Wikipedia:Reference desk/Archives/Science/2018 March 1

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March 1

How "fine tuned" is the exponent of E=mc²?

No one knows if there's other Big Bangs with other exponents like E=mc^1.9 but if there are what's the anthropic principle range? Sagittarian Milky Way (talk) 04:07, 1 March 2018 (UTC)

The c² simply makes the units work out — see dimensional analysis. In natural units, the equation is just E=m.

Personally I can't make any sense of the idea of a possible world where E=mc^1.9, but maybe someone else can. --Trovatore (talk) 04:27, 1 March 2018 (UTC)

• It came as a shock to me when I finally realized (after the age of 60) that for equations describing the real world, dimensional analysis pretty much restricts exponents to being integers, except for exponents of dimensionless numbers. Furthermore, the exponent is also dimensionless. -Arch dude (talk) 06:34, 1 March 2018 (UTC)

  Per archdude, the exponent just means E = m * c * c , it's not a magical constant or anything, it's just a function of basic mathematical notation. --Jayron32 12:04, 1 March 2018 (UTC)

Exactly as Jayron wrote its all just simple math. Its not "fine tuned" but absolute and exact by definition because every part of the equation is exactly defined. Energy mass and speed are the same in your home, somewhere a million lightyears away in space and even close to a black hole tho our understanding of physics reportedly gets odd the closer you go in. Luckily you stayed away from that with your question. --Kharon (talk) 19:01, 1 March 2018 (UTC)

• Not quite. The speed of sound is square root of (gamma*R*T) in the correct units. I am sure there are other examples.Greglocock (talk) 23:15, 1 March 2018 (UTC)

$c_{\mathrm {ideal} }={\sqrt {\gamma \cdot R_{*}\cdot T}}$ at Speed of sound#Speed of sound in ideal gases and air is only possible because the multiplication under the square root produces units which can be expressed as squares, e.g. giving ${\sqrt {\tfrac {m^{2}}{s^{2}}}}$ = m/s. We could also write the relationship as $c_{\mathrm {ideal} }^{2}=\gamma \cdot R_{*}\cdot T$, just like E = mc² is equivalent to $c={\sqrt {\tfrac {E}{m}}}$. In practice we just tend to write formulas so the value we are most likely to want to compute is isolated. PrimeHunter (talk) 02:15, 2 March 2018 (UTC)

Agreed. In which case the statement at the top of this sub-thread is meaningless. If you exponentiate (?) both sides of the equation enough times you can get rid of square roots, always, and if you take the square root of both sides you can always get fractional powers. Greglocock (talk) 04:09, 2 March 2018 (UTC)

I'm confused by the claim that the exponent has to be exactly 2, at least as a question of physics. E=mc² comes from a certain mathematical model where the exponent is exactly 2, and the model apparently closely matches experimental observation. But maybe the model is only a close approximation. I don't see an inherent reason why the exponent couldn't be 1.9998 or whatever. Is it somehow different from the exponent 2 in the inverse-square law of gravity, which could conceivably be approximate?[1] I also remember something about Gauss trying to confirm the gravitational inverse-square law using astronomical observations.

Regarding Einstein's discovery of the exponent 2 in his famous equation: see here. 00:32, 2 March 2018 (UTC) — Preceding unsigned comment added by 173.228.123.121 (talk)

What exactly would it mean for the exponent to be 1.998? In (say) SI units, you would be saying that the energy in kg*m^2/s^2 would be equal to the mass in kg times c to the power 1.998, which gives you units on the right-hand side of kg*m^1.998/s^1.998. What on Earth is that supposed to even mean? --Trovatore (talk) 00:40, 2 March 2018 (UTC)

We might say that energy more fundamentally is a conserved quantity with certain physical characteristics, from which experimental observation leads to that formula you gave. But the formula itself could also be an approximation rather than an exact specification of what energy really is (i.e. the quantity in the formula might turn out to not have the exact properties we want energy to have in the physical universe). "Spacetime swimming"[2] is an example of how energy and momentum conservation are more subtle than they seem.

