# Length contraction of falling things

1. Jun 17, 2012

### jartsa

1: Does an observer, standing on the moon, see a brick, that is falling straight down, to contract?

2: Does an observer, standing on the moon, see a light pulse, that is falling straight down, to contract?

2. Jun 17, 2012

### GAsahi

Yes.

The answer to this question is much more complicated, the detailed mathematical explanation can be found here, I just posted it a few days ago.

3. Jun 17, 2012

### jartsa

Oh yes, the contraction is proportional to blue shift. And the simplest blue shift case is very simple.

Here's something people here may disagree with:

The contraction happens because the font of the light pulse moves slower than the rear of the light pulse. And that happens because light slows down in a gravity field.

4. Jun 17, 2012

### GAsahi

I don't know where you get this but it is wrong.

You are making up stuff.

5. Jun 17, 2012

### jartsa

6. Jun 17, 2012

### GAsahi

You obviously cannot follow the math. Nor do you understand the physics.

7. Jun 18, 2012

### pervect

Staff Emeritus

If you're trying to understand SR better, dragging GR into it isn't really a good way to proceed. You're much better off handling the simpler case in flat space-time where there's no gravity to confuse the issue.

If you're trying to understand GR better, it would be a good idea to sketch exactly how you propose to measure the length of the object remotely. I can warn you in advance that the notion of "distance" in general relativity is ambiguous, mainly due to differing notions of simultaneity. Some people understand the warning, some don't, and I get a bit frustrated trying to explain the problem to those who don't. This is really a SR issue, by the way, but it's an important one to understand before you try to tackle GR.

I can tell you how in general I would go about "setting up" the problem. The first step is to decide what notion of simultaneity you want to use, and why. So I would start by assuming that the moon is the most significant gravitating body, and that we can ignore the Earth. Right away, I can see a possible conflict with your question, perhaps you didn't intend the moon to be the most significant gravitating body? In that case, the issue of how to determine the notion of simultaneity would require some thought. If the moon is the most significant source of gravity, however, there is an obvious way to proceed, that is to use a clock synchronization scheme compatible with "static observers". It's less obvious how to proceed if the moon is not the most significant source of gravity.

Once you have decided on the notion of simultaneity you're going to use, the simplest procedure is to construct a spatial geodesic curve. (Not a space-time geodesic, but a spatial geodesic!). Constructing this geodesic will require you to use the metric induced on your spatial hypersurface by your space-time geometry. Constructing the geodesic will require some knowledge of the geodesic equations, and how to solve them. Solving them directly is usually difficult, and taking advantage of conserved quantities via means of Killing Vectors or some equivalent procedure, is highly recommended.

Next you need to confirm that the problem that you've set up is really one-dimensional. If it is, then you mark on this spatial geodesic curve the position of the front end of the rod, and another mark for the rear end of the rod, "at the same time", according to your notion of simultaneity. If both ends of the object aren't on a single curve, then your problem isn't one dimensional. But let's assume that it is. Then the length along the curve (the spatial geodesic) between the marks on the front and rear will be the "length of the rod".

I'm afraid I can't give an exact formula for the result you'd get if you followed this procedure, I'd actually have to work it. My own view is that the exercise of thinking about what you want to measure is considerably more helpful than going through the detailed calculations of calculating the number. Furthermore, if you don't understand what all the steps I've outlined mean (and some of them might require a certain amount of expertise), just giving you the number won't really accomplish much.

Last edited: Jun 18, 2012
8. Jun 18, 2012

### jartsa

I could use some new facts in my thought experiment building hobby.

Let's say the light pulse is catched into a box which is just large enough. This is done at many different altitudes, with many light pulses.

Then all boxes are moved into same place and compared.

I would guess the boxes from lower altitudes are smaller, and light in these boxes has more energy.

9. Jun 19, 2012

### pervect

Staff Emeritus
If you raise or lower a meter stick, oriented normally to the gravitational force so the change in stress doesn't cause the length to vary, I can't see any reason for it to change it's length.

So I would say that the same applies to a box, say a microwave cavity filled with one wavelength of microwave radiation. The local physics will be the same, the cavity/box won't change length, and a local frequency counter would measure the same frequency, the constant length of the cavity divided by c, no matter what the height of the box was.

The energy-at-infinity would change as you raised the box, but I think this could eventually be traced back to the work done in raising the radiation in the box. It will take more work to raise a box full of microwave radiation than one that's empty. I'm not aware of ay formal p roof of the matter, but I don't see how it could be otherwise.

10. Jun 19, 2012

### harrylin

People can disagree with anything - and often it very much depends on definitions of words. :tongue2:
But it's a simple fact that if according to a distant observer a stationary meter rod is length contracted and a clock at one end is ticking slower, that then with that reference system the return speed of light can only be measured as c if the light according to the distant observer is slowed down with gamma squared (using a standard and consistent meaning of words).

