You need a pocket mirror, a laser pointer and a counter. Then measure both the up-down time (or distance) and the down-up time (or distance). The two are different.
This means, taking light-radar as a reliable measuring device, that the two measured heights are different. As far as I know, the experiment has never been done in spite of its simplicity.
Why is it worth doing? This “V-Lambda” experiment can also be called “WM” experiment, with the two letters printed on top of each other. You then get XXXX. Very regularly, no shifts. That is, upper and lower time intervals interlock even though being different.
You can do the same experiment between earth and a neutron star (provided a mirror can be deposited on its surface). Then the two time intervals that interlock differ by a factor of about 2.
So while the tip of the Eiffel tower is only minimally closer to the bottom (upwards distance shorter than downwards distance), the distance of earth from a neutron star is half as large as the distance of the neutron star from earth.
I find that cute.
The strong interlinking between the two levels — top and foot of the Eiffel tower — is a physical constraint in the Einstein equation.
Well this is also known as the Shapiro delay, isn’t it?
The justly famous Shapiro time delay is related in its spirit.
The perfect “intercalation” between the up-down and the down-up light pulses at one place, alluded to above, is a very much easier experiment.
Its extension to a parallel horizontal experiment — both upstairs and downstairs with the right distance between mirrors to choose for a clearcut intermeshing also horizontally — is a straightforward possibility that you just triggered in my mind as being important to be done in reality.
Here, even making a clear-cut prediction as to what will turn out to be the horizontal distance that intermeshes perfectly downstairs (upstairs it is trivial) seems to be open to me right now.
Thank you for the dialog.