Spacetime transformations or not?

[matheinste]

Hello all.

I asked this question as a sub-question in another thread where it was perhaps inappropriate. It is very basic but the more i try to understand relativity the nearer to the absolute basics i need to go. The more i learn the less i seem to actually understand.

When length and time are transformed between inertial frames in relative unaccelerated motion are the underlying dimensions of spacetime transformed. Does the way i have put the question make any sense. Of course this question leads naturally on to other questions regarding the "reality" of length contraction and time dilation but i'll take it one step at at time.

Thanks Matheinste.

 

[HallsofIvy]

What do you mean by the "underlying dimensions of spacetime"? I think the best answer to your question is that there are no "underlying dimensions"! One can, of course, calculate the "proper length" of the trajectory in spacetime. How one measures length and time from that would depend upon the frame in which they were measured.

 

[matheinste]

Hello HallsofIvy.

I mean when we carry out a Lorentz transformation between frames do we transform the basis vectors, or axes, of the one frame to get a (different) set of basis vectors or axes for the other frame. Spacetime diagrams suggest geometrically that this is the case.

Matheinste.

 

[JesseM]

I mean when we carry out a Lorentz transformation between frames do we transform the basis vectors, or axes, of the one frame to get a (different) set of basis vectors or axes for the other frame. Spacetime diagrams suggest geometrically that this is the case.

The axes are rotated relative to one another, if that's what you're asking. A timelike path with constant spatial position in one frame would have a non-constant position in the other (this is true of the Galilei transformation in Newtonian physics too, just because different inertial observers are in motion relative to one another and each defines themselves to have a constant position in their own rest frame), while a spacelike path with constnat time coordinate (or an entire plane of simultaneity with constant time-coordinate) in one frame would have non-constant time-coordinate in the other (this is different from the Galilei transform, where different frames don't disagree about simultaneity).

 

[matheinste]

Hello JesseM.

Am i correct in thinking that the Lorentz contraction and time dilation which result from a frame being in relative inertial motion with respect to another frame, are due to the measurement in the moving frame as seen from the "staionary" frame being referred to these transformed axes. Or are the changes in dimensions of an object, and its local clock rate a direct result of the transformation itself.

I hope i have phrased this in a meaningful way.

Matheinste.

 

[JesseM]

Am i correct in thinking that the Lorentz contraction and time dilation which result from a frame being in relative inertial motion with respect to another frame, are due to the measurement in the moving frame as seen from the "staionary" frame being referred to these transformed axes. Or are the changes in dimensions of an object, and its local clock rate a direct result of the transformation itself.

Well, if we assume that identical physical clocks at rest in different frames will each have the same amount of coordinate time between ticks in their rest frame, and that identical rulers at rest in different frames will each have the same coordinate length in their rest frame (both of which follow from the first postulate of relativity which says all laws of physics should work the same in every frame), then both the time dilation equation and the length contraction equation can be derived purely from the Lorentz transformation which relates the coordinates of different frames (and which is where we get the notion of one frame's axes being skewed relative to another frame's axes). Not sure if this is what you meant by "changes in the dimensions of an object, and its local clock rate a direct result of the transformation itself"...if not, maybe you can elaborate (though I'm going on vacation thursday and probably won't respond for a little over a week).

 

[CaptainQuasar]

Does the concept of http://en.wikipedia.org/wiki/Minkowski_space" help? Minkowski spacetime is basically a four-dimensional coordinate system that sort of has the Lorentz transformation embedded in it. Ignoring any GR effects, SR scenarios can be evaluated within it while retaining a single coordinate system for all observers.

So although the transformations result in different lengths and clock-ticks for different observers, it's not really some sort of cosmic subjectivity, it's just different perspectives on a unified underlying reality. The weird effects are basically caused by space and time overlapping and edging in on each others' turf. (Which they're forced to do because nothing can go faster than the speed of light.)

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[matheinste]

Hello again.

