I've been reading about how black holes can effect both time and light with gravity. So I was wondering, doesn't something have to have mass to be effected by gravity? And if so, does this mean both light and time have mass? And if not, how can gravity effect something that has no mass?
Sunday, July 2, 2017
lateral thinking - A pilot's messages
A friend of mine works as a pilot, so he travels a lot and I don't see him that often. But every year around the same time he sends me a message. He has been doing that ever since 1997.
Here is a list of all the messages he sent:
1997: 0.6 - LAFRBRAV - Kg
1998: 1.8 - BVOHOTXY - Is
1999: 5.8 - JULITTROMOIRRA - wdn
2000: 2.6 - LFOOOVB - Dk
2001: 2.4 - CHRLPPHL - sn
2002: 5.0 - NGOMM - v
2003: 4.8 - OMOINDIAXA - Tuk
2004: 3.0 - DSTGO - Uk
2005: 4.2 - INDIAHOVITO - G
2006: 4.0 - TGOHOTE - id
2007: 2.4 - HOTLCHOLM - Srb
2008: 4.8 - BVOECHOGOLF - ussi
2009: 3.2 - DELTMIKEECH - Nrwy
2010: 5.8 - OSCSILI - Gny
2011: 2.8 - DLTUFOMS - zbj
2012: 5.2 - GOLFYAKLTA - Sw
2013: 3.6 - IIXY - Dn
2014: 2.0 - CHLEPPHOEL - us
2015: 4.6 - VICTORIIACHO - Sw
2016: ???
Unfortunately I lost last year's message. Can you help me figure out what it was?
Note: a good starting point for this is to try and make sense out of the other messages.
Answer
Two things jumped out at me as soon as I saw this puzzle.
Firstly, the mixed-case letter strings on the far right. Several of them look like
country names with certain letters removed. For instance, Srb -> Serbia, ussi -> Russia, and Nrwy -> Norway.
Secondly, the all-caps letter strings in the middle. Several of them start with what look like
words from the
NATOIACO phonetic alphabet. Reading down, we can immediately spot JULI(E)T, INDIA, HOT(E)L, DELT(A), GOLF, VICTOR.
Looking at it a little more closely, I noticed that
the letters missing from the country names seem to be identical in each line to the letters removed from the
NATOIACO phonetic alphabet words.
So let's fill in some of those letters and see where we get:
1997: 0.6 - DELTA / UNIFORM / BRAVO - United Kingdom
1998: 1.8 - BRAVO / HOTEL / XRAY - Israel
1999: 5.8 - JULIETT / ROMEO / SIERRA - Sweden
2000: 2.6 - ALFA / ROMEO / NOVEMBER - Denmark
2001: 2.4 - CHARLIE / PAPA / HOTEL - Estonia
2002: 5.0 - TANGO / LIMA / LIMA - Latvia
2003: 4.8 - ROMEO / INDIA / XRAY - Turkey
2004: 3.0 - INDIA / SIERRA / TANGO - Ukraine
2005: 4.2 - INDIA / ECHO / VICTOR - Greece
2006: 4.0 - ALFA / TANGO / HOTEL - Finland
2007: 2.4 - HOTEL / ECHO / LIMA - Serbia
2008: 4.8 - BRAVO / ECHO / GOLF - Russia
2009: 3.2 - DELTA / MIKE / ECHO - Norway
2010: 5.8 - OSCAR / SIERRA / LIMA - Germany
2011: 2.8 - DELTA / UNIFORM / SIERRA - Azerbaijan
2012: 5.2 - GOLF / YANKE / DELTA - Sweden
2013: 3.6 - MIKE / MIKE / XRAY - Denmark
2014: 2.0 - CHARLIE/PAPA/HOTEL - Austria
2015: 4.6 - VICTOR / INDIA / ECHO - Sweden
On each line, the all-caps part deciphers to
three letters, which together with the context of the message suggests airport codes.
Furthermore,
the airport given by this three-letter code is in the country given by the right-most part of the previous line! For instance, in 2008, BEG is Nikola Tesla Airport in Belgrade, Serbia - the country mentioned in 2007.
Having noticed this connection, I actually used it quite a lot to help me find some of the missing letters to reconstruct the above.
So now we have the following list of airports:
Dublin (DUB), Ireland
Birmingham (BHX), United Kingdom
Atarot (JRS), Israel (Palestine?)
Stockholm Arlanda (ARN), Sweden
Copenhagen (CPH), Denmark
Tallinn (TLL), Estonia Riga International (RIX), Latvia
Istanbul Ataturk (IST), Turkey
Kiev International (IEV), Ukraine
Athens International (ATH), Greece
Helsinki (HEL), Finland
Belgrade Nikola Tesla (BEG), Serbia
Domodedovo International (DME), Russia
Oslo Gardermoen (OSL), Norway
Dusseldorf (DUS), Germany
Heydar Aliyev International (GYD), Azerbaijan
Malmo (MMX), Sweden
Copenhagen (CPH), Denmark
Vienna International (VIE), Austria
Thanks to Sconibulus, these correspond to
the locations of the Eurovision Song Contests in the respective years. (This explains why the airport in one line matches the country in the previous line - because each year's winner becomes the next year's host.)
Looking at the linked table, we can now spot that the number on the far left of each line in the puzzle corresponds to
the date of the ESC (you did mention that the messages come "every year around the same time"). E.g. for 2010, the number 5.8 equals 29/5 and the ESC was on the 29th of May.
So in the row for each year, we have
the date, location, and winner of that year's Eurovision Song Contest (the date encoded by simply treating the slash as division; the location encoded by an airport IATA code in the IACO phonetic alphabet, and the location and winner encoded by removing their common letters).
