Wikipedia:Reference desk/Archives/Science/2024 March 18

Welcome to the Wikipedia Science Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 18

Special Relativity. Is it possible to calculate the velocity of a system, composed of two bodies, we given their different inertial masses and velocities?

An equivalent question: What's the system's (relativistic) inertial mass, we bearing in mind that the conservation of (relativistic) mass, as a general rule, does not hold in Relativity Theory.

2A06:C701:427F:A900:B552:C10D:40A2:7B31 (talk) 16:58, 18 March 2024 (UTC)

There is an article on the general problem Two-body problem in general relativity. Good luck with it! NadVolum (talk) 19:08, 18 March 2024 (UTC)

Oh, I forgot to point out that my question was only about Special Relativity, the given masses being the inertial ones (rather than the gravitational ones). Due to your comment, I've just added above, that I'm only interested in finding the system's velocity (or inertial mass) in Special Relativity. 2A06:C701:427F:A900:B552:C10D:40A2:7B31 (talk) 19:34, 18 March 2024 (UTC)

Given the four-momenta of the two bodies, $p_{i}=(E_{i},\mathbf {p} _{i})=(\gamma _{i}m_{i},m_{i}\mathbf {v} _{i})$ ($i=1,2$; the last expression only for bodies with finite mass), the combined four-momentum is $p_{\mathrm {tot} }=p_{1}+p_{2}$. The mass of the two-body system is given by $m_{\mathrm {tot} }^{2}=p_{\mathrm {tot} }\cdot p_{\mathrm {tot} }=(E_{1}+E_{2})^{2}-(\mathbf {p} _{1}+\mathbf {p} _{2})^{2}$. The centre-of-mass velocity is given by $(\mathbf {p} _{1}+\mathbf {p} _{2})/m_{\mathrm {tot} }$. (I've set $c=1$; spatial (three-)momentum is denoted by $\mathbf {p}$). For photons use $E=|\mathbf {p} |=h\nu$. --Wrongfilter (talk) 09:58, 19 March 2024 (UTC)

For being more explicit, let's assume that the bodies: have masses m,n, and move at velocities u,v (respectively) on the same line. What's the system's velocity, using m,n,u,v only? 2A06:C701:427F:A900:B552:C10D:40A2:7B31 (talk) 11:13, 19 March 2024 (UTC)

I told you everything you need to know, you just have to restrict to one-dimensional motion and use your notation. Why the anonymity today? --Wrongfilter (talk) 11:29, 19 March 2024 (UTC)

if you suggest that I study maths, then I accept the idea. For the time being, could you calculate it for me (using m,n,u,v only) ? I'm bad at mathematics. (If you also suggest that I register, then you are not the first one to suggest that, but I'd rather avoid registering.) 2A06:C701:427F:A900:B552:C10D:40A2:7B31 (talk) 11:54, 19 March 2024 (UTC)

SOS. Despite my bad maths, I think I've found a contradiction in your method. Please help.

I'm checking two cases, each of which involves two bodies, that have the same mass, and that move with the same speed in opposite directions, so according to your formula of the "combined four-momentum", we get:

First conclusion: $p_{\mathrm {tot} }=(\mathbf {p} _{1}+\mathbf {p} _{2})=0,$ right?

By combining, the first conclusion, along with your formula of "centre-of-mass velocity", we conclude:

Second conclusion: The velocity of the two-body system is $0$ as well, right?

By the second conclusion, which tells us that the two-body system is at rest, we conclude:

Third conclusion: The mass of the two-body system, $m_{\mathrm {tot} },$ is simply its rest mass, which is supposed to be invariant, right?

Fourth conclusion: In the first case checked out, the speed of both bodies is low, so $E_{i}$ is low for both bodies, so according to your formula of the "mass of the two-body system", along with the first conclusion, we conclude that $m_{\mathrm {tot} }$ (proportional now to the energy only) is small. While in the second case checked out, the speed of both bodies is high, so $E_{i}$ is high for both bodies, so according to your formula of the "mass of the two-body system", along with the first conclusion, we conclude that $m_{\mathrm {tot} }$ (again proportional now to the energy only) is big, right?

How can we avoid the contradiction between the third conclusion and the fourth one? Where is my mistake?

