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March 23[edit]

Why do Blattella germanica cerci point up then out?[edit]

Not adult: they're in a vertical plane, adult: in horizontal plane. Sagittarian Milky Way (talk) 05:43, 23 March 2024 (UTC) Okay that's not very accurate but it sure does look that way from above. They have different orientations before and after sexual maturity. Sagittarian Milky Way (talk) 06:16, 23 March 2024 (UTC)[reply]

Courtesy links: Blattella germanica, cerci.-Gadfium (talk) 05:46, 23 March 2024 (UTC)[reply]

Hours of daylight vs latitude vs day of year[edit]

@SebastianHelm: modified a vectorisation I made of File:Hours_of_daylight_vs_latitude_vs_day_of_year.png. Yesterday, @Episcophagus: posted some questions which I couldn't answer:

Why are the borders to constant night and constant day labelled with ‘1 hour’ and ‘23 hours’, respectively?
Why isn't ‘equal day length’-latitude south of the equator, as the rate of change in the equation of time affects the daylength (the steepness of the ecliptic at the equinoxes make the sun to move slower in rectascension and thus the daylength is shorter than at the solstices)?

Can anyone help them, please? Thanks, cmɢʟee⎆τaʟκ 12:11, 23 March 2024 (UTC)[reply]

I don't know but it increases ~~asymptotically~~ "exponentially" cause geometry, the 1 and 0 hour lines would be very close at this scale. I don't know if it's an exponential function but the layman meaning of exponential. Half of 12 hours is most of the way to the polar circle 1 hour would be over 2 more halvings, at the polar circle refraction and Sun radius add at least 50 nautical miles of midnight Sun above simple geometric anyway. Very close it could rise after set while still going down or vice versa (at extremely glancing angle) just from refraction fluctuations. And with a Titanic-like extremely calm sea horizon you could float in the water and climb 5 feet when it set and it rises, float in the water again and it set repeat (eyeballs 5 feet above ocean lowers the horizon about 0.1X Sun radii with ever diminishing returns with height. 10 times more horizon dip than 5 feet takes hundreds of feet, 1,000 times more dip takes millions of feet and 2,000 times more dip takes over infinity light years cause the horizon can't dip more than 90 degrees). At medium latitude this period only lasts seconds.
At the equator the Sonnar goes up about 3 and one third minutes before the center would be risen without refraction. Daytime would be about 12 hours 6 minutes and 40 seconds at the equator if there was no equation of time.

Sagittarian Milky Way (talk) 15:57, 23 March 2024 (UTC)[reply]

Thanks, @Sagittarian Milky Way: cmɢʟee⎆τaʟκ 14:29, 24 March 2024 (UTC)[reply]

British physician who debunked Lourdes miracles[edit]

Hi, I've read in a book by Piero Angela (Italian science writer) an interview with William A. Nolen who said a British physician once wrote a book debunking alleged Lourdes miracles. Does anyone know who he was? Thanks.-- Carnby (talk) 12:39, 23 March 2024 (UTC)[reply]

Probably Donald J. West in Eleven Lourdes Miracles (1957). See this URL Mike Turnbull (talk) 12:47, 23 March 2024 (UTC)[reply]

Thank you!-- Carnby (talk) 12:57, 23 March 2024 (UTC)[reply]

The definition of relativistic momentum[edit]

As opposed to Newtonian Mechanics, which defines momentum as $P=mv,$ Relativity theory defines momentum as $P=\gamma mv,$ i.e. as $P=m(dx/d\tau ),$ but I wonder why $d\tau ,$ rather than $dt$ - thus sticking to the original Newtonian definition of momentum...

If the reason for preferring the "rest" time $\tau$ is because also the mass $m$ being referred to is the "rest" mass, then also the velocity referred to should be the "rest" velocity, i.e. $v=0,$ shouldn't it?

On the other hand, since the velocity being referred is not the rest velocity $v=0$ but rather the real veloctity, so isn't it usually defined elsewhere as $dx/dt,$ rather than as $dx/d\tau ?$

On the third hand, If the reason for defining the relativistic momentum as $P=m(dx/d\tau )$ is our desire to let Newton's second law $F=dp/dt$ remain invariant under the Lorentz transformations, then why do we replace the expected definition $P=m(dx/dt),$ by the less intuitive definition $P=m(dx/d\tau ),$ rather than by any other definition (e.g. $P=2m(dx/d\tau )$ or whatever), that lets Newton's second law remain invariant under the Lorentz transformations? 2A06:C701:7478:1B00:1F2:1345:432B:F844 (talk) 22:21, 23 March 2024 (UTC)[reply]

τ represents proper time, as distinct from coordinate time represented by t. See those two articles for more details, although to be honest, neither article does a great job of explaining the difference. CodeTalker (talk) 01:26, 24 March 2024 (UTC)[reply]

Yes, I know that, but how does that answer my question? isn't the proper velocity usually defined elsewhere as

dx/dt,

rather than as

dx/d\tau ?

