Talk:Time dilation/Archive 2013

Archive 2010

Archive 2011

Archive 2012

Archive 2013

Archives by year: 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021

Am I Immortal From the View-Point Of the Photon?

According to SR velocity is relative. I am sitting here in the cahir. With respect to an ant in the room I am moving with the speed of one centimeter per second; with respect to the vehicles passing-by on the near-by road I am moving with the speed of 40 km per hour; with respecr to the aeroplane flying above my head I am moving with at the velocity of 500 km per hour; and with respect to the photons coming from the Sun I am moving at the speed of light. All the above observers find that my wrist-watch is running slow. The photon from the sun finds that I am not ageing at all. So, my question is: " Am I immortal from the wiew-point of the photon coming from the Sun?210.56.144.108 (talk) 12:04, 1 February 2013 (UTC)

Here we discuss the article, not the subject. See wp:Talk page guidelines. Someting for wp:Reference desk/Science. Good luck. - DVdm (talk) 18:04, 1 February 2013 (UTC)

Incomplete Equalities

It looks like something's wrong with the equalities in the section Experimental confirmation - Velocity time dilation tests. The first row of formulae ends up with unfinished (equation?) term "and {f_rest} \over {1+v/c}", and in the second row there's incomplete term "and f_rest \sqrt{(1-v/c)/(1+v/c)}". — Preceding unsigned comment added by 95.180.45.253 (talk) 22:54, 7 February 2013 (UTC)

That is horrible indeed.  Fixed and simplified, with a source — see [1]. Feel free to hone. - DVdm (talk) 17:08, 8 February 2013 (UTC)

"Equatorial Earth Radius" in Figure Caption beside Eqn (1)

Specifying that the earth clock is located on the equator would be an improvement but this is not stated. The caption currently only specifies that the VALUE used for Re should be the earth equatorial radius, which actually makes the caption worse in my opinion.

1.) A better caption would state "The clock rates agree at an orbital radius of r/Re = 1.497 for an earth clock at the equator. The clock rates also agree at an orbital radius of r/Re = 1.5 for an earth clock at the north pole." If the caption must stick to only 1.497, the caption should specify that the earth clock is on the equator. 2.) Mentioning a distinct "equatorial" Re gives the false implication that taking the earth shape eccentricity into account yields a 0.003 correction, when in fact this correction is primarily an earth rotational velocity effect. The caption should be consistent with the surrounding article where it is placed, where no eccentricity effects are introduced and thus there is only one average earth radius.

For the case of the earth clock being on the equator, the current caption appears to depend on a reference giving the break even orbital radius at APPROXIMATELY 9545 km. Given respective polar, average, and equatorial earth radii of 6357, 6371, and 6378, the equatorial 6378 km was arbitrarily chosen, and three decimals were obtained by round(1000*9545/6378)/1000 = 1.497. I calculated 1.4974*Re, where Re is the simple average earth radius. This gives 9540 km, and also rounds to 1.497 for three decimal places. There is "some" justification to a relunctance to eliminate this ad hoc replacement of average Re with equatorial Re, because going to a normal average earth radius gives round(1000*9545/6371)/1000 = 1.498 which is no longer accurate to the third decimal place. Thus getting the third decimal place correct requires deviating from the Ashby reference, which gives 9545 km instead of 9540 km. The Ashby reference appears to expand terms into series and drop smaller terms, so it may be inappropriate for this amount of resolution. An appeal to simply use (3/2)Re for a caption may also be controversial, since some may argue that earth rotational velocity induced time dilation is within the scope of the article.128.244.42.5 (talk) 18:20, 25 February 2013 (UTC)

As I said on my talk page User talk:DVdm#Time dilation

We do not have to explicitly tell that the earth clock must be on the equator. The Ashby source doesn't say it either, because the clock can be anywhere. All clocks on the geoid are synchronised. As I have shown before, a clock on the pole is synchronous with a clock on the equator (remember that 1.498 above), and in fact with every clock in between, provided it rests on the ground and rotates with the earth. The one on the pole does not rotate but it is closer to the center. The two effects cancel everywhere. - DVdm (talk) 18:29, 25 February 2013 (UTC)

And of course the earth rotational velocity induced time dilation is within the scope of the article. It is in fact part of the subject of the section itself (Time dilation#Time dilation due to gravitation and motion together) - DVdm (talk) 18:33, 25 February 2013 (UTC)

