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I’ve been trying to grok the “Bell’s Spaceship Paradox” lately. Unless you REALLY know your special and general relativity, that one can really get your head spinning !! It’s a mental experiment meant to show that your intuition can be confounded by space-time relativity in more ways than you thought.

PART ONE — THE PARADOX

Bell’s Spaceship Paradox starts off with two spaceships that have equivalent weight and configuration. They will blast off and accelerate away from you, the observer, in your “frame at rest”. You have a ruler, which you use to measure the lengths of the ships and the distance between them just before they leave. Each ship has the same kind of rocket firing, and feels exactly the same force for the same length of time.

Oh, and the rockets are moving longitudinally away from you along the same line, one in front of the other. They are not side-by-side, they are moving in line. This is an important detail that isn’t always made clear in the layman’s explanations of this problem that you find on the web. Anyway, you have to imagine that the thrust from the lead rocket somehow does not affect the rocket behind it. Well, we can imagine some futuristic arrangement where that might be possible.

These rockets can go really, really fast, approaching the speed of light. And you know that weird things happen at that point. Basically, the two ships start out separated by certain distance that you, the observer, have measured before they gun their engines and accelerate (once again, in the same direction and with the same force). To make this more specific, the separation distance is between the centers of gravity of each ship, basically somewhere in the middle of the ships. Once these ships get up near the speed of light, the observer in the rest frame (i.e, you) gets the chance to make another observation of these ships from your “rest frame”, while the ships are flying along at near light speed (relative to you). You make this observation with some kind of really high-speed camera using another measuring stick. This second stick, along with the camera, is also at rest relative to you.

(How this would be done is usually not specified very well in the various explanations of this problem on the web — another problem with most layman’s explanations of Bell’s Spaceship Paradox. Maybe there is a set of remote sensors way out in space somewhere along the path of the two rockets, and the measuring stick and camera are attached to them. Thru radio signals you know that these sensors are not moving relative to you, they are in the same frame, i.e. the “rest frame”. Thru those remote sensors, the observer at the starting point — you — can measure the two rockets as they speed by at near light speed, using the super-camera and measuring stick. Well, let’s imagine that anyway, and get on with the thought experiment.)

By the basic laws of physics, the observer in the rest frame (you) expects that no matter how far and how fast the two rockets are from you, they would stay just as far apart as when they started, so long as the rockets were exactly the same and experience the same forces over time. And according to relativity theory, this works out — that is what you would see from your remote observation camera and ruler as the two rockets zoomed past.

Here is where it gets weird — suppose that there were a string or rubber band connecting the centers of the two spaceships. Let’s say that this string or rubber band runs longitudinally in the direction of acceleration and travel of the two ships. (That fact is once again quite important, but not usually made clear in the layman explanations on the web). The question is, does the string break or not; or if a rubber band, does the band get stretched out as the rockets accellerate from rest in your observation frame to near light speed? You know that the distance that you would measure between the two ends of the string or rubber band have not changed despite all the acceleration of the two ships. Does that mean that the string would not break or the band not get stretched?

NO! Relativity says that a string WOULD break or a rubber band would get stretched while and after accelerating to near-light speed (and you could see that as the ships zoom past your remote observation point). This has something to do with the fact that each ship and the string or rubber band itself have their own “time-space frames”, and since they are moving very fast relative to your resting frame, their length compresses in the direction of their motion (relative to you and your frame).

OK, fine — but then, why did not the overall distance that I see between the centers of the two ships then also compress and shrink, since that space moves along in the same direction of acceleration and motion, with equivalent velocity?

Here are some sites that attempt to give intuitive explanations of this mental exercise and the paradox involved —

www.youtube.com/watch?v=lQps_kPjgTQ

www.physics.utoronto.ca/~aephraim/201/Lecture14-30Oct12-del.key.pdf

Most sites that deal with this problem, including the Wikipedia site, don’t even try to explain this in intuitive layman’s terms, they just stick with the formal math equations. This paradox is not all that easy to convey without the math — and it takes a while to get trained in understanding the math. Most busy laypeople are NOT going to take the time to do that.

PART TWO — MY OWN EXPLANATION

But I am now going to make my own attempt to intuitively explain the Bell Spaceship Paradox without any math (which I don’t fully understand either). In order to do that, I am going to re-imagine the thought experiment, and eliminate the rubber band or string. They seem to make the whole thing more confusing. I’m going to attach a long, rigid ruler to the top of each spaceship, weird though that might seem, and let the point be made by observing that ruler.

