Can different time-slices of an object interact?

[gabeeisenstei]

In Special Relativity the example is given of a pole that alternates color, as seen in its own rest frame. Someone in a different frame is supposed to see the pole as striped, because he sees a different slice through 4D spacetime--a slice that includes past and future states of the pole mixed together.
My question is whether such a striped moving pole can have other properties that wouldn't occur in the pole's rest frame. What if it was alternately composed of materials that would explode upon contact? Or waves of different frequencies in a medium, how would they join together? Is my question missing something essential about the process of changing the frequency or the material composition? Or is it missing something about the privilege of proper time in physical processes? Or have I taken the "different slices through spacetime" idea too seriously?

 

[WannabeNewton]

Sure it can. Consider a slightly modified scenario in which the pole is rotating with constant angular velocity about the origin of an inertial frame. If we boost to an inertial frame moving to the right then at any given instant of the new frame, the rotating pole won't look like a straight line anymore but rather like a complicated sinusoidal function, as can be easily shown using a Lorentz boost, and therefore very near the origin it will look like a parabola. The reason for this is the same as what you mentioned as the reason for the striped colors of the pole: in the moving inertial frame the simultaneity slices are at an angle to the simultaneity slices of the original inertial frame i.e. they intersect the tangent field in space-time of the rotating pole in a different manner.

Any frame-dependent, and more specifically simultaneity dependent, property of the pole such as periodic alternation of color and spatial shape will, unsurprisingly, change from frame to frame because of the relativity of simultaneity i.e. the fact that different inertial frames generate different space-like foliations of space-time. But obviously if the pole breaks in one frame then it will do so in all frames-the frame dependence is in the "when", the "where", and the "why".

EDIT: just to clarify, when I say the pole "looks" like such and such I mean its geometric shape on any given simultaneity slice relative to an inertial frame and not what you or I would see in an actual photograph of the pole.

 

[gabeeisenstei]

Seeing a different shape seems just like seeing the stripes; it doesn't seem problematic to me.
My question is about physical processes, like the explosion in my example, generated by the contiguity of features in the moving frame that are not contiguous in the rest frame.

Or in my varying-frequency wave example, will I see a single wave with varying distances between peaks, or disconnected fragments of waves (which would presumably have different properties than a single wave)?

But here's a question about shapes: could two time-slices have topologically different 3D shapes, like a 2-sided strip becoming a 1-sided Mobius strip?

 

[WannabeNewton]

But here's a question about shapes: could two time-slices have topologically different 3D shapes, like a 2-sided strip becoming a 1-sided Mobius strip?

If you use the $\epsilon = \frac{1}{2}$ simultaneity relation relative to inertial frames then no obviously not: the time-slices are always hyperplanes orthogonal to the 4-velocity. If you allow for $\epsilon$ to be a function of coordinates then the time-slices relative to an inertial frame can be very complicated geometrically and topologically-at a basic metrical level they don't even need to be Euclidean-the same goes for time-slices relative to non-inertial frames and the time-slices can change geometrically from one instant to the next discontinuously for example if we accelerate an inertial observer in some arbitrary fashion and demand the use of Marzke-Wheeler simultaneity.

 

[pervect]

Seeing a different shape seems just like seeing the stripes; it doesn't seem problematic to me.
My question is about physical processes, like the explosion in my example, generated by the contiguity of features in the moving frame that are not contiguous in the rest frame.

[add]My quick summary. Different time slices can interact, but they must interact at light speed or less, that's part of relativity.

Everyone will agree about frame-invariant properties like whether or not the pole explodes. Simultaneity - which is dependent on the observer, is different from cause and effect - which is not dependent on the observer and modeled in SR by lightcones. So a hypothetical mechanism that makes the pole explode or not explode depending on who watches it isn't possible in special relativity. (As it shouldn't be -what if two different people watch it?). If you think you actually have such a mechanism, we can address why it doesn't work within the context of SR. If it's just something you're worried about, the basic answer is that it doesn't happen. If you think you have such a mechanism and have already explained it, you'll have to be more detailed, I'm not quite following your line of thought.

Or in my varying-frequency wave example, will I see a single wave with varying distances between peaks, or disconnected fragments of waves (which would presumably have different properties than a single wave)?