2A06:C701:427F:A900:B552:C10D:40A2:7B31 (talk) 14:27, 19 March 2024 (UTC)

The total mass of the 2-body system comprises the masses of the two components and their kinetic energies with respect to the common centre-of-mass rest frame. This is now to be considered as internal energy, i.e. energy in internal degrees of freedom. The total mass is invariant with respect to changes of reference frame. --Wrongfilter (talk) 14:45, 19 March 2024 (UTC)

Thank you. May I formulate it as follows: A rest mass is invariant in one-body systems only (e.g. an electron), rather than in multy-body systems, because of the internal energy that may change if several bodies are involved in the whole system? 2A06:C701:427F:A900:B552:C10D:40A2:7B31 (talk) 14:52, 19 March 2024 (UTC)

In an isolated system the internal energy will not change. --Wrongfilter (talk) 14:54, 19 March 2024 (UTC)

Thank you. 2A06:C701:7478:1B00:1F2:1345:432B:F844 (talk) 13:28, 27 March 2024 (UTC)

Aliens: from cavemen to NASA

I have been reading many books about extraterrestrial life lately. Although there is no actual example to study, the usual approach is to study which things are required to happen or be there, which ones can be likely, which ones not so much, and the feasibility of achiving similar results in other ways (such as life being made of an element other than carbon, or with a solvent other than water). I have seen this approach used to see the chances of a given planet to turn into a Earth-like planet, and for life, once formed, to survive and evolve.

But there is a missing step: once we get full ecosystems with diverse animals, how likely would it be for some animal to evolve into a human-like intelligence? Which would be the requirements and obstacles? And if one actually does, how likely would it be to go all the way from rocks and sticks to space programs? I'm not asking to discuss that ourselves, but to point me some author that did. The article Extraterrestrial intelligence is almost devoid of content and does not help with this search. Cambalachero (talk) 19:44, 18 March 2024 (UTC)

Are you familiar with the Fermi paradox? 41.23.55.195 (talk) 05:39, 19 March 2024 (UTC)

There is a strong bias in the question itself. Which is more intelligent: A species of humans who invented war and terrorism or a species of dolphins that chat and play most of the day? What if a planet of "intelligent" creatures stopped at having fun and didn't feel the need to progress into colonialism and repression? It leads to two common points on the topic. The first is that we will meet technology from other space-exploring societies before we meet the society just as others will see our satellites and transmissions long before they meet a human. The second is that any species that travels through space to meet us isn't doing so to be nice. They see our planet as a source of profit in some way and will want to exploit it as much as possible. 12.116.29.106 (talk) 18:46, 19 March 2024 (UTC)

“For instance, on the planet Earth, man had always assumed that he was more intelligent than dolphins because he had achieved so much—the wheel, New York, wars and so on—whilst all the dolphins had ever done was muck about in the water having a good time. But conversely, the dolphins had always believed that they were far more intelligent than man—for precisely the same reasons.” ― Douglas Adams, The Hitchhiker’s Guide to the Galaxy I'm betting you were making a reference, but figured this was a good chance to introduce the unenlightened to Adams. --User:Khajidha (talk) (contributions) 13:19, 20 March 2024 (UTC)

Dolphins probably never advanced towards a civilization because, as an underwater species, they have no access to fire, the one element that allowed all human technology in the first place. And the lack of versatile limbs, such as our hands with opposing thumbs, probably does not help either. Those two are the kind of points I'm sure some author must have thought and organized in a related book. Cambalachero (talk) 14:04, 20 March 2024 (UTC)

Why would dolphins need fire? ←Baseball Bugs ^{What's up, Doc?} carrots→ 16:56, 20 March 2024 (UTC)

To cook all the fish we gave them. --User:Khajidha (talk) (contributions) 17:32, 20 March 2024 (UTC)

Maybe they prefer sushi. ←Baseball Bugs ^{What's up, Doc?} carrots→ 02:26, 21 March 2024 (UTC)

Good luck getting to the copper age or further without fire. Cambalachero (talk) 17:25, 21 March 2024 (UTC)

Why would dolphins need metals? ←Baseball Bugs ^{What's up, Doc?} carrots→ 20:10, 21 March 2024 (UTC)

How else could they become our tyrannical overlords, forcing us to frolic and have fun every single day? Clarityfiend (talk) 00:51, 22 March 2024 (UTC)

Evolution of human intelligence discusses how some apes got human-like intelligence. It includes links to cephalopod intelligence and cetacean intelligence. Also, possible extraterrestrial life has no obligation to develop intelligence, space pollution or Rubik cubes. To believe otherwise is probably a misleading use of teleology in biology. --Error (talk) 23:34, 19 March 2024 (UTC)