By the way, since indeed Ashby says "The effects cancel at a ≈ 9545 km", in the spirit of wp:CALC I centainly agree to replace "At r = 1.497" with "At r ≈ 1.497" in the caption. No problem. - DVdm (talk) 19:08, 25 February 2013 (UTC)

I am now persuaded that a better caption would not give distinct north pole and equator results, when both results may be the same at sea level. There still is an issue regarding the caption's inference that an ellipsoidal correction has been made. Perhaps the disagreement is whether an earth velocity correction also REQUIRES making an ellipsoidal potential correction. In my opinion, it does. An ellipsoidal correction should not only allow non-average R values, but also replace GM/R with an ellipsoidal surface gravitational potential. I do not see a referenced source using the ellipsoidal potential. For the spherical surface GM/R potential used in Eqn (3) of the current Wikipedia article and its references, plugging in different non-spherical non-average earth radii values for R is invalid. For example, at the north pole "R" is smaller, but using a smaller R value in GM/R assumes an earth density higher than the actual earth density, given that a fixed M and lower R is being plugged into the spherical GM/R surface potential. Adding an earth velocity effect should not be done independently of correcting the earth's spherical potential approximation with an ellipsoidal potential. 128.244.42.5 (talk) 21:20, 25 February 2013 (UTC)

I don't see the caption inferring that an ellipsoidal correction has been made. It seems to be backed by the source, so I see no reason to change it. - DVdm (talk) 16:18, 26 February 2013 (UTC)

In leiu of a source that asserts that the Figure's x-intercept was obtained with a 55 degree inclination, it is still possible to modify the caption wording to enable readers trying to recover the caption result, and make it true. Append the caption with the sentence, "Time dilation has a weak dependence on the orbital inclination angle, which is set to 55 degrees to precisely interpolate the GPS data point."

This sentence avoids explicitly claiming that the inclination angle effect is big enough to make 1.497 wrong at other inclination angles. The sentence also avoids explicitly claiming the plot was generated with a quadrapole term. The facts in this sentence are backed by many good sources, including Ashby. If 1.497 is supposed to be valid for all inclination angles, then it should also hold for 55 degrees. Telling the world Ashby thinks it works for all inclination angles could be too much information. It is not clear that 9545 km cannot be interpreted as some sort of average over inclination angles.

The figure depicts the average circular orbit time dilation relative to earth as a function of only the orbital radius. But this function also depends on inclination angle. A unique inclination cannot be inferred from the figure, which includes geostationary (0) and GPS (55) data points. This is not an issue in the plot, where the inclination effect is too small to be visible.

The caption, however, presents a value for the Figure's x-intercept as accurate to 4 significant figures. For this degree of accuracy to be valid, the caption should associate the given x-intercept with an inclination angle of 55 degrees, but fails to do so. The dependence on inclination angle can be determined from Eqn 85 of the Ashby reference.

People are more likely to know the geostationary inclination rather than GPS. Those plugging in 0 degrees inclination fail to recover the caption's 1.497*Requator. Only those who plug in 55 degrees can get 1.4965114*Requator=9544.955 km.128.244.42.5 (talk) 14:48, 28 February 2013 (UTC)

By all means, if we can find good independent-Wikipedia-kinda-non-original-research-source that says that "Time dilation has a weak dependence on the orbital inclination angle, which is set to 55 degrees to precisely interpolate the GPS data point", then we could add it to the caption :-) - DVdm (talk) 16:24, 28 February 2013 (UTC)

My wrist watch clock rate is one tick per second, or 1 Hz. This is a frequency. Ashby's Eqn 85 gives the shift of this frequency on a satellite. Here the right hand side has an i0 term, where i0 is the angle of inclination. Ashby describes the term as small and even negligible for (modern) GPS satellites that have i0=55 degrees. If you misrepresent a term in a (clock rate) equation, you get less precision (in the clock rate). Thus Ashby is an adequate source. 128.244.42.5 (talk) 19:25, 28 February 2013 (UTC)

Note - Having had the last word without a reply, does not mean that you can go ahead and make the change according your taste, specially if it does not comply with the source. See above. We have policies about wp:consensus. See also wp:BRD. - DVdm (talk) 16:24, 7 March 2013 (UTC)

A consensus is a good idea. To that end I propose deleting the discussion prior to "In lieu of..." since too much material discourages potential reviewers.

