Why do we hold umbrellas upright while walking?

Recently I encountered this (quite interesting) question:

Rain is falling vertically. A man running on the road keeps his umbrella tilted but a man standing on the street keeps his umbrella vertical to protect himself from the rain. But both of them keep their umbrella vertical to avoid the vertical sun-rays. Explain.

The answer that I thought is as follows:

The speed of man is comparable to that of the rain and hence his act of walking cause the relative velocity vector to change in direction and magnitude. Whereas his speed is nowhere comparable to that of light (about $$3 \times 10^8\ \mathrm{m\ s^{-1}}$$) and hence there is no significant deviation of the relative velocity vector of light.

But an obvious question that occurred to me was:

• What if the speed of man was quite comparable to that of light (say $$0.9c$$, just to give some figures!) then would there be a change in direction and hence the need of tilting the umbrella?

I think there might not be any deviation at all as the speed of light is invariant in all inertial frame of references (though this conclusion might not be correct as my understanding of SR is quite superficial.)

• Go even faster and it's black but check your unbellra is it UV proof. – user6760 Dec 23 '19 at 8:04
• – PM 2Ring Dec 23 '19 at 15:23
• I tilt my umbrella forward to shield me from walking face first into rain and wetting my head. But incident light rays are of no concern so the umbrella stays vertical to block direct light from heating me. Am I completely misunderstanding the question? – Cell Dec 24 '19 at 3:20
• @Cell the question is about relative velocity and the change in the angle of incidence of light from a moving reference frame. – user249968 Dec 24 '19 at 4:56

This is a simple but important point. While the speed of light is invariant among all inertial observers, the direction of light is not. Let's show this explicitly. Let's say that light is propagating in the negative$$-y$$ direction w.r.t. observer $$\mathcal{O}_1$$ who is standing still. Now, consider an observer $$\mathcal{O}_2$$ running at a uniform speed $$v$$ in the positive$$-x$$ direction w.r.t. $$\mathcal{O}_1$$. The relativistic relation between the velocity $$\vec{u}$$ of an object as observed by $$\mathcal{O}_1$$ and the velocity $$\vec{u'}$$ of the same object as observed by $$\mathcal{O}_2$$ is the following

$$u'_x=\frac{u_x-v}{\sqrt{1-\frac{v^2}{c^2}}}$$

$$u'_y=\frac{u_y\sqrt{1-\frac{v^2}{c^2}}}{{1-\frac{vu_x}{c^2}}}$$

In our case, we are concerned with the transformation of the velocity of light which is $$c$$ in the negative$$-y$$ direction as observed by $$\mathcal{O}_1$$. Thus, $$u_x=0,u_y=-c$$. Putting in these values in the above formulae would give you

$$u'_x=\frac{-v}{\sqrt{1-\frac{v^2}{c^2}}}$$ $$u'_y=-c\sqrt{1-\frac{v^2}{c^2}}$$

Thus, in the reference frame of $$\mathcal{O}_2$$, light makes an angle of $$\arctan\frac{u_x}{u_y}=\arctan\big(\frac{v/c}{{1-{v^2}/{c^2}}}\big)$$ with the vertical. So, indeed, you are correct. If $$v/c$$ is not ridiculously small, the person would have to tilt their umbrella noticeably.

You can easily verify that the speed of the light is still $$c$$ in this reference frame.

• A side comment: I have no idea why HC Verma has included this question in an introductory chapter on simple Newtonian kinematics. I presume that the intention is to intrigue the reader's curiosity in the kinematic behavior of light from an early stage. – Dvij D.C. Dec 23 '19 at 8:11
• However, I can't let go of the possibility that he simply had a blissfully ignorant reader in mind and he wanted the reader to treat light non-relativistically (in which case, you actually arrive at the same answer that if the person is running at speeds comparable to the speed of light then they would have to tilt the umbrella--however, your estimation of this tilt would be different from the correct relativistic result). – Dvij D.C. Dec 23 '19 at 8:12
• Is this similar to the elevator thought experiment that lead to GR (in that a different frame, though in that case a non-inertial one, results in a different direction of light)? – yshavit Dec 23 '19 at 19:53
• AND the colours shift :-) – Russell McMahon Dec 23 '19 at 22:43
• @yshavit Well, no, not really. As you notice, the name of the game is non-inertial observers in that case. It changes everything. In particular, even the coordinate speed of light changes in that case--not just the direction. You might find this old answer of mine partially relevant: physics.stackexchange.com/a/337952/20427. – Dvij D.C. Dec 23 '19 at 23:55