And the final answer is
2016: 2.8 - LFOMOOVMB - Uki, because the 2016 ESC was held on 14/5 in Stockholm and won by Ukraine, giving "ALFA/ROMEO/NOVEMBER - Ukraine", from which we remove the letters R, A, N, and E (thanks to Henning Makholm for realising that the letters removed are precisely the ones found in both strings).
Saturday, July 1, 2017
energy conservation - Does the mass lost by merging black holes depend on how they merged?
We've all heard the news about the detection by gravitational waves of two black holes, one 29 solar masses and the other 36 solar masses, spiraling into each other to create a single black hole of 62 solar masses.
For me, the loss of three solar masses into the gravitational waves over a fraction of a second is the most amazing aspect of this. I can understand how the waves were created (and mass changed into broadcast energy) by the spiraling motion of the black holes as they did their final do-si-do, and it's clear that losing that energy was a necessary part of their joining; otherwise they would have orbited forever.
However, does the amount of mass lost depend on just how the two holes merged? For instance, if they had rammed each other head-on, would they still have somehow lost that amount of energy? I expect that less energy would have been diverted into the resultant black hole's spinning, but I don't know how that would have changed the resulting mass.
Answer
Yes. The amount of energy radiated as gravitational waves will depend on the details of the two black holes before the merger. The answers to questions like:
- Was the orbit circular or elliptical?
- Were they spinning?
- Were the spins of the black holes aligned with the orbital plane?
will affect the energy of the gravitational waves. The most import detail in terms of energy radiated is the mass ratio of the two precursor black holes.
The final mass of the system $M_\mathrm{fin}$ is always less than the original total mass ($m_1 + m_2$), since some of the mass energy gets converted to gravitational waves, $M_\mathrm{rad}$.
$$M_\mathrm{fin} + M_\mathrm{rad} = m_1 + m_2 $$
Because of something akin to the second law of thermodynamics the final black hole must be bigger than the biggest original black hole. Basically, you can't radiate so much energy in gravitational waves that a black hole shrinks.
$$M_\mathrm{fin} > m_1 \quad \mathrm{and} \quad M_\mathrm{fin} > m_2$$
We can define the fraction of mass radiated aways as:
$$ e = \frac{M_\mathrm{rad}}{m_1 + m_2} $$
This is sometimes called the efficiency of radiation.
If the black holes have about the same mass (as they did in the LIGO detection), about 5% of the total mass will be radiated away. This is the most efficient possibility.
On the other extreme imagine the case where one black hole is way more massive than the other: maybe 1 million solar masses and 1 solar mass. In order to follow the two rules stated above, $M_\mathrm{fin}$ is less than 1 million and one solar masses and greater than 1 million solar masses. In this case the efficiency would be about $e=10^{-6}$ or 0.0001%. Extreme mass ratios produce the weakest gravitational waves.
newtonian mechanics - Solving for motion of rotating rod using only Newton's laws?
I have a question that's been bothering me for years.
Given a rod of uniform mass distribution with total mass $M$ and length $L$ that lies on a horizontal table (with one end fixed to the table around which the rod is free to rotate in the horizontal plane, and a force F applied perpendicular to the rod at the other end), how do you solve for the motion of the rod (and the internal forces) using only Newton's laws and the assumption that the rod is a rotating rigid body? By that I mean only using the most basic conception of Newton's laws and the system's constraints, without the ideas of torque and moment of inertia, energy and momentum, and even without the idea that the net force on the rod gives the acceleration of the center of mass--so only using Newton's laws for point particles, or in this case infinitesimal $dm$ sections of the rod.
I've tried to solve this by breaking the rod down into these small $dm$ components and using an idea I've seen (at least I think I've seen) where you set $F(x+dx)-F(x)=dm(a)$ and then are able to find $F'(x)$ and integrate and then use the boundary conditions on the force. I did this for both tangential and radial components, with radial acceleration equal to $x(\omega(t))^2$ and tangential acceleration equal to $\omega\ '(t)x$, but was unable to obtain the right answer. I used the force at one end of the rod as a boundary condition (is this correct?), but was unable to even solve for the force at the pivot, let alone the angular velocity as a function of time, and have no idea if this technique is even valid. I feel like at a certain point it might be that my force equation switches sign--as the net force that accelerates the infinitesimal mass starts coming from the inward instead of outward side.
I'd also be interested to know more generally how to solve for internal forces and motion of a rigid body using only these most basic assumptions, such as for a free uniform square on a horizontal table with a force applied perpendicularly to one side into one corner.
general relativity - What is the entropy of the universe today?
What's the entropy of the universe today? How does one go about calculating this? I've heard the statement that black holes account for the bulk of the entropy in the universe today, but don't know why this would be true or the relationship between black holes and entropy.
newtonian mechanics - In projectile motion, what does it mean for the motion along the $x$ and $y$ axis to be independent?
In projectile motion, what does it mean for the motion along the $x$ and $y$ axis to be independent?
This question is referring to the concept of projectile motions.
biophysics - Factors affecting pain of static electricity shock
Referring to How to avoid getting shocked by static electricity?, someone I knew suffered much pain from static electric shocks when touching door knobs or car doors where I didn't feel anything doing the same thing in the same environment. What are the importance of factors determining how painful (or not) shocks are, e.g. one's pain threshold, shoes and clothes one wears or even how one walks?
Answer
Everyone is likely to have a similar pain threshold to electric shocks.
If your friend wears leather soled shoes he will conduct static electricity to earth because they offer much less resistance to earth than rubber or plastic soled shoes. Wearing synthetic clothes may cause a similar effect.
If he/she has lower resistance than you (by having sweaty hands for example) that may make things worse for him/her.
Possibly your friend lives in a house with carpets made from polyester or some other synthetic material which builds up static electricity as he/she walks on it.
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