• In May 2002 p.45 Physics Today issue Ashby gives 9550 km, indicating Ashby does not always attribute 4 digits of precision to a 4 digit result.
• The Physics Today article makes a stronger inference that GPS orbital parameters are used, ie, "Effects that contribute significantly to the integral [time] on the right include ... quadrapole terms. If the GPS orbit were perfectly circular....9550 km"
• Currently there is nothing to make the Wiki reader aware that the difference between 1.5 and 1.497 is due to a non-spherical rotating earth time dilation effect, and thus relevant to the subject topic. This detail dilutes the emphasis on introducing how satellites have combined gravity and velocity effects.
• Pulling in 4 digits requires also specifying earth equatorial radius rather than earth radius, and 55 degree inclination angle. But the caption is already too long and detailed for an appropriate level of coverage.128.244.42.5 (talk) 20:39, 7 March 2013 (UTC) — Preceding unsigned comment added by 128.244.42.5 (talk) 20:35, 7 March 2013 (UTC)

Ashby does not mention a breakeven point in his Physics Today caption for this Figure. While omitting this is best, if one must be given here are 3 references that support approx (3/2)R for a caption;

1.WGV Rosser’s “Intro to Special Relativity” 1991 p.235 agrees with (3/2)*R by stating “about R/2 above sea level”.

2.Wikipedia Gravitational time dilation. For a non-rotating spherical black hole with Schwarzschild radius r, a circular orbit radius must be above (3/2)*r. The stationary clock stops while approaching r and the orbiting clock stops while approaching (3/2)r, thus the clock rates agree. Making a connection to this might be nice in this article, but maybe not jammed into a caption.

3.http://www.askamathematician.com/2012/03/q-satellites-experience-less-time-because-theyre-moving-fast-but-more-time-because-theyre-so-high-is-there-an-orbit-where-the-effects-cancel-out-is-that-useful/

The issue is un-polished caption writing, not references. Only a limited amount of new info can go into a good Figure caption. Presenting the 4th decimal x-axis intercept and all the details to make it valid to this degree of precision is bad. Reviewers should not waste time wondering if the Ashby Physics Today article stipulates a 55 degree inclination clearly enough. The focus should be what interesting caption material is being displaced while PhDs debate the 4th decimal impact of earth rotation, equatorial bulge, a gravitational quadrapole term, and orbital inclination.

A typical reader will be drawn to this article after taking an interest in the paradox about the YOUNGER twin astronaut. What is currently in a prior Figure caption (of this Wiki time dilation article) confirms that space travel can slow down aging (for ISS astronauts). This reader will find it interesting to learn;

1. Some astronauts will be slightly OLDER than their twin. (This is not obvious from a prior Wiki figure caption that ignores the effect of relative motion.)

2. Clocks run FASTER in a GPS satellite (relative to earth).

3. There is no time dilation at SOME orbital altitude where the gravitational blue shift and velocity red shift cancel.

I ranked these by how interesting they seem. The current caption only attempts to clarify the 3rd ranked concept, and does so by stating “approximations cancel” (???). 108.15.23.67 (talk) 14:17, 9 March 2013 (UTC)

The Physics Today reference "if the GPS orbit ... 9550 km" introduces sufficient doubt about 1.497*Requator being accurate to 4 digits for a 0 degree inclination circular orbit. Statements in Wikipedia should be verifiable. Is there any qualified source or even a person that will explicitly verify 1.497 for a 0 degree inclination? [Not me. Using Ashby Eqn 85 I get R/Req of 1.4948 for an inclination angles of 0 degrees, and a 1.4961 average over all equally spaced inclination angles.]128.244.42.5 (talk) 01:15, 13 March 2013 (UTC)