Semi-related: we're taught in school that F=ma, alternatively written F=m dv/dt, implicitly assuming m is constant. But I've heard that Newton's Principia actually says (translated to modern notation) something likeF=d(mv)/dt, i.e. Newton left open the possibility that m varies with velocity, as later predicted by relativity. I haven't checked out the claim though. 173.228.123.121 (talk) 01:22, 2 March 2018 (UTC)

You haven't responded to the request for clarification. What would it even mean for the exponent to be something other than 2? On the face of it, it's like you're asking whether the mass of the Sun could be Omaha, Nebraska. --Trovatore (talk) 01:27, 2 March 2018 (UTC)

Trovatore, maybe there's a philosophical disconnect. I'm positing that the physical universe has various symmetries, which by Noether's theorem lead to corresponding conservation laws. Those laws will hold even if there are no humans in the universe trying to guess at what they are. One of those symmetries might pertain to a quantity I'll call "X", which is conserved in physical systems like planetary orbits or elastic collisions. Now humans come along and make observations, and eventually develop a theory (i.e. a mathematical model) in which planetary motion conserves something they call "E" (for energy). That theorized E is very close to but not exactly equal to the quantity X that's conserved in the actual physical universe.

So what is the intended referent of the word "energy"? If energy is conceptually the thing specified in the human formula, then energy is by definition exactly E. But if it's conceptually the physically conserved quantity such that yada yada, we should say that energy is actually X, where E is just an approximation to X because the human mathematical model is slightly off: to be perfectly accurate the formulas we have for energy might have to be tweaked by a small correction factor, like changing 2 to 1.999998 and maybe a few other things.

I may be missing something but I don't see an obvious argument that the exponent should be have to be 2. Sure, the 2 is necessary to make the (human-theorized) units work, but the same can be said about the gravitation law $F=G{\frac {m_{1}m_{2}}{r^{2}}}\$. Yet we already know (see citation above) that the 2 in the gravitation law has not always been considered exact. Am I missing something here, that gives the 2 in the energy equation a more fundamental character than the 2 in the gravitation law? Maybe I am, but it doesn't seem obvious. I don't see a way to write the gravitation law with exponent 1.999998 while still having the units work, but physicists apparently still aren't certain that the exponent is exactly 2. So I'm left open to the possibility that we live in a universe in which the units don't quite work. Assuming that they work and concluding that the exponent is 2 is circular reasoning. 173.228.123.121 (talk) 02:52, 2 March 2018 (UTC)

That's completely different. Your r is a variable here. It's at least logically possible that you could have a force depending on r^2.02; you would fix up the units in the proportionality constant G.

The c in E=mc² is not a variable. It's a constant. There's no other constant around to fix up the units with. --Trovatore (talk) 03:11, 2 March 2018 (UTC)

Couldn't there be a multiverse with a universe where c is 101% this universe's c? All other things being the same antimatter should make a bigger boom there even though c is still 1 in real units. Sagittarian Milky Way (talk) 03:38, 2 March 2018 (UTC)

Maybe there is another universe that only has 1 dimension so there is no time and everything is stuck in a line with no chance to ever move because there is no time for that. You ask like if it is really a line or maybe a giant circle but there are no circles in 1 dimension because circles need 2 dimensions. So my offer is you draw me a 1 dimensional circle then i will show you that c is always and forever exactly c. Pinky Promise! --Kharon (talk) 04:11, 2 March 2018 (UTC)

A 1-D circle is a degenerate circle aka a point. Sagittarian Milky Way (talk) 04:31, 2 March 2018 (UTC)

Does not make sense to call a point a circle, nomatter how many dimensions. I could aswell call a question an answer and tag this section as solved. --Kharon (talk) 05:01, 2 March 2018 (UTC)

This is off-topic, of course, but actually a one-dimensional circle is an ordinary circle. Moving along a circle, you can go only forwards or backwards, same as on a line. A two-dimensional circle is a sphere, on which you can go east-west or north-south, same as on a plane. --Trovatore (talk) 20:23, 2 March 2018 (UTC)

