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Einstein’s Thought Experiments

October 2, 2015

 

Albert Einstein loved thought experiments. He famously arrived at both Special and General Relativity not through the usual mathematical route, but through the much more accessible logic of the gedankenexperiment. As a result, Einstein’s own explanation of both theories is only a few pages long; anyone can understand them and appreciate how they were derived.

 

We will use thought experiments liberally in these pages. We’ll follow a few simple but strict rules. First, we start with a statement on which everyone can agree. This is an axiomatic “water is wet” kind of truism that grounds the foundation for the logical progression of the experiment.

 

We then layer further statements on top, with the proviso that everyone must agree that any new statement logically flows from the previous, and does not contradict known science. Bit by bit, we progress until we’ve reached a perhaps surprising truth.

 

Now, the economy of not actually performing the experiment in real life comes with a downside. You can get it wrong. There’s no Universe pushing back at your impeccable logic. Fortunately, there’s always a curmudgeon in the back row who raises his or her hand and objects. We must listen to this person, for they keep us honest.

 

Speaking of which, there is a glaring hole in Einstein’s logic that I think tarnishes the brilliance of his conclusion. He should not have proceeded past a certain step, and I raise my hand as a curmudgeon to object. More on that in a bit.

 

Let’s take a look at how Einstein came to the General Theory of Relativity, arguably the greatest scientific achievement of the 20th century. Einstein’s gedankenexperiment is not long or difficult; we can summarize it here. I’ll embellish it a bit and I’ll also point out the hole in the logic.

 

Why Space Curves

Imagine yourself in an elevator. (Note the “water is wet” beginning – nice!). The elevator is motionless on the ground floor. You stretch out your hand and let go of an apple (I think apples are de rigueur when talking about gravity; it would be so wrong to use oranges or rocks, don’t you agree?)

 

The apple falls, of course. More precisely, the apple accelerates at 9.8 meters per second squared towards the center of the Earth. (Ok, the common center of gravity of the Earth-apple system – don’t write to me about it! And that’s about as much math as you’ll get in these pages, so quit complaining.)

 

Now imagine yourself in that same elevator, but transported to outer space, far from any planets, stars, or other gravitational influences. You stretch out your hand again and let go of the same apple. Of course, the apple doesn’t fall. It floats, as do you and all your possessions, because there is no gravity. You can immediately tell where you are, on Earth or in space, by the lack of gravity. So far so good.

 

Now, what if we hooked up a rocket to the elevator and started to accelerate it, with you inside, at precisely 9.8 meters per second squared? You again let go of the apple and the apple still floats in space, but the floor is now accelerating towards the apple at 9.8 m/s2. To you, floor and apple meet in short order – the apple falls!

 

The acceleration of the elevator in space is therefore indistinguishable from the acceleration of gravity on Earth. There is no way to tell if you’re in space or back on Earth. Read the last two paragraphs again if you’re not clear on this idea – it’s critical.

 

Now, imagine shining a flashlight “horizontally” from one side of the elevator to the other. The light consists of photons – particles of light traveling at finite speed. While these photons are in transit, the elevator is accelerating up, faster and faster. You observe the photons starting at the level of the flashlight, but by the time they get to the opposite wall, the floor has moved up and closed some of the distance. To you, it looks like the light beam is drooping towards the floor.

 

Now, since we already said that you can’t tell whether you’re in space being accelerated by a rocket or in Earth’s gravity sitting motionless, therefore – and this is Einstein’s infinitely audacious move – light beams that appear to droop in an accelerating space elevator will also droop in a gravitational field.

 

So Einstein predicted that this effect could be measured; light passing by a sufficiently massive object, such as the Sun, should be bent. Observe the stars behind the Sun during a total eclipse and you’ll see them slightly spread apart, because the Sun’s gravity has “bent” space and caused light to travel in a curved path.

 

Once this experiment was confirmed (in 1919!), Einstein’s fame rocketed, and today posters of his disheveled hair with some corny philosophical musing is required wall adornment in college dorms.

 

But Einstein cheated. He made a leap that is not supported by science or the logic of the thought experiment. There’s actually a very easy way to tell if you’re in an accelerating elevator or a gravitational field: tidal deformation. Stand in a gravity field and your feet get pulled more than your head. On Earth, the difference is small, but try standing on a neutron star and your feet will disappear, Wile E. Coyote-style, well before your head comes crashing down.

 

Another way to tell is to drop two apples, one in each outstretched hand. If your hands are long enough (and the elevator wide enough) you will see that the two apples don’t fall in parallel lines. They converge slightly, because the floor is closer to the center of the Earth than your hands.

 

But in an accelerating space elevator, there is zero tidal deformation, and apples fall in precisely parallel lines no matter how far apart. As far as I’m concerned, Einstein’s most famous gedankenexperiment should not have proceeded past this point.

 

A century later, no one has ever disproved any part of Relativity. In spite of many attempts, no experiment has shown any discrepancy between theory and observation. Even the Bell Theorem (Google it, kids) does not give permission for faster-than-light communication. Einstein still stands untouched today. But he got away with a whopper in 1919.

 

Afterthought

Einstein has one thin thread to hang on to. If we assume objects are point masses, the problem goes away, because we have confined the experiment to an infinitely small volume of space. (Einstein sidestepped the issue by postulating a "homogeneous gravitational field", which doesn't exist; it's the philosophical equivalent to a point mass).

 

Niels Bohr, the quantum physics guy, might have had a few thoughts on how to use this to join the realms of the very large (gravity) with the very small (individual particles). Perhaps the seed of reconciliation between Relativity and quantum physics will be found in this question – does an electron experience gravitational tidal stretching?

 

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