CaptainQuasar i have some knowledge of linear algebra and hence inner product spaces in general and Minkowski spacetime in particular but not a detailed knowledge. I will be working at it.

JesseM have a nice holiday. Rest your brain.

I am more at home now with what is going on with Lorentz transforms but bear with me although i am taking it slowly,step by step . Just focusing on length contraction for now. Is it possible in principle for an observer in B to use his own measuring rods to measure directly or indirectly lengths in A.

If a rod in B of rest length in X as measured by an observer in B is made to pass a rod of rest length X measured in A by an observer in A, next to it in space i believe we can compare them and see a difference in length.
Also am i correct in believing that each observer (frame A and frame B) will see the other's rod contracted, there must be symmetry as neither frame is preferred or regarded as at absolute rest. I cannot remember seeing the mutual contraction ever stated explicitly although the comparable result for clock rates is often presented as a "paradox", which of course it is not.

The direct comparison of length would convince me that the contraction is real rather than just an optical effect (which i never believed) or due to a transformation in axes that the measurment is referred to (which i was unsure of).

Thankyou for your patience Matheinste.

 

[CaptainQuasar]

CaptainQuasar i have some knowledge of linear algebra and hence inner product spaces in general and Minkowski spacetime in particular but not a detailed knowledge. I will be working at it.

One thing though, I'm not saying that it would be advantageous in any way to work out these scenarios in Minkowski spacetime. I'm just pointing out that you don't have to regard the coordinate system as arbitrary - despite all the transformations that go on there's a basically fixed spacetime framework underneath it all.

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[JesseM]

JesseM have a nice holiday. Rest your brain.

Thanks! But I do have time to comment on your most recent post:

I am more at home now with what is going on with Lorentz transforms but bear with me although i am taking it slowly,step by step . Just focusing on length contraction for now. Is it possible in principle for an observer in B to use his own measuring rods to measure directly or indirectly lengths in A.

If a rod in B of rest length in X as measured by an observer in B is made to pass a rod of rest length X measured in A by an observer in A, next to it in space i believe we can compare them and see a difference in length.

Yes, in fact the coordinate systems of SR are based on the idea that a given observer assigns coordinates to events using only local readings on a system of rulers and clocks at rest with respect to themselves, with the clocks synchronized in their own rest frame (because of the relativity of simultaneity, different observers disagree on what it means for clocks to be synchronized--each synchronizes their clocks using the assumption that light signals move between the clocks at a speed of c in their own frame--so a given observer will find that the clocks of another observer are out-of-sync as measured in the first observer's rest frame). For example, if I see a distant event occurring right next to the x=50 light-seconds mark on my ruler, and the clock in my system that is sitting right next to that mark reads t=20 seconds at the moment the event occurs in its local neighborhood, then I assign that event coordinates x=50 l.s., t=20 s in my coordinate system. The "length" of a moving object is defined by looking at two events at either end of the object which have the same time-coordinate in my system, so if the back of an object is passed next to the x=35 l.s. mark on my ruler when the clock at that mark reads t=19 s, and the front of the object passes next to the x=38 l.s. mark on my ruler when the clock at that mark reads t=19 s, then I can say that the moving object is 3 l.s. long in my frame.

Also am i correct in believing that each observer (frame A and frame B) will see the other's rod contracted, there must be symmetry as neither frame is preferred or regarded as at absolute rest. I cannot remember seeing the mutual contraction ever stated explicitly although the comparable result for clock rates is often presented as a "paradox", which of course it is not.

Yes, it's symmetrical, if we have two observers in motion relative to one another, each observer's ruler/clock system will measure the marks on the ruler of the other observer to be shrunk relative to their own. This may be a bit hard to visualize, so you might want to take a look at the diagrams I posted in this thread a while back, showing how each ruler/clock system can see the other one's marks shrunk and the other one's clocks slowed down, yet they always agree on which markings/times on one ruler coincide with which markings/times on the other...it has to do with the relativity of simultaneity again, and the fact that each system measures the other one's clocks to be out-of-sync as well as slowed down.