So line and circle are just the next lower dimensional analogues of topological spheres and mugs? Sagittarian Milky Way (talk) 21:13, 2 March 2018 (UTC)

I'm not totally sure what you're asking. The circle is the one-dimensional analogue of both the sphere and the torus ("coffee mug"); you'll see it notated S¹ or T¹, depending on what the author wants to emphasize. --Trovatore (talk) 21:28, 2 March 2018 (UTC)

Oh I see that "number of holes" is far less fundamental than the S or T categories and is called genus (mathematics). Sagittarian Milky Way (talk) 01:21, 3 March 2018 (UTC)

(ec) Oh ok, I see what you mean: in relativity theory c is constant and m as the mass of a multi-particle system is additive between particles, etc. So sure, in the math of that theory, the exponent must be 2. Physically though, maybe we have something like E=mc^f(m) where f(m) is extremely close to 2 in the regimes we've been able to experimentally observe, but there's still some potential discrepancy. And relativity itself is the subject of many experimental verifications, some of whose purpose is to validate the hypothesis that c is constant, up to some amount of precision. I don't claim to be smart about this stuff, but I don't see a way to be sure that we'll eventually stop being able to find more wiggle room at the "non-crazy physics" level (i.e. the level where we are now, where maybe we're open to a r^2.02 gravitational force but can experimentally rule out r^2.05). At the purely mathematical level, of course, eliminating the wiggle room is impossible even in principle. 173.228.123.121 (talk) 04:01, 2 March 2018 (UTC)

Since in the right units the equation is just E=m, as already pointed out, what you're essentially asking is if it's possible that E is not exactly m. Or in other words, does E=km (or E=kmc²) where k is a proportionality constant that is not quite 1. That is, can the inertial and/or gravitational mass of a closed and isolated system change due to a mass-energy conversion? Our article on mass-energy equivalence describes a variety of experiments done to test this theory, and the best data cited on Wikipedia would bound k at 1±0.00004. Someguy1221 (talk) 04:14, 2 March 2018 (UTC)

(ec) No, you're still not following. It's not about c being constant or not in theory. People have proposed theories with a variable speed of light, though it's tricky to say what you actually mean (in SI, for example, the meter is defined in terms of the second and c, so it's actually logically impossible for c to be anything other than what it is).

That's not the point. The point is that c is a constant in the equation in question. The equation does not express E as a function of m and c. It expresses E as a function of m, period. It can be right or it can be wrong, but it can't be wrong in terms of the exponent of c. The c² is just a conversion constant, not a variable. --Trovatore (talk) 04:17, 2 March 2018 (UTC)

Trov, I interpreted the OP question as being about the natural universe, while the SI is just a way to organize the numbers in a particular mathematical model that aims to describe the universe but might not do so exactly. It could be that there's no way to make the math exactly describe the universe while still keeping the SI intact. So "changing the exponent is inconsistent with the SI" might only mean that the SI is unphysical. Sure, c is constant in the equation, but the question was about the universe and not about the equation. 173.228.123.121 (talk) 05:38, 2 March 2018 (UTC)

Well, there's certainly plenty we don't know about the universe, or that we think we know but we could be wrong. However, I don't see how you can express any of that by asking about how precisely we know the exponent in E=mc². If it's anything other than exactly 2, then the whole underlying theory makes no sense at all and you have to start over from scratch. It's not at all like asking about empirically measured quantities like the fine structure constant. --Trovatore (talk) 06:10, 2 March 2018 (UTC)

You know what, I expressed myself too softly in the struck-out part above. It simply doesn't make any sense to talk about the exponent being anything other than exactly 2. Forget changing the underlying theory. An exponent other than 2 simply does not state any claim. If the units are wrong, then it's like an ungrammatical utterance; it doesn't have any meaning whatsoever. --Trovatore (talk) 06:14, 2 March 2018 (UTC)

Well, I guess we're in dead horse territory and I'm still not totally convinced, but maybe I'm just clueless. Thanks and I'll stop posting about this. 173.228.123.121 (talk) 09:24, 2 March 2018 (UTC)

Here is a brief version: E is energy but mc^1.9 is not a measure of energy (or of anything meaningful) so it doesn't make sense to discuss whether it's equal to E, or could be equal in another universe.