The Physics Today reference "if the GPS orbit were perfectly circular...9550 km" may have had the intent of setting eccentricity to zero and inclination to 55 degrees. Technically, however, only a 0 degree inclination orbit can traverse a quadrapole potential and stay in a *perfectly* circular orbit. This detail can be swept under the rug in the context of approximations in a long derivation for improving the prediction of GPS clock drift. But in Wikipedia this circular orbit contradition can be taken at face value, as a contradiction. This is another example of how extracting an isolated number from Asbhy's long derivation opens too many can of worms. The topic of getting a tiny earth rotational velocity and associated gravitational quadrapole correction is inadequately addressed with a single number (1.497) and is thus best not addressed at all in a short article. The article should focus on communicating the dominant effects of satellite's velocity and reduced gravity from earth. 128.244.42.5 (talk) 14:17, 13 March 2013 (UTC)

Third opinion

Hi, I saw a request at WP:Third opinion for help, so I'm here to do what I can. Can both editors please briefly state what their proposal is? Please identify a reliable source or two, if available, to backup your proposal. Thanks. --Noleander (talk) 15:35, 15 March 2013 (UTC)

• Proposal from DVdm:

The caption of the figure in section Time dilation#Time dilation due to gravitation and motion together now says:

"... where rs is satellite orbit radius and re is the equatorial Earth radius. At r ≈ 1.497 there is no time dilation, i.o.w. the gravitational and velocity approximations cancel."

The relevant originating source for this is given in the caption of the firgure. See Ashby, Neil (January 2003). "Relativity in the Global Positioning System on page 16. It says:

"The effects cancel at a ≈ 9545 km"

Using the fact that the equatorial radius of Earth is ≈ 6378 km, per wp:CALC we can use 9545/6378=1.497, and use that result in the caption. - DVdm (talk) 16:31, 15 March 2013 (UTC)

Thanks for the information. I'll wait for the other editor to supply their thoughts. --Noleander (talk) 17:17, 15 March 2013 (UTC)

Note - FYI, the discussion started with the first comment by 128.244.42.5 ^{timestamp 19:26, 21 February 2013 (UTC)} on my talk page here, and then continued on the talk page of this article. Cheers. - DVdm (talk) 19:22, 15 March 2013 (UTC)

• Alternate caption text proposal;

"Daily time dilation as a function of r=rs/re, where rs is a satellite's circular orbital radius and re is the earth radius. In low orbits time passes slower than on earth, because the satellite speed is the dominant effect. In high orbits time passes faster than on earth, because the reduced earth gravity is the dominant effect."

The current caption introduces excessive info by attempting to establish the "break-even" altitude to 4 digits of precision. Getting the 4th decimal correct requires nailing down too many things outside the scope of this article. These things include a distinct earth average and equatorial radii, the earth geoid and lopsided sea level, the earth's quadrapole moment, and the satellite's orbital angle of inclination.

References indicating the 4 digit result is taken out of context;

1.) May 2002 p.45 Physics Today (Ashby) presents significant doubt that the 4 digit result is accurate in lieu of a stipulating the orbital angle of inclination as being 55 degrees. The current Figure gives a geostationary data point, and these satellites' orbital angle of inclination is 0 degrees.

References indicating a 3 bit breakeven is more appropriate for a breif Wikipedia article, if any breakeven result must be given at all;

1.) WGV Rosser’s “Intro to Special Relativity” 1991 p.235 agrees with (3/2)*R by stating “about R/2 above sea level”.

2.) Wikipedia Gravitational time dilation. For a non-rotating spherical black hole with Schwarzschild radius r, a circular orbit radius must be above (3/2)*r. The stationary clock stops while approaching r and the orbiting clock stops while approaching (3/2)r, thus the clock rates agree. Given a ratio other than (3/2) muddies the water for this nice connection to be made.

3.) http://www.askamathematician.com/2012/03/q-satellites-experience-less-time-because-theyre-moving-fast-but-more-time-because-theyre-so-high-is-there-an-orbit-where-the-effects-cancel-out-is-that-useful/ 128.244.42.5 (talk) 22:13, 15 March 2013 (UTC)

Thanks for the input. It sounds like one editor wants three digits precision (which is 1.50, right?) and the other editor wants four digits (1.497). Is that correct? Also: is anyone proposing to include the 9,545 value in the caption (which seems like a useful number for readers to grasp the matter)? --Noleander (talk) 15:55, 17 March 2013 (UTC)

My point is, if the source had said that "the effects cancel at a ≈ 9567 km", then we could write ≈1.5 or even ≈1.50, but the source says "the effects cancel at a ≈ 9545 km", so I see no problem writing ≈1.497 instead of ≈1.5. Rounding the number from that source would be —at least i.m.o.— subtle orginal research.