PS: As Shapiro's phrased it, "the speed of a light wave depends on the strength of the gravitational potential along its path" - http://prl.aps.org/pdf/PRL/v13/i26/p789_1 [Broken] .
See also discussions on "Shapiro time delay" and note that according to GR the rod will not be contracted if held parallel to the surface.

Last edited by a moderator: May 6, 2017
11. Jun 19, 2012

### Staff: Mentor

Similarly to your misconception about clocks slowing down, in GR light (in vacuum) also does not ever slow down in a coordinate independent sense. It always travels along null geodesics.

Even in flat spacetime, it is possible to write down coordinates in which the coordinate speed of light is faster or slower than c, but these are obviously coordinate-dependent statements. Anywhere in any spacetime that you find a set of coordinates where light slows down there are also other sets of coordinates where it does not.

12. Jun 19, 2012

### Q-reeus

Agree entirely with that observation. And it agrees I believe with what #10 was intending to convey. One just needs to be careful in defining precisely the relative nature of 'length change' in such situations.

13. Jun 19, 2012

### Q-reeus

In SC's transverse length in coordinate measure expressly is invariant wrt potential, but not if that rod is radially oriented (and that's with 'stress' subtracted out). But 'radially shortened' is imo only of value in the sense that integrating over an extended radial path r2 to r1 (coordinate measured), there is more proper distance covered in the interval r2-r1 than if gravity were switched off. We all agree that locally no length or time distortions can logically be evident.
Which is distinctly different to OP's scenario - free-falling light pulse measured at different heights by a 'hovering ruler' (the box in effect).
Sure, and that gets back to arguments I was making elsewhere re 'charged BH' and elsewhere - EM field energy associated with charges/currents of a non-free-fall system, is depressed in a gravitational potential by redshift factor. Not so for geodesic propagating light beam.

Last edited: Jun 19, 2012
14. Jun 19, 2012

### harrylin

Size yes, but energy I don't know - that's a tricky topic. My guess would be that the total energy doesn't change.

15. Jun 19, 2012

### HallsofIvy

Even in "thought" experiments, you cannot just ignore basic facts of physics. You cannot "catch" light in a box and move it.

16. Jun 19, 2012

### harrylin

17. Jun 19, 2012

### jartsa

Well, then I guess I must be saying things in a coordinate dependent sense. Is that a bad thing?

18. Jun 19, 2012

### jartsa

Well of cource it changes. Light that falls, and bounces back, does not change. Light that falls, and is lifted back, does change.

19. Jun 19, 2012

### Staff: Mentor

No, it isn't a bad thing as long as you understand what you are doing and are clear about it. Otherwise it leads to confusion, which I believe is reflected in your writing.

Last edited: Jun 19, 2012
20. Jun 19, 2012

### Staff: Mentor

My guess would be the same as harrylin's. I don't think that the "of course" is warranted, unless you have a completely foolproof derivation in three lines or less.

Last edited: Jun 19, 2012
21. Jun 19, 2012

### pervect

Staff Emeritus
It is possible to say things correctly in a coordinate dependent manner, just difficult. For instance, you need for starters to specify what coordinates what you're saying is valid in. This usually leads to rather long explanations. Coordinate independent explanations don't need all the background, so they can usually be much shorter.

So, if you don't like writing a lot, and in great detail, it's a big advantage to talk in coordinate independent terms. It's also easier (for the most part) for the reader to follow.

Of course, it's also possible to say things in a coordinate dependent manner that are just plain wrong. (It's possible to say wrong things in a coordinate independent manner too, of course).

I'll use an example I've used before of saying something that's just plain wrong. Suppose you're talking about length contraction, in the context of the surface of the Earth, and you start saying that "distances are smaller near the poles". Hopefully, it's obvious that this is just plain wrong.

But lets examine what you might be thinking, a related concept that isn't wrong.

Now, you might be thinking in a coordinate dependent manner, and what you are really trying to say is that one arc second of longitude near the equator is longer than one arc second of longitude near the poles. This would be correct, unlike saying that "distances are shorter near the poles".

The fundamental mistake that was made here is conflating (confusing and combining) differences in coordinates, i.e. changes in lattitude, and distances. The two concepts are different. The first concept is not a distance. You can, however convert it to a distance by using the local metric coefficients. After you've performed the appropriate connection, it becomes a distance.

I often see the same type of error made in relativity, in relation to the Schwarzchild R coordinate. Conceptually, it's a coordinate, not a distance. It doesn't become a distance until you apply the metric to it.