It would be like discussing whether 5 kg can be equal to 3 s. A measure of mass cannot be compared to a measure of time. A formula like E = 0.9mc² could be discussed since 0.9mc² is a measure of energy, just 10% less energy than Einstein says. PrimeHunter (talk) 14:38, 2 March 2018 (UTC)

• I think that, while all of the responses here are correct, most of you are getting lost in the weeds in trying to explain the OP's misconceptions here, which are largely due to a misunderstanding of mathematics rather than of physics. You're all providing physics explanations for what is a fundementally mathematical mistake. Let me try this again: As with all equations, there are two different bits: there are values and there are mathematical operations. A value is something like energy or speed or mass. Some of these values may be constants, and some may be variables, but that's not really relevent to our discussion; the speed of light is a constant, but for the purpose of this discussion, it needn't be. All that matters is the equation has three actual values: energy, mass, and speed. What we do with those values is set up a relationship using mathematical operations, and the relationshop is this: energy is equal to the mass multplied by the speed of light multiplied by the speed of light again. For notation purposes, when we multiply something two times we right a little "2" as a superscript next to that value in the equation. But, and I can't stress this enough which is why I'm going to put it in bold, that little 2 is not a value, it's a mathematical operation that means multiply by this value twice. It's completely nonsensical to change the value to anything except a whole number here (there are times when non-integer exponents are meaningful and useful; this is not one of them). If we change the number to anything except 2, that's a completely different equation. If there is "fine tuning" to be done, it's with the "c" value. Changing that little "2" is not even wrong. --Jayron32 15:28, 2 March 2018 (UTC)

The problem is I never decomposed it all the way down to (kg·(m/(s·s)))·m = kg·m/s·m/s or realized it's basically a tautology before Trovatore mentioned dimensional analysis. I just thought of it as the joules to kg conversation rate of this universe (or ergs to gram). Maybe it's an integer in all universe(s) cause reality seems to like elegance but the most fundamental deep reasons for everything are unknown so fractional or irrational isn't ruled out right? But it is. Sagittarian Milky Way (talk) 21:13, 2 March 2018 (UTC)

Well it could change but so much of physics would have to change it would be a total and complete overhaul. The question is a bit like asking if speed went up as distance divided by time to the power of 0.9. I believe it would require factional dimensions so we lived in something like a 3.1 dimensional world for instance and what that would be like I haven't the foggiest idea. Dmcq (talk) 18:57, 2 March 2018 (UTC)

One issue brought up in the thread is that dimensions don't have to use integer powers of the units in the measuring system. E.g. if the inverse-square gravitational force turns out to really have exponent 2.02 rather than 2, then (per Trovatore) we'd have to redefine the gravitational constant G and it would fractional units. Or in the era before SI, the electrostatic force was F=q₁q₂/r² where the q's are charges (note the absence of a multiplicative constant like ¹/_4πε₀). Knowing that force=mass*distance/time², that meant the units of charge were mass^1/2*distance^3/2*time⁻¹. But the OP has now clarified they meant the question in a simpler way, so everything's cool. 173.228.123.121 (talk) 01:39, 3 March 2018 (UTC)

I have to agree. We know well that fractal systems exist in mathematics. It is entirely conceivable that 3d space could be embedded in a fourth dimension in some complicated way that makes it not precisely three dimensions any more, in some sense. Whether that can be made to reconcile with the physics and observations we know is certainly above my pay grade, but I should note that there are a lot of M-theory type things with lots of "coiled up extra dimensions" that have little impact at large distances (like an angstrom) - I'm not sure if that is relatable to this or not. Wnt (talk) 16:01, 7 March 2018 (UTC)

Cold air, colder water

Apologies if this has been asked before - I skimmed the archive but couldn't find it. Imagine it's a warmish summer day of about 20°C. You jump into a swimming pool with the water at about 27°C, and the first thing you do is feel cold. Why? Is it because your body warms a thin layer of air around itself like a blanket, which is stripped away when you enter the water, or is it simply that water (or liquid in general perhaps) is better able to take heat from a warm solid than air (or gas in general)? Thanks in advance, Grutness...wha? 13:51, 1 March 2018 (UTC)

