know this is trash-can material from day one, don't we? Like most other users I just wanted to do my part in bringing it out. True colours showing. We've got the full spectrum, from kefir to non-clay bricks, plus tidbits of old-time religion. All topped with insult, instead of arguments. I think I'm done.

You see genius in a seed because there are billions of years of incremental improvement in this marvel of coordinated chemical actions. Mountains of evidence is precisely what's helped us understand what a seed is. When it develops, it recapitulates the history of the Earth, so in a way, a seed has chapters of the history of the Earth written in it. It took centuries of human curiosity to end up in Darwin's great insight to explain that "genius of a seed" that you extol without understanding. There are hundreds of billions of planets in the universe where nothing like the genius of a seed has come to fruition, for reasons easy to understand, not because a petty god (obsessed with being worshipped by small vulnerable things above anything else) decided those planets weren't worthy of his handiwork. You are blind indeed. The worst kind of blindness is lack of will to see. It's not that you don't know. It's that you don't want to know.

The word "point" in itself does not tell you what it is. A point on the real line: \( x \in \mathbb{R} \) A point on the real plane \( \left(x,y\right) \in \mathbb{R}^2 \) ... etc. Edited: A point is a locus, location, place in a set. When you say "point" normally you imply some kind of position (distance-->geometry, topology...). When you say "element" or "member" you normally imply just set theory. There's context missing. And as Ahmet suggested, "light cones", "faster than light", "hyperplanes"... That has nothing to do with your drawing or the concept of points. The impression I get is, again, you're trying to connect too much in one simple concept. Points don't need light in order to be defined.

(My emphasis.) You've given no answer to any of my points. I've provided you with references and reasons why many of the things you hold as true about the past simply cannot be correct. Then you engage in an argument about bricks by using 16/17th-century language. The fact that you desperately try to attack the man, "you're a cynic", while fleeing from the argument tells me I must be doing something right. People always do that when they're logically cornered. "Debating is always cynical" is the bit that I've decided to leave uncommented because it needs no further comments from me. I don't know what to say. You might as well say "reason is always cynical". Maybe you simply don't know what "cynical" means.

A message from the Bronze Age from an invisible being, compiled by people from the Iron Age, written in English from the 16th century. Not very illuminating to me, I'm sorry. If I want to be understood, I use 21st-century English. That's why safety warnings, for example, are not written in 16th-century language: Being understood could be a matter of life and death in that case.

I stand by every word here. There are many sources of possible mistakes. A very common one is forgetting that \( g^{\mu \nu} \) is the inverse of \( g_{\mu \nu} \). It is always the best idea to go over the calculation again, instead of believing you've found a shortcut to the Nobel Prize. It's a natural rite of passage. You do it in the abstract, with indices; you do it with polar coordinates on the plane, you do it with hyperbolic polar coordinates. You convince yourself that it's correct. You turn on and off the contravariant to covariant "switch". It still works... Oh my. It must be true. That's the path.

Muons orbiting nuclei? Lifetime of a muon is 2.2 ms. That's a mighty ephemeral magma.

Physical four velocities are not made of arbitrary numbers. As Markus said, in just a bit more explicit notation and rephrasing what he said, although I think it was clear enough, \[ \left( v_t, v_x, v_y, v_z \right ) \] but constrained to, \[ \left( \frac{c}{\sqrt{1-v^2/c^2}}, \frac{v_x}{\sqrt{1-v^2/c^2}}, \frac{v_y}{\sqrt{1-v^2/c^2}}, \frac{v_z}{\sqrt{1-v^2/c^2}} \right) \] Same with 4-momenta. 4-momentum is the product of \( u^{\mu} \) --the 4-velocity-- times the mass. So it doesn't do to fill arbitrary numbers in the slots, so to speak.

\[ \left( \frac{1}{\sqrt{1-v^2/c^2}}, \frac{v_x}{\sqrt{1-v^2/c^2}}, \frac{v_y}{\sqrt{1-v^2/c^2}}, \frac{v_z}{\sqrt{1-v^2/c^2}} \right) \]

"Par excellence" after a noun meaning "a very good example of something": https://www.oxfordlearnersdictionaries.com/definition/english/par-excellence_1?q=par+excellence No excellence implied in the literal sense.

I haven't had time to review in detail. The numeric relation seems correct to me. As to the claimed contradiction, I haven't got around to it yet. Contradiction with what exactly? Don't have time to read Markus' comments. Maybe later.

There is no conflict in the operation of taking covariant derivatives of any tensor, at any order of derivation. Again, can you explain here, instead of linking to a document, please?

OK, the fact that you mention Adam and his immediate offspring as factual is enough for me to know this discussion is not leading anywhere useful. Some biblical myths are inclusions from Babylon. Ezra re-edited the Torah, because it had been lost after Nebuchadnezzar II destroyed the Second Temple. The myth of the flood from the Epic of Gilgamesh is very recognizable. Another one is the story of a man whose wife cannot conceive, so that they arrange that it is the slave who is going to play that role --Abraham--. The latter story is foreshadowed in the Nuzi tablets over and over, and over again. Also in Mari --Mesopotamia. It is more than likely that they picked it up by the rivers of Babylon, because they came to know it was a common Babylonian story. Also you say all deities in the Bible stand for God. We already know this cannot be true -beyond any reasonable doubt. There are also inscriptions speaking of Yahweh and his Ashera (his wife). The existence of a pantheon of gods is very clear in the archaeological record. Baal is not, as you seem to suggest, another representation of Yahweh, but the bull god that appears in many places of the Middle East and features prominently in Exodus, different in name and in the statuettes --in the human form, El, sitting and serene; while Baal, aggressive, in smiting position, and using his strength. It's Baal-Zebub, the Lord of the Flies, that in Christian iconography became to be known as Satan. Yahweh, in the Sinaitic depictions, looks nothing like El. Why would he? They are different gods from different regions. Are you going to believe what a book which was copied again and again, recompiled hundreds of years later after its partial destruction, probably recited at some points; or are you going to believe the fragments of script that are dug from the ground and tell us what the Canaanites of that time probably believed? Faith-based religion is not like a message passed down in its pristine form generation after generation; it is more like a game of Chinese whispers played throughout the centuries in which you never know what the message is going to become. That much we already know beyond any reasonable doubt. Understanding the process, rather than the details, makes it very easy to see how you can throw in a new element and make it part of the broth, keeping some words but changing the meaning, etc. Like your kefir. It is the lack of logical strictures which allows to do that. This appears not to be true: https://weareisrael.org/spiritual-seed-2/male-child/betulah-vs-almah/ But, when Eliezer recalls his story to Rebekah’s father (Bethuel), he calls Rebekah a young woman (עלמה, al-mah’), a sexually mature woman at the prime age for work, because he was not privy to her actual sexual status. Helenization of Roman Jews started in the 4th century BCE, but found strong resistance that culminated in the Maccabean revolt during the Seleucid rule. So they were not completely Helenized, especially considering the Maccabeans were successful, unlike the rebels of Masada. It was after the diaspora that most transcriptions of Torah appear only in Greek, at least in Europe. But the sect of Qumran still copied the Bible in Hebrew, and I'm sure the rebels of Masada also did so, at the time of the Jewish revolt. Some of the parchments have been found to appear to have been dropped on the ground of the caves in Qumran as the legions came to arrest them. Those texts range from old copies to contemporary copies --at the time. They are all in Hebrew.