22. Jun 20, 2012

### harrylin

It isn't really coordinate dependent, however it's "reference frame" dependent. And while some people seem to think that that's a bad thing, others don't. For example, many physicists don't think that it's a bad thing to talk about "high energy" electrons. Of course, in such cases the reference is clearly implied.
A material object "falls" by accelerating towards the gravitating body, and in that process we can imagine that potential energy is transformed into kinetic energy. I think that the total energy of such a two body system in isolation must remain constant, as determined with a non-local reference inertial system. In contrast, and as you suggested yourself, light that propagates directly towards the gravitating body does not accelerate towards it but decelerates; moreover it looks to me that it has only kinetic energy which should remain constant from that perspective as long as the box is not moved (I could be wrong there, but for sure its frequency is unchanged). Next, moving the box demands more analysis.
So, in view of those points, please elaborate your thought experiment - which, it seems, has little to do with your topic. :tongue2:

Last edited: Jun 20, 2012
23. Jun 20, 2012

### jartsa

We can drop light into a box, lift the box, take light out, move box back down, drop the light into the box, lift the box ... energy is used, it must go somewhere, the light is the only alternative.

UNLESS light is weightless. If light is weightless form of energy, we can very easily build a perpetual motion machine. So it's not weightless.

Let's define weight as exchange of momentum with a gravitating object. We probably agree that light bends in a gravity field, and momentum change happens when light bends.

What happens to light that moves straight up or straight down ... that's an interesting question.

24. Jun 20, 2012

### Naty1

I think you mean some basic facts from which to draw conclusions and make thought experiments?

A few to get started:

[1] All local clocks [of good quality] tick at the same rate. A clock here and a clock there tick at the same rate, but neither may appear to do so to a DISTANT observer. Another way to say this is that clocks record proper time, and that is not dependent on the coordinates used, nor the path you take. Any clock you carry with you records your local age and local processes along your worldline...your path through space and time. In GR, clocks don't slow down in any coordinate-independent sense; they measure proper time along their worldline, your worldline if you are carrying the clock, which is an invariant quantity.

[2] The local speed of light is always c. This means light always travels at the same speed 'c' right where you are. The speed of light as observed in curved spacetime varies, that's general relativity, but right where you are [locally] spacetime is flat and light is always observed at speed 'c'. This seems somewhat 'crazy' at first because no matter your speed relative to anything else, light still zips by you at the constant 'c'. That idea takes a while to get used to.
In special relativity, meaning flat spacetime, all inertial observers see light at speed 'c'. This means distant and local observers. But for accelerating observers, things change... observations vary.

Last edited: Jun 20, 2012
25. Jun 20, 2012

### Naty1

Jartsa:
WHOAAA!!!! You have way too much stuff going on...ambiguous terminology.....too many conflicting ideas with erroneous statements and conclusions.....slow down, dude!!!! seems like ideas pop into your head and you write them down.....instead, take ONE idea and give it some thought as to the consequences of what you are posting ....

First off; light has no mass...but you cannot willy nilly draw conclusions from that such as 'no mass [no weight] so no work' with your box questions .....and 'no mass' does NOT mean because it is 'weightless' you can build a perpetual motion machine...that's mostly gibberish. [Energy is equivalent to mass via E =mc2 and light has energy; this means it is gravitationally attractive like everything else we know.]

So before we talk about light in boxes going up and down let's look at a few basics:

[These have been fully discussed elsewhere in these forums, so if I can remember them, they'll be correct and maybe we can even find all the detailed discussions if required]

Let's say a bunch of observers have identical light sources...all have the same reference color [frequency]. A series of local observers at different gravitational potentials, say at fixed distances in a uniform gravitational field, would each subsequently observe a photon passing their own location to have different red shift [different energy, different color] relative to a local reference photon. Each measures velocity 'c'. [from my post above]
This means the kinetic energy of observed photons has changed....such KE increases as gravitational potential decreases. Light further down becomes blue shifted.

[Different coordinate perspective: From the viewpoint of a free falling observer at rest in the freely falling frame, there is no "gravitational field". She is weightless, feeling no force, so to her, local physics looks the same as in free space with no gravity. Photons in free space with no gravity do not redshift. So the motion of the observer is critical to what KE they measure.]

Another example: Suppose a flashlight shines down into a deep gravitational well with a lead shield at the bottom: could it penetrate the lead shield?

Answer is yes: The light picks up kinetic energy [is blueshifted] and loses potential energy as it falls: that means a flashlight beam becomes like gammas rays [lots more KE if the gravitational potential is strong enough]. So in theory it can punch thru the shield.

I think this is also accurate: Another view: If you move a lead shield rapidly enough against a flashlight beam, the beam can also in theory punch through that shield: the shield 'sees' the light as extremely energetic....really short frequency.....these views are equivalent....

Ok: so what do you conclude will be the characteristic of such light 'boxed' by different observers. Right where they make the observation. Then maybe we can figure what happens when the light is removed outof the gravity well??

Last edited: Jun 20, 2012