Because water is about 3 orders of magnitude more dense than air, so it is better at conducting heat away from your body. It isn't just the difference in temperature, it is how fast the energy is transferred from one material to another that matters here. In order to lower the temperature of your body, molecules of something else have to bump into your body and have some of that energy transferred to them. There's about 1000 times more collisions with the water molecules than the air molecules. Of course, when you do finally equilbrate with the surroundings (well, when your corpse does) the final temperature of your body in 20 degree air is colder than in 27 degree water. However, your body will lose heat faster in the water, resulting in you feeling colder. Also, water has a very high heat capacity, meaning that it is very efficient at soaking up heat energy. Qualitatively, this is all pretty intuitive if you understand what heat energy is (it is just the kinetic energy of individual molecules), quantitatively this is very complex, however, and requires some understanding of Heat transfer, which requires far more calculus than I'm prepared to go into. --Jayron32 13:58, 1 March 2018 (UTC)

Some of each: which is why both wet-suits and dry-suits are used, for example. All the best: Rich Farmbrough, 14:02, 1 March 2018 (UTC).

Another data point here and a couple of references to back this up: water can feel cold, but obviously not always, for my family prefers swimming in the mid-Alantic ocean surf in the late summer and early fall when the water is warmer than the air, for after the initial sensation of coldness when entering the water by diving in our bodies quickly acclimate to the water to the point that it is like taking a warm bath because the water is warm (lukewarm starts at about 27°C and can feel cold and the water temps we swim in are between 27°C and 28°C) and we towel off in a hurry because the evaporation makes us colder anytime the air is cooler than room temp. That the water feels between cold or lukewarm but can feel warmer over time (if it's not too cold to start with) is because the body adapts to the different temperatures, see table 3 of this extensive reference. For instance, at around 28°Celsius the skin vasodilation increases. On page 17 it states "When water temperatures rise between 30°C to 34°C, there is intense vasodilatation[sic], strong blood flow, an increase in the peripheral and core temperatures, and a decrease in [athletic] performance. These temperatures are appropriate foe[sic] exercise activities for infants and children, as well as for the elderly (65, 55). When temperatures drop between 25°C to 20°C, there is pronounced vasoconstriction, an increase in oxygen consumption, a decrease in peripheral and core temperatures, and a drop in maximum performance. These temperatures are appropriate for endurance swimmers (12, 57, 45)". As you can see, YMMV (your mileage may vary) because there are different responses to different water temperatures for different individuals. -Modocc (talk) 23:54, 1 March 2018 (UTC)

Thanks all. @Modocc:, I take it that also explains why your pulse rises when you get into a hot bath? I've noticed that my pulse is often around 120 when I'm taking a bath, compared with my normal resting rate in the high 60s. Grutness...wha? 04:00, 2 March 2018 (UTC)

Er, I am not free to speculate as to why your pulse increases as it does, per the reference desk guidelines at the top of the page which state "We don't answer (and may remove) questions that require medical diagnosis or legal advice." Thus I cannot answer your question regarding your experience and medical condition. In addition, I have not read the entire source that I provided above nor its references nor do I have any additional references which adequately address the actual quantitative magnitude of the effects of hot water on individuals so as to know or understand what is considered a normal change or not. However, I did find a vague description of the general effects of hot water on people from Cleveland Clinic here. My very limited understanding from this one source in particular is that vasodilation of the peripheral blood vessels moves blood out of the body's core which lowers the blood pressure causing the heart to pump faster and harder to keep it near normal. -Modocc (talk) 14:14, 2 March 2018 (UTC)

:) I wasn't asking for a diagnosis! I've just looked it up elsewhere online and yes, it seems to be the reason and is perfectly normal and - as I suspected - is caused by the same phenomenon. Grutness...wha? 01:15, 3 March 2018 (UTC)

What do you call...

...those cards that you use in photographs to give an index to colour and scale.

