General Relativity Question

[phildonnia]

He rigs a system whereby as soon as the tail of the ship passes by the South doorway an intense laser will activate which can cut through the ship, in the plane perpendicular to the path of the ship.

"As soon as" implies an absolute simulteneity, which is one of the casualties of special relativity.

The distance between the events "Laser fires" and "Ship hits back of barn" is interpreted by the ship as some extra time between "laser fires" and "ship hits back of barn".

Here's my space-time diagram of the situation:

 

[Beerina]

Well, forget about sensors and lasers.

Suppose the far barn door is simply closed, and the ship physicist flies into the barn anyway. Suppose the barn is 90 meters long with the ship 100 meters. Thus the relative ship size is 50 meters, with 40 meters to play with.

As soon as the tail of the ship goes in (need only observe the head entering, the rest is math and no measurements necessary) he closes the entry door to the barn.

Of course, I don't really understand why this would be a problem. If it shortens, it shortens, end of story. If the ship stopped suddenly (thanks, Larry Niven!) it would expand and bust out the barn doors?

 

[Skeptic]

As soon as the tail of the ship goes in...he closes the entry door to the barn.

Well, there's your problem: two events that are simulaneous in one reference frame (the barn's) are not necessarily simultaneous in another (the ship's). "As soon as"--e.g., simultamous--is a meaningless statement without telling us in which reference frame you're in.

If it shortens, it shortens, end of story.

Er, no. The ship only shortens according to someone in a specific reference frame. Like "As soon as", "it shortened' is meaningless unless one tells you in what reference frame one is.

If the ship stopped suddenly (thanks, Larry Niven!) it would expand and bust out the barn doors?

No. Why, however, is more complicated. For starters, special relativity only deals with inertial systems, not with stopping or starting, which imply acceleration.

 

[Walter Wayne]

Well, there's your problem: two events that are simulaneous in one reference frame (the barn's) are not necessarily simultaneous in another (the ship's). "As soon as"--e.g., simultamous--is a meaningless statement without telling us in which reference frame you're in.

In this case they be simultaneous to both observers. The tail of the ship is at the entrance, and the barn door at the entrance closes Thus distance between tail and closing entrance equals zero, time between tail cpassing entrance and door closing is zero. Therfore the interval, s²=x²+t², is zero. Given that interval between events is zero, the time passed as observed by everyone is zero.

Walt

 

[Ziggurat]

As soon as the tail of the ship goes in...he closes the entry door to the barn.

Well, there's your problem: two events that are simulaneous in one reference frame (the barn's) are not necessarily simultaneous in another (the ship's). "As soon as"--e.g., simultamous--is a meaningless statement without telling us in which reference frame you're in.

No, that statement's fine, because the event in question (closing the barn door and the tail of the ship passing the threshod) happen at the same spot in space-time.

If the ship stopped suddenly (thanks, Larry Niven!) it would expand and bust out the barn doors?

No. Why, however, is more complicated. For starters, special relativity only deals with inertial systems, not with stopping or starting, which imply acceleration.

Actually, this is wrong, it's a common misperception. Special relativity can handle acceleration just fine (though the math can get ugly), it's just gravity that it can't handle. And yes, as a matter of fact, the ship WOULD try to "expand" as it stopped - information doesn't travel through a solid instantaneously, it's limited by the speed of sound in the material, and one consequence of SR is that the speed of sound in a material cannot exceed the speed of light. Which means in SR, everything is by necessity at least somewhat squishy, you're not even allowed to talk about something infinitely rigid.

 

[69dodge]

Originally posted by Ziggurat
Special relativity can handle acceleration just fine (though the math can get ugly), it's just gravity that it can't handle.

How would this work? Isn't the whole idea of general relativity that acceleration and gravity are indistinguishable? (Well, locally, anyway. Is that the crux of it?)

I wouldn't be surprised if this isn't the sort of question that can be answered in a few sentences. But it can't hurt to ask, I suppose.

 

[(S)]

It is my understanding that Special Relativity is the special case of General Relativity where you assume that space is flat, that there are no gravity wells.

 

[epepke]

It is my understanding that Special Relativity is the special case of General Relativity where you assume that space is flat, that there are no gravity wells.

Or when there's no acceleration. Although a lot of problems with acceleration can be solved by SR alone, including the so-called "twin paradox."

 

[Ziggurat]

How would this work? Isn't the whole idea of general relativity that acceleration and gravity are indistinguishable? (Well, locally, anyway. Is that the crux of it?)

I wouldn't be surprised if this isn't the sort of question that can be answered in a few sentences. But it can't hurt to ask, I suppose.

Well, I'll give a partial answer in a few sentences. First, the equivalence principle that you make reference to is the idea that standing still in a *uniform* gravitational field is indistinguishable from accelerating. But gravity is not uniform, and SR can't handle that non-uniformity.