But yes, adding the source value of ≈9545 km would be ok for me, even if a single click brings the reader directly to the page where they can verify. Perhaps even mentioning the equatorial radius ≈6378 km as well would be ok. - DVdm (talk) 17:24, 17 March 2013 (UTC)

Okay, I think I have a good understanding of the issues. Here is my suggested caption:

Graph of daily time dilation gain in microseconds (loss, if negative) as a function the ratio rs/re, where rs is satellite orbit radius and re is the equatorial Earth radius. Dilation is zero at satellite orbit radius 9,545 km, where the ratio is approximately 3:2. International Space Station (ISS) astronauts fly below that neutral orbit, whereas GPS and Geostationary satellites fly above.[1]

I think the caption needs to tell readers that the 3:2 ratio is generally the correct answer (although there may be many variables that make the actual value slightly different from 1.5). Using anything other than 1.5 (such as 1.501 or 1.499) would deprive the reader of the key fact that the 3:2 ratio is underlying the fact. But, since 1.5 is not going to be precisely the correct value, it must be qualified with "about" or "approximately". If editors want readers to have more detail, a footnote could be added to the caption. Per WP:V, the facts in the captions need a citation ... and since the 9,545 fact is not cited in the article body, a footnote must be included in the caption. --Noleander (talk) 23:42, 17 March 2013 (UTC)

The 3rd opinion proposed caption includes 9545 km, indicating that this number is not taken out of context due to the omission of a 55 degree angle of inclination. The proper context is a derivation of GPS clock drift. Just in case access to Physics Today is an issue, I quote the relevant paragraph;

"If the GPS orbits were perfectly circular, the corrections would include just a few constant contributions: for the gravitational potential differences between the satellites and the geoid, and for the second-order Doppler differences between the orbiting clocks and the reference clocks on the ground. Figure 4 shows how the relativistic frequency shift depends on the circular orbit’s radius. At a radius of 9550 km, about 3000 km above the ground, the gravitational and Doppler effects cancel. Because the GPS orbits are higher than that, the gravitational blueshift is the largest contribution. So the net frequency correction for a GPS satellite is negative, amounting to 4.4645 parts per ten billion."

After using Ashby Eqn (85) of the original Ashby reference I see that; for a 0 degree angle of inclination the breakeven rs = 9534 km, for a 55 degree angle of inclination the breakeven rs = 9545 km. Perhaps it would help (to get a consensus) to state in the caption that "The earth's rotation shifts the orbit where time dilation is zero to slightly below r=(3/2)." 128.244.42.5 (talk) 14:19, 18 March 2013 (UTC)

Because it is a caption, the text should be brief and summary-ish. Details can be in the text next to the picture, or in a footnote. How about this:

Graph of daily time dilation gain in microseconds (loss, if negative) as a function the ratio rs/re, where rs is satellite orbit radius and re is the equatorial Earth radius. Dilation is zero at satellite orbit radius of approximately 9,550 km, where the ratio is approximately 3:2. International Space Station (ISS) astronauts fly below that neutral orbit, whereas GPS and Geostationary satellites fly above.[1]

where the footnote explains the distinction between 9550 and 9545 and also describes the orbit inclination influence.--Noleander (talk) 16:20, 18 March 2013 (UTC)

Adding a footnote indicating 55 degree angle of inclination adequately resolves my objections. I assume footnotes are OK for Wikipedia, and are somehow distinguishable from references. Is that the case?

If a restriction to 55 degrees angle of inclination is stated somewhere, then my preference would be to stick to 1.497 since

.001*round(1000*9550/6378) = 1.497

.001*round(1000*9545/6378) = 1.497

and there is no longer any need for the footnote to also explain the 9550 vs 9545 discrepancy. 128.244.42.5 (talk) 16:57, 18 March 2013 (UTC)

Ok, since it follows the source perfectly, I suggest we stick with the current caption, and you add the remark about the 55 degrees and its source in a footnote. - DVdm (talk) 17:32, 18 March 2013 (UTC)