All the best: Rich Farmbrough, 13:58, 1 March 2018 (UTC).

Like this: [3]? I can't find any standardized name other than "calibration chart" or "calibration card". --Jayron32 14:25, 1 March 2018 (UTC)

I suspect you want the Munsell color system, but it may be another color model. Wnt (talk) 15:20, 1 March 2018 (UTC)

• Photographic scale[4][5] (archaeological) or ABFO scale (criminal) [6]
• Kodak gray card

Andy Dingley (talk) 15:59, 1 March 2018 (UTC)

Thanks both, looks like "color scale card" is close to what I was thinking of, but the one that fits in a wallet seems ideal. I also found these nice 1cm cubes. [7] All the best: Rich Farmbrough, 16:57, 1 March 2018 (UTC).

It is also common to refer to the color calibration chart by the vendor name - as there is at least some merit to the idea that a vendor's proprietary quality-control methodology makes their chart different from any other particular "standard" color chart. After all, these are meant to be used for calibration - one hopes that every aspect of the chart is calibrated - from the ink and printing process all the way down to the paper's optical and surface properties! One hopes that the optical properties of the calibration chart is more accurate and precise than the optical equipment that is photographing or measuring the chart!

So, for example, you can buy a Macbeth chart, or a Munsell chart, or a Pantone chart, or a Kodak chart; these are among the most commonly-used (and commonly-copied) proprietary form of the color chart. As it turns out, economic forces are at play, and as of 2018, almost all of these proprietary calibration methodologies are now owned by one conglomerate, X-Rite Pantone, an American optical instrument company. They sell, for example, the ColorChecker Classic, which you may know as the GretagMacbeth chart (named for the original conglomerate who bought the conglomerate who bought the rights to the chart and its methodology...)

Personally, I'm a fan of the more interesting angular resolution charts, which are not proprietary; but beware! There's no value in a poorly-manufactured optical calibration knock-off! (Tip: whether you're calibrating color, intensity, or resolution, don't simply download a JPEG and print it ... calibration of a calibration system is... a bit involved)!

Nimur (talk) 17:30, 1 March 2018 (UTC)

Coloured cubes are pretty handy, but mostly for setting up lighting, rather than the camera itself. It can be a nuisance for some jobs if your lights from different directions have different colour temperatures (typically artificial fill-in under directional natural light).

Resolution test targets, like the USAF one, are a different thing.[8] They're easy enough to print yourself, if you want a large one to test some telescopic system - it's microscopic one which test the limits of your production system. I did one myself nearly 30 years ago, as an exercise in learning PostScript coding.

The USAF one is a bit limited though. It's no use for demonstrating astigmatism, or basic setup of an optical system. For that you're better with something like the IEEE target [9] or just something with rosette targets on it [10] Andy Dingley (talk) 20:02, 1 March 2018 (UTC)

Very true! There are proverbial truck-loads of special-purpose calibration charts for specific use-cases, lenses, adaptive-optics systems, and so on! As I frequently do, I refer the interested reader to Applied Photographic Optics ... if you actually want to get into the meat of real, modern lens design! Nimur (talk) 23:21, 1 March 2018 (UTC)

I was kind of hoping Lenna would lead somewhere, but the closest I get is Standard test image, neither of which are really what's needed. Matt Deres (talk) 17:46, 1 March 2018 (UTC)

@Rich Farmbrough. You mentioned cubes. At first, I was a little disappointed when I revived my Spydercube. It was smaller than the illustration suggested. I thought I had wasted my money on a bit of plastic. In use though, for ' when one just wants to take a quick photograph ' and perhaps correct it after in Photoshop or GIMP. It provides a good and quick datum. The little silver ball on top, even indicates any highlight that may burn out. Take it that you know how to use your camera's gammer display in order to expose to the right. This cube makes it easier, without having to think too much about what one is doing. --Aspro (talk) 21:25, 1 March 2018 (UTC)

Lots to think about there! The card Andy Dingly mentioned is what I a going with, when I take pictures of small objects for Commons. All the best: Rich Farmbrough, 23:55, 3 March 2018 (UTC).