In more than a few sentences: Take it as a given that special relativity can handle acceleration without gravity (trust me, it can). It shouldn't be too hard to see that this is going to cause serious problems, though, when we try to treat gravity with SR, at least on a global scale. Consider, for example, two people standing on opposite sides of a gravitational body (forget rotation). Now we consider treating them both as if they're accelerating. In our SR treatment of the problem, they should be accelerating away from each other, but of course they aren't, they're stationary with respect to each other. Uh-oh. We've run into a problem, because gravity is pointing in different directions in different places, and we can't just patch our local SR descriptions to each other and make a globally coherent picture of what's going on without some serious tinkering. And that's where GR steps in.

I haven't proved here that SR can handle acceleration (but it can), though I think you may be willing now to accept that handling gravity is more difficult than handling acceleration.

 

[Gestahl]

Gestahl,

The people here have an ...shall we say acerbic wit? If You had approached the question from a different direction rather then a challenge wrapped in a riddle , and asked for help. There are world class specialists that subscribe to this board and they would have been happy to help. There are also other boards that help with homework like http://www.physicsforums.com/index.php?. The hints and scenarios that people have stated here are BTW correct information.

Fair enough, I have quite a large one myself. That does not excuse people being rude, and implying I am lazy. I have a very short temper for people who are both not helpful and scornful. Don't waste the web-space. Plus, he *assumed* that I knew less than I did, and I really hate that.

 

[Gestahl]

Re: Re: General Relativity Question

A photon isn't a valid inertial reference frame. Not to say you're wrong, more that it's just not really relevant to anything.

No but I do find it useful to keep things straight about which directions things deviate... the universe shortens around you and time dialates.

Plus, it is just idle musing...

 

[Gestahl]

Or when there's no acceleration. Although a lot of problems with acceleration can be solved by SR alone, including the so-called "twin paradox."

Actually, SR can solve *constant* acceleration problems (note 0 is still a constant), just as it can solve *constant* gravitaitional field problems, I am pretty sure. GR is just where you make the accelearation/g-field a parameter, rather than a constant.

 

[Ziggurat]

Actually, SR can solve *constant* acceleration problems (note 0 is still a constant), just as it can solve *constant* gravitaitional field problems, I am pretty sure. GR is just where you make the accelearation/g-field a parameter, rather than a constant.

No, that's still not correct. Acceleration that is changing in time can still be handled with SR, but that's not at all equivalent to spatially varying acceleration in GR.

Perhaps a better example is what happens WITHOUT acceleration (acceleration is just changing inertial reference frames). If standing still in a uniform gravitational field is like accelerating, then freefall in a uniform gravitational field is like no acceleration. A falling reference frame is therefore an inertial reference frame. Nothing strange is that gravitational field is uniform, but if it's not, then trajectories for different inertial reference frames (say, different orbits around a planet) aren't going to make sense in SR. To handle it, you need to use a non-flat space-time. But varying accelerations in flat space-time can be handled quite directly in SR.

 

[TeaBag420]

ummm, acceleration is BY DEFINITION "changing in time". Are you differentiating between change in speed and change in direction?

No, that's still not correct. Acceleration that is changing in time can still be handled with SR, but that's not at all equivalent to spatially varying acceleration in GR.

Perhaps a better example is what happens WITHOUT acceleration (acceleration is just changing inertial reference frames). If standing still in a uniform gravitational field is like accelerating, then freefall in a uniform gravitational field is like no acceleration. A falling reference frame is therefore an inertial reference frame. Nothing strange is that gravitational field is uniform, but if it's not, then trajectories for different inertial reference frames (say, different orbits around a planet) aren't going to make sense in SR. To handle it, you need to use a non-flat space-time. But varying accelerations in flat space-time can be handled quite directly in SR.

 

[Ziggurat]

ummm, acceleration is BY DEFINITION "changing in time". Are you differentiating between change in speed and change in direction?

No, that's not correct. Let's run through our definitions.

x = position.
dx/dt = velocity. if dx/dt=0, then position is constant.
d^2x/dt^2 = acceleration. If d^2x/dt^2 =0, then velocity is constant.
d^3x/dt^3 doesn't have a name, but if d^3x/dt^3 = 0, then acceleration is constant.

Changing directions is changing acceleration (these quantities are all properly vectors), but even in one direction, acceleration can be constant or it can be changing.

 

[Gestahl]

No, that's not correct. Let's run through our definitions.

x = position.
dx/dt = velocity. if dx/dt=0, then position is constant.
d^2x/dt^2 = acceleration. If d^2x/dt^2 =0, then velocity is constant.
d^3x/dt^3 doesn't have a name, but if d^3x/dt^3 = 0, then acceleration is constant.

Changing directions is changing acceleration (these quantities are all properly vectors), but even in one direction, acceleration can be constant or it can be changing.

I think that x''' (x triple prime, easier notation for web) is most normally described as "jerk" or "shock". I.e. the more the change in acceleration, the harder the "jerk". In fact:
http://math.ucr.edu/home/baez/physics/General/jerk.html]
here[/url] is a decent refrence, and evidently is coverd by an ISO standard. I thought there was a name for it, but didn't bother to look it up till today... learning is fun!