I have read various dubious explanations as to why people often wear black in the heat, from cultural to somehow encouraging the evaporation of sweat (unconvincing). So, does anyone know what, if any benefit there is to black clothing in hot dry conditions? It is certainly counterintuitive.
[Physics] Why do people wear black in the Middle East
everyday-lifethermal-radiationthermodynamicsvisible-light
Related Solutions
The color of a surface doesn't reliably indicate the emissivity at non-visible wavelengths. The color in the visible spectrum is more of a side effect than anything. Most thermal radiation around body temperature or room temperature happens in the infrared region, not the visible, and that's not reliably indicated by visible color:
The visibly transparent glasses are opaque to the body's infrared emissions, while the visibly opaque trashbag is transparent to infrared. So one property has no relation to the other.
The emissions of the sun occur mostly in the visible region, which is why white clothing reflects solar energy and stays cool while black clothing absorbs solar energy and gets hot:
But your body's thermal radiation is in the infrared, so this rule doesn't apply to the inside of clothing (unless your body is hot enough to radiate visible light, but then you have bigger problems).
Your basic idea would work, though, if you found a material that reflects visible light while transmitting infrared light (but that material would probably have the same properties on the inside and outside, and thus be visibly white on the inside, too).
For example, white paint is quoted as having an absorptivity of 0.16, while having an emissivity of 0.93. This is because the absorptivity is averaged with weighting for the solar spectrum, while the emissivity is weighted for the emission of the paint itself at normal ambient temperatures. ... The white paint will serve as a very good insulator against solar radiation, because it is very reflective of the solar radiation, and although it therefore emits poorly in the solar band, its temperature will be around room temperature, and it will emit whatever radiation it has absorbed in the infrared, where its emission coefficient is high. − Kirchhoff's law of thermal radiation
NASA uses such materials, which they call "selective surfaces", and are used to cool the Hubble telescope:
These surfaces can be designed to reflect solar radiation, while maximizing infrared emittance, yielding a cooling effect even in sunlight. On earth cooling to -50 °C below ambient has been achieved, but in space, outside of the atmosphere, theory using ideal materials has predicted a maximum cooling to 40 K!
Wikipedia's article on selective surfaces describes the opposite effect: Transmitting sunlight and blocking infrared from escaping, to capture the sun's energy.
Physics answer
A black body at a temperature T, area A (on either side) will radiate heat by: $$\dot{Q}_{radiated ,downwards}=\sigma A T^4$$. If incident radiations are of power $P$, then $$\dot{Q}_{body}=mc\frac{dT}{dt}=P-2\sigma AT^4$$ (the 2 comes into place because it radiates in both directions)
We can integrate this if we want, but we mainly want to qualitatively analyse it. From this equation, we can see that as time passes, temperature will increase to an equilibrium value ($T_{eq}=(P/2\sigma A)^{1/4}$). I am assuming that $P>2\sigma A T^4$ in this analysis (I've never seen an umbrella that feels hotter than direct sunlight). At this equilibrium value, you will receive heat $\sigma A T^4$, i.e., $\frac{P}{2}$.
For a non-black umbrella, assuming no transmitted heat, we have $e=a=1-r$ (coefficient of emission,absorption and reflection respectively). Out of incident heat power $P$, $aP$ is absorbed. The body radiates heat on either face as $e\sigma AT^4=a\sigma AT^4$. So we have $$\dot{Q}_{body}=mc\frac{dT}{dt}=aP-2a\sigma AT^4$$. From this, we get the same value of equilibrium temperature as for a black body. But, radiated heat power is $eA\sigma T^4$, so it will radiate less heat than the black umbrella onto you (Since both $T=T_{eq}$ are the same).
So paradoxically, we get that a black umbrella is the worst thing to use on a sunny day.
IMHO
Black seems to be the majority because of a cultural bias. I'm assuming you're from India (by looking at your name). Since India was once a British territory, we can trace the use of black umbrellas back to the Brits. The Brits used to use black because black was considered formal attire (Black suits are actually the most uncomfortable from the physics point of view). This must have proliferated to India (Most British traditions have). Umbrellas before the British must've been colorful (See the painting here). If you take a look at Japanese umbrellas, they're all colorful. So black is really not the majority worldwide.
Best Answer
As it was explained in one of Halliday's books, the reason is that the black dress heats the air inside it up, thus causing a continuous flow of air in between the skin and the dress. The cold air flows in from below, gets heaten up, and gets out from above, providing a continuous ventilation.