Therefore, here is my set-up: In the drawing below, we have two spaceships — a red ship and a blue ship (yea, OK, they more like boxes than spaceships, but this is the future, so who knows what spaceships will look like!)

.
Both ships have a ruler attached to them up top, starting at their center of gravity (let’s assume that this is the mid-length of the ship). The rulers are marked “0” at the center of gravity of ship. These rulers are a part of each ship, they are rigidly attached.

These rulers are longer than the ships, but that is OK, since we imagine that these ships are already floating in space; there is no air or earth’s gravity to mess them up. The center of the red ship is 8 units from the center of the blue ship. We see another ruler at bottom, our own ruler in our rest frame; our ruler confirms the measurement from the red ship ruler, i.e. 8 units.

Next, let’s assume that both red and blue ships start their propulsion rockets at exactly the same time and move exactly in the same direction from the rest frame that you are observing from. Let’s once again assume that the ships have exactly the same weight and configuration, and their engines provide exactly the same force to both of them. The engines are somehow pointed so that the thrust from the blue ship does not affect the red ship, does not cause any drag or opposing force to it. So both ships feel exactly the same force for the same length of time.

At the moment that the ships start moving and run past our stationary ruler, the centers of both ships are 8 units apart, and they appear to stay that way as the two ships move out — given that the ships weigh the same, have the same mass configuration (including rulers at the top of the ships), and are feeling the same equivalent forces.

Let’s also say that these ships have really powerful engines from the future, and can get up to light speed in relatively short length, say only a few million miles, like going from earth to Jupiter. Let’s say they are going straight ahead, and we know the path they will be on — no gravity curvatures along the way. So out at 5 million miles or so, we have an observation post containing a ruler, one that is equivalent to the one we had at the starting point. We can use this ruler to measure the length between the ships, using the camera that we previously discussed. And we can verify that this ruler and camera are stationary with respect to our first ruler, thru the radio signals from the observation post sensors.

By this point, the two ships should be getting up past half the speed of light, they are really moving. They might still be accelerating at this point, or they may have turned off their engines to cruise along at say 3/4 light speed. The final outcome is the same; acceleration at the time when you photograph them out past Jupiter or whereever is NOT the issue (as I had at first thought). BUT, the acceleration “experience” of the ships leading up to that point are indeed important to the paradox.

So here comes the two ships, and you the observer have your super-high speed camera (or maybe a radar device), that will give you images of the ships as they pass your stationary-frame ruler deep in space. When the measurement images get to us, we are not surprised that both ships appear to have contracted along the direction of motion — as per special relativity. And since the rulers at the tops of the ships are a rigid part of each ship, these rulers have also contracted. It looks like 8 units on the ship rulers are now as long as 4 units are along our stationary ruler at bottom.

Next, we remember that the red ruler originally showed the two centers of the ships to be 8 units apart, just before they left. BUT NOW, the red ruler ends right about half way between the centers of the ships. It looks like the centers are now about 16 units apart, from the perspective of the rulers attached to these fast-moving ships.

BUT — from our ruler at rest in our Jupiter observation post, we see that the centers of the two ships are STILL 8 units apart. That’s what we expected from the basic laws of physics regarding force and mass and distance and acceleration and speed.

SO — the sentient beings onboard the blue and red ships would measure the distance between their two ship centers to be 16 units at this point as they fly along near light speed. They still measure their own ship lengths to be about 4 units, just as before they started moving. Everything about their ships looks normal to them, nothing seems to have changed. BUT — they both think that their two ships have drifted apart; they were 8 units apart from center to center when they left, but they are now 16 units apart. Remember, the rulers on these ships look all scrunched up to us from our rest observation frame; but to them, their rulers look just fine.

Thus, to us, the observers at rest, the ships never varied in their center-to-center distance — they started at 8, and at our Jupiter observation post, we still measure their center-to-center distance at 8, even at their very high speed.

BUT – why do the ships and their rulers scrunch up in the direction of travel as they move faster and faster (relative to us), and thus the distance between the centers of the rockets appears greater when measured by their scrunchy rulers — and yet, the center distance stays exactly the same for us with our stationary rulers, no matter how fast they go?

BASICALLY — this is because the ships are in a different frame from our rest frame, AND BECAUSE EACH SHIP IS IN ITS OWN SEPARATE FRAME. They are NOT in the SAME COMMON FRAME!! You might think that they SHOULD BE, since they started very close together and had the same forces against the same weight and weight configuration.

