2. PHYSICS IN DAILY LIFE: Capricious suntime. VER +

  3. CURIOSIDAD MATEMÁTICA. ¿Dónde está el cuadrado que falta? VER +

  4. NOVEDAD EDITORIAL. Ciencia, tecnología y torres gemelas. VER +


Dear Editor:

As usual, the “Search and Discovery” piece by B. Schwarzschild (Physics Today Dec. 2011, pp 14-17) on the 2011 Nobel prize was compact and informative. In Fig. 2b a picture with confidence contours for Λ>0 in the ΩmΛ plane shows convergence near Ωm= 1/4, OmegaΛ=3/4, using information inferred from high-z type supernovae, cosmic microwave background and baryon acoustic oscillations. It implies that there may be three times as much dark energy density as visible plus dark matter energy density.

I would like to point out that an alternative interpretation is possible using the Ωm, Ωk plane (where k<0 stands for space-time curvature) instead of the the Ωm, ΩΛ plane. It is well known that Friedmann’s solutions of Einstein cosmological equations for closed (k>0), flat (k=0) and open (k<0) universes were obtained assuming Λ=0 (probably because Friedmann realized that k<0 and Λ>0 play similar roles). Hence Ωmk =1 in the Ωm, Ωk plane results, in exactly the same downwards straight line from (0,1) to (1,0) as it does (in Fig. 2b) in a flat (k=0) universe. It is easy to check that using the Friedmann-Lemaître solutions for an open universe with Λ=0, both Ωm and Ωk are time-dependent. Therefore the straight line from (0,1) to (1,0) describes in this case cosmic evolution from Ωm≅1 (the big bang) to Ωm ≅ 0.044 (now, t = 13.7 Gyrs) and beyond, with Ωm=0 in the distant future.

(a) Ωk vs. Ωm, and (b) ΩΛ vs. Ωm


Light proceeding from high-z supernovae with z ≅ 1 is arriving to us now but may have been emitted at a time somewhere before t = tSN ≅ 7.5 Gyrs ago, corresponding to Ωm(tSN) ≅ 0.10 and so forth for more distant supernovae. Beyond some z+ = 20.7, cosmic Schwarzchild radius (time) no galaxies (no stars) are observed for obvious reasons. To correlate quantitatively Ωm(t) with z(t) and with t itself is presently straightforward having into account that the relevant cosmic quantities (T0=2.726 K ± 0.01%, t0= 13.7 Gyrs ± 2%, H0= 67 Km s-1Mpc-1 ± 5%) are known with fair precision after the COBE and WMAP satellites, as well as knowing of the present cosmic equation of state (RT= const.) and that radiation and matter densities become equal at atom formation (taf ≅ 1 Myr, Taf ≅ 3000 K, Haf ≅ 6.48×105 Km s-1Mpc-1).

In other words this alternative interpretation might well be useful to substantiate the elusive “dark matter”-“dark energy” problem.


Julio A. Gonzalo - UAM/U. San Pablo CEU, Madrid

PHYSICS IN DAILY LIFE: Capricious suntime.

At what time of the day does the sun reach its highest point, or culmination point, when its position is exactly in the South? The answer to this question is not so trivial. For one thing, it depends on our location within our time zone. For Berlin, which is near the Eastern end of the Central European time zone, it may happen around noon, whereas in Paris it may be close to 1 p.m. (we ignore the daylight saving time which adds an extra hour in the summer). But even for a fixed location, the time at which the sun reaches its culmination point varies throughout the year in a surprising way. In other words: a sundial, however accurately positioned, will show capricious deviations through the seasons: the solar time on the sundial will almost always run slow or fast with respect to the “mean solar time” on our watch. It’s all determined by the rotation of the earth around its axis, combined with its orbit around the sun.

The first thing we realise is that, from one day to the next, the earth needs to rotate a bit more than 360 degrees for us to see the sun in the South again. The reason is obvious. During a day, the earth moves a bit further in its orbit around the sun and thus needs to turn a little extra to bring the sun back to the same place (remember that the rotational direction of the earth around its axis and of its orbit around the sun are both counterclockwise). Now, if the earth were well-behaved, and would move in a circular orbit around the sun, with its rotational axis perpendicular to its orbital plane, this would be the end of the story.

But there are two complications, both of which cause deviations. The first one is the elliptical orbit of the earth. In fact, the earth is 3% closer to the sun at the beginning of January than at the beginning of July. So, the globe must rotate just a bit longer in January to have the sun back in the South than in July; just think of Kepler’s law. The result is that the solar time will gradually deviate from the time on our watch. We expect this eccentricity effect’ to show a sine-like behaviour with a period of a year.

There is a second, even more important complication. It is due to the fact that the rotational axis of the earth is not perpendicular to the ecliptic, but is tilted by about 23.5 degrees. This is, after all, the cause of our seasons. To understand this “tilt effect” we must realise that what matters for the deviation in time is the variation of the sun’s horizontal motion against the stellar background during the year. In midsummer and mid-winter, when the sun reaches its highest and lowest point of the year, respectively, the solar motion is fully horizontal, so its effect on time is large. By contrast, in spring and autumn, the sun’s path also has a vertical component, which is irrelevant here. But it makes the horizontal component smaller in these parts of the year, and so its effect on time. This gives rise to a sine-like deviation having a period of half a year.

The two contributions are shown in the graph. Superposition of these “single and double frequency” curves yields the total deviation of the ‘solar noon” from the “mean solar noon” on our watch. We see that around February 11 the sun is about 15 minutes later than average, and around November 3 about 15 minutes earlier.

So, a simdial in our front yardmay be quite charming, but understanding its readings requires a scientist.



L.J.F. (Jo) Hermans, Leiden University, Tha Netherlands





Corre el rumor entre los matemáticos de que el área de un triángulo es base por altura dividido entre dos. Pues bien, en la imagen podemos ver dos triángulos con la misma base y la misma altura, que sin embargo tienen áreas distintas. Baste ver que están formados los dos por las mismas figuras, y, sin embargo, hay un cuadrado que falta (o sobra, según cómo se mire)

¿Cómo es posible?









Darwin afirmó en una ocasión que Cristo nada tiene que ver con la ciencia, sino que más bien se desvanece a la luz de un pensamiento crítico aguzado por el trabajo científico. Muy lejos de esta pretensión, el hecho de abrazar una u otra religión ha determinado en gran medida el avance científico de sus escenarios.

Con Ciencia, teología y torres gemelas, Stanley L. Jaki sitúa el nacimiento de la ciencia moderna en el occidente cristiano y aborda el papel de Cristo como salvador de la misma. Todo ello, bajo diferentes perspectivas, desde el atentado de las Torres Gemelas (con la tecnología occidental implicada en su propia destrucción) hasta el teorema de Gödel, o el análisis de las suras que citan a Jesús. Una obra que derribará muchos mitos.





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