While far less critical issues remain, the current caption still has "i.o.w." and "approximations cancel". Completely reverting from the 3rd opinion version (which removed these problems) back to the current caption resurrects these problems. The concept is that orbits exist for which the average time dilation relative to earth is zero because the real physical effects of *motion* and *reduced gravity* precisely cancel. Mathematical approximations may also cancel, but the topic is physics and not math. Something like "the effects of motion and reduced gravity cancel" would be nice.128.244.42.5 (talk) 18:25, 18 March 2013 (UTC)

Hm... I don't think that there are separate "real physical effects of motion and reduced gravity". There is one physical effect that, as far as we know, can, with best precision, be calcuated from the metric. Only due to the smallness of the Earth mass, the calculation of the effect can be approximated by pretending that there are separate effects of motion and gravity. But that's a discussion about the subject and this is not the place for that. Anyway, your suggestion is ok wih me. - DVdm (talk) 20:03, 18 March 2013 (UTC)

It sounds like you both are in agreement now, and I no longer need to be involved. A final note: Someone above asked about footnotes vs citations: Yes, WP supports both. Most footnotes are for citations; but WP can have a footnote that is just remark, as explained at Help:Footnotes#Explanatory_notes. If I can help in any other way, just post a note on my talk page. --Noleander (talk) 00:30, 19 March 2013 (UTC)

Thanks, Noleander. I attempted to put the new Physics Today reference and citation inside the new footnote. After numerous error messages I appended the new Physics Today reference and citation just after the new footnote reference in the caption. If someone else can move the Physics Today reference to the footnote, OK by me. I am now home where I will get a different IP signature.108.15.23.67 (talk) 01:13, 19 March 2013 (UTC)

Yes, thanks, Noleander. Good job! Ditto, Anon(eil?) ;-) - DVdm (talk) 15:47, 19 March 2013 (UTC)

Thanks, DVdm. I finally got the new Physics Today reference into the footnote. Special commands are required to nest references.128.244.42.5 (talk) 12:40, 20 March 2013 (UTC)

That's a pretty high-tech footnote construction we've got there now :-) - DVdm (talk) 13:20, 20 March 2013 (UTC)

Time dilation vs relative velocity

According to SR time dilation would occur if a space ship traveled near the speed of light from Earth. But according to SR velocity is only relative to other objects. So why would time progress faster on Earth, and not the other way around? This seems to be a huge problem not addressed in the article. --81.170.234.195 (talk) 09:19, 19 May 2013 (UTC)

It is addressed in the section Time dilation#Relative velocity time dilation and deeper in the article Twin paradox. If you have problems with understanding the subject, you can go to our wp:Reference desk/Science (see wp:Talk page guidelines) - Good luck. - DVdm (talk) 09:26, 19 May 2013 (UTC)

Greek Letters

In the section "Time dilation due to relative velocity" Greek letters are used in the formulas. I recognized the little triangle as delta from my time spent in calculus, but the Y looking character? What the heck is that? I suspected it's a Greek letter, but the Wikipedia page about the Greek Alphabet does not show any such animal. I had to go to Google to find out that it's gamma. If you are going to use these obscure symbols it would help if you named them and perhaps gave an example of how to read these equations aloud, or at least provide a link to a page that would explain this stuff. Pergelator. — Preceding unsigned comment added by 50.43.12.61 (talk) 21:54, 28 July 2013 (UTC)

It's gamma. Ginsuloft (talk) 21:56, 28 July 2013 (UTC)

I have changed the wording of that section to make this clearer. Gandalf61 (talk) 14:38, 29 July 2013 (UTC)

Moreover, it's a lowercase gamma character, γ, which looks quite different to the uppercase gamma character, Γ -- 79.123.73.175 (talk) 14:40, 29 July 2013 (UTC)

Dilate and Burn up?

If you dilate 100x and fly by the Earth (forget the impossible energies), I think I have concluded that you would also experience a 100x increase in solar flux. Agreed? Frank Layden (talk) 20:10, 19 November 2013 (UTC)

Maximum v?

Has anyone run across restrictions on velocity other than c? Given an object with mass, but all the energy you need, do we eventually have to worry about the Uncertainty principle, Plank length, quantized v, or anything else ... as we approach c? Frank Layden (talk) 20:19, 19 November 2013 (UTC)

Good questions for our wp:reference desk/science. Here we can only discuss the content and format of the article, not the subject. See wp:talk page guidelines. Good luck at the ref desk. - DVdm (talk) 20:30, 19 November 2013 (UTC)