Relative to us, in our own rest frame, the RELATIONSHIP between the red and blue ship frames regarding their center-to-center distance STAYS THE SAME, even as they accelerate and move very fast relative to us. Relative to our rest frame, the basic laws of physics apply — i.e., the same mass and acceleration keeps the two ships at the same distance as when they started.

But to them, THAT RELATIONSHIP CHANGES! Because of that small difference in the position where they started from, because they are experiencing their own acceleration from their own starting points, a significant change happened to their own overall relationship, even though everything else was exactly the same for them. (Remember, their rulers now look scrunched to us, but to them, they are just fine!).

PART THREE — BUDDHIST PERSPECTIVES?

OK, so that’s my attempt to explain this. BUT, I’m not going to stop here; I want to ponder something more: i.e., what does all of this stuff about different frames and relative ways of experiencing things say about our own daily world of human relationships? What might it teach us about our lives and our own little worlds?

Well, ponder this. You might sometimes THINK that another person shares your exact frame of reference because of all the things and experiences that you have shared together; and thus, you conclude that things between the two of you are the same as always, and will so remain. BUT NO!!! WE ARE ALL IN OUR OWN INDIVIDUAL FRAMES !!! AND THE RELATIONSHIP BETWEEN THOSE FRAMES CAN CHANGE EVEN AFTER MANY, MANY COMPLETELY COMMON EXPERIENCES!!!

Even if all of our experiences along the way are exactly the same, just a small difference in where we started from will cause something between us to change, as our shared experiences unfold. We think that shared experiences will draw us closer, OR AT LEAST KEEP THINGS FROM CHANGING BETWEEN US. But they might draw us apart eventually, as with the 2 spaceships, given that we have those seemingly tiny differences in where we started out from.

Although once again, it depends on perspective — from the moving FRAMES, it seems clear that they have drifted apart. But if they could somehow talk with the stationary observer who photographed them as they whizzed by, they would be told NO, NOTHING HAS CHANGED BETWEEN THE TWO OF YOU!!!

Who is right? Which frame is right? Which represents reality, and which could be called an “illusion” (as the Buddhist people like to say in their great wisdom)? The Spaceship Paradox suggests to me that “Reality is Relative to the Relationship”. The 3 R’s !! Another implication pertains to the Buddhist “doctrine of no-self”. Basically, the Buddhists teach that we believe to possess individual selves, but that’s ultimately an illusion; what we think of as “ourself” really isn’t all that important in the grand scheme of things.

But to me, the Paradox suggests that we DO have a “fundamental self”, given the individual frames that each observer occupies in spacetime according to the physics of relativity. We get unique “frames” that affect all of our relationships. The characteristics of our “frames” are set by our past, by what we’ve been thru, what we’ve experienced. In Buddhist terms, you might say that our frame is defined by our unique “karma history” or “karma train” over time. Yea, the Buddhists like to say that individual frames (i.e., the self) are ultimately an illusion; but the spaceship paradox shows that our past, our “accumulated karma”, helps to set the terms of how our future relationships go. And that is ultimately very real!

In the thought experiment, different past histories i.e. different karmas would put the spaceships at different distances and orientations when they blast off from the rest frame. Thus, they would experience different separations and separation measurement, even if their karma-distance is tiny. Their ongoing relationship as they both experience acceleration would vary, based on how they start their relationship. And where and how we start our own relationships is very much a function of where we’ve come from, how our karma has accumulated over time (and likewise for our relationship-partners). Buddhism may need to learn some new lessons from science about the importance of relationship, and the effect upon on such relationships of the accumulated identity (i.e. “self”) of the relationship partners. Relationship gives the individual its identity, its life in the universe; but the individual gives the relationship its character!

The bottom line is that Buddhism, despite its great love of paradoxes, might still be in for some unanticipated and paradoxical revelations about reality from science. Despite claims from Buddhist defenders that the Buddha-world completely embraces modern science and is fully consistent with it, Buddhist teachings will be shown to be a bit too simple, and sometimes even a bit wrong and in need of revision.

Thus far I have not encountered any wise Buddhist teachers who dare to challenge the traditional notions of “the dharma” based on what science tells us about reality. I’m not sure that Buddhism is going to do a whole lot better than Christianity and the other “Abrahamic religions” did when early cosmology (Galileo, Copernicus) and 19th century biology (Darwin, etc.) started to contradict their ancient teachings. In order to survive, most forms of Christianity learned to adopt their teachings over time. It is now the Buddhist’s turn to start embracing their own unanticipated “paradoxes from space”, courtesy of scientific teachings such as Bell’s Spaceship Paradox!

◊   posted by Jim G @ 7:46 pm      
 
 


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