2A06:C701:7478:1B00:1F2:1345:432B:F844 (talk) 01:29, 24 March 2024 (UTC)[reply]

Four-momentum

{\vec {p}}=m{\vec {U}}

and four-velocity

{\vec {U}}={\tfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} \tau }}

have to be covariant, i.e. frame-independent, vectors; this is only possible by taking the derivative with respect to a frame-independent quantity,

\tau

, but not with respect to a frame-dependent coordinate time

t

. Proper time

\tau

parameterises the space-time trajectory

{\vec {x}}(\tau )

of the particle in question, and

{\vec {U}}(\tau )

is the tangent vector to that curve. In the non-relativistic limit these definitions reduce to the standard Newtonian momentum and velocity, so there is no contradiction. --Wrongfilter (talk) 09:09, 24 March 2024 (UTC)[reply]

If the only reason for defining, the four-momentum as

{\vec {p}}=m{\vec {U}},

and the four-velocity as

{\vec {U}}={\tfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} \tau }},

is our desire to let the four-momentum and the four-velocity be covariant, then why do we replace the apparently-more-intuitive definition

P=m(dx/dt),

by the less intuitive definition

P=m{\tfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} \tau }},

rather than by any other less intuitive definition (e.g.

P=2m{\tfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} \tau }}

or whatever), that lets the four-momentum and the four-velocity be covariant? I'm pretty sure there are infinitely many options for the four-momentum and four-velocity to be covariant, aren't there? 2A06:C701:7478:1B00:1F2:1345:432B:F844 (talk) 12:59, 24 March 2024 (UTC)[reply]

Most people would want the relativistic momentum to coincide with the Newtonian momentum in the non-relativistic limit. That puts a lot of restrictions on the relativistic form (not sure whether that makes it unique). Your factor 2, for instance, just wouldn't go away. --Wrongfilter (talk) 13:14, 24 March 2024 (UTC)[reply]

Well, your remark in the parentheses is the most important one so far, in my eyes. I'm really curious to know (even though you are not sure), if the common relativistic form of momentum is unique, for it to be frame independent and to coincide with the Newtonian momentum in the non-relativistic limit. If the relativistic form of momentum is really unique, then there must be a rigorous proof that derives this relativistic form somehow, so this proof is probably mentioned in textbooks, but it's not, AFAIK. On the other hand, if the relativistic form of momentum is not unique, then I wonder what other forms the relativistic momentum could have, while still being frame independent and coinciding with the Newtonian momentum in the non-relativistic limit. 2A06:C701:7478:1B00:1F2:1345:432B:F844 (talk) 13:39, 24 March 2024 (UTC)[reply]

Bold 2601:2C6:4700:1530:457A:81D9:783:FC83 (talk) 20:21, 2 April 2024 (UTC)[reply]

This is the unique definition that (1) reduces to the Newtonian momentum for small speeds, and (2) makes conservation of momentum frame-independent. See [1]. --Amble (talk) 20:59, 26 March 2024 (UTC)[reply]

Thank you. However, I noticed that (3.6) and (3.7) in the chapter, are definitions rather than conclusions.

Anyway, also the continuity of the function should be assumed, unless it's considered to be included in your first assumption (1). 2A06:C701:7478:1B00:1F2:1345:432B:F844 (talk) 13:30, 27 March 2024 (UTC)[reply]

Continuity follows from (1) and the Lorentz transformations, since a discontinuity anywhere will be a discontinuity at v=0 in some frame of reference. Eq. 3.6 is a definition, but it just attaches a convenient label to a particular quantity. Equation 3.7 defines relativistic momentum and shows that this definition satisfies the relevant criteria, while p=m(dx/dt) does not -- which I believe was your original question. Quoting from [2]: "Indeed, the underlying philosophy is that energy and momentum are nothing else than functions of mass and velocity that, under suitable conditions, happen to be conserved. This is why we treat in a special way those functions, rather than others." --Amble (talk) 04:53, 28 March 2024 (UTC)[reply]