I'm not sure what you are envisioning. I envision the pole as consisting of a number of rotating wheels, half of which are painted red, and half of which are painted green. Or LED's on a timer, which emit red or green on a cyclic basis.

I'm concerned that you may be envisioning something involving something faster than light (FTL) in your modelling of the pole. The rotating wheels/cycling LED's clearly don't have any FTL characteristics, and also clearly (to me) don't show any paradox.

 

[JVNY]

. . . I'm not sure what you are envisioning. I envision the pole as consisting of a number of rotating wheels, half of which are painted red, and half of which are painted green. Or LED's on a timer, which emit red or green on a cyclic basis. . .

The OP likely refers to LED's on a timer. Here is a typical description of the scenario (observer A is at rest with respect to the pole, and observer B is in relative motion to them):

Let the rod again be at rest in A's reference frame . . . There are lights mounted on the end and middle points of the rod. Every instant the color of the lights changes simultaneously in A's reference frame: an instant before the meeting of A and B all three lights are green . . . [A]t the moment of the meeting . . . the lights are red, and an instant after the meeting they are blue . . . http://philsci-archive.pitt.edu/2408/1/Petkov-BlockUniverse.pdf (page 14).​

For B, the lights on the rod do not all have the same color at the same time. Simultaneously for B, one of the lights on the rod is red, one is blue, and one is green.

 

[JVNY]

. . . My question is whether such a striped moving pole can have other properties that wouldn't occur in the pole's rest frame. What if it was alternately composed of materials that would explode upon contact? . . .

The pole contracts proportionately all along its length, so no part of the contracted pole can touch any other part that it does not touch in the pole's rest frame. If a moving train with 10 cars length contracts by 40%, think of the train shrinking proportionately all along its length (resulting in 10 proportionately shorter cars), rather than removing cars 4-7 and then attaching car 3 to car 8 (creating a train of 6 regular length cars, and allowing two cars to touch that do not touch in the train's own frame).

So the explosion on contact problem does not arise.

There might be other interesting hypotheticals. Say some event occurs (like an explosion) whenever two materials on the pole get sufficiently close to each other. The explosion can only occur because of some interaction between the materials (a transfer of heat from one to the other, for example). If the materials are sufficiently distant in their rest frame to not interact and explode, then they will not interact and explode in the frame in which their distance is length contracted. Hopefully another contributor can explain the reason for this.

 

[WannabeNewton]

Hopefully another contributor can explain the reason for this.

Your description is still very vague. How is the heat generated between the points? What device is implemented to use the heat as a measure of distance between the points and trigger an explosion if the distance is sufficiently short? How does the heat register the distance between the points in every possible reference frames as opposed to just the rest frame of the pole? Don't use heat, I guarantee you it will become overly complicated-the Lorentz transformations for heat flow are highly non-trivial. Instead, fix a mirror to one point and a light source + photodetector to the other point; come up with a scenario using these, perhaps by attaching a detection counter to the photodetector and associating an explosion with a particular detection frequency.

 

[Dale]

My question is about physical processes, like the explosion in my example, generated by the contiguity of features in the moving frame that are not contiguous in the rest frame.

The laws governing all known physical processes can be written in a form which is independent of the coordinates chosen. The explosion you describe is rather vague, so I am not sure what physical process you envision. But if you correctly analyze the explosion process you are envisioning then it is simply not possible to get different results in different frames.

 

[PAllen]

Let's say the molecules of the pole alternate between two states. Somehow, the timing is such that in one frame, they all switch at exactly the same time. The period of oscillation is exact, so the state repeats. Let's say if two molecules of different state interact, they react, with a flash of light. The key point is that to react, the information that one molecule is state A has to reach another molecule in state B at no greater than c. Let us say, this is sufficient to cause reaction (in the one in state B, for example). Then, even though another frame 'sees' the pole as bands of different molecular state, there is still no explosion because the boundaries of state transition move greater than c (the transition space of events is a spacelike hypersurface; this feature is true in all frames, even though it is not a simultaneity surface in all frames).

To get a conundrum, you would have to posit a superluminal influence, which violates SR.

 

[gabeeisenstei]

Thanks to all responders.
I wrote this before seeing PAllen's post:
(
pervect's point about FTL interactions is along the lines of what I was looking for: I wanted someone to take a hypothetical interaction between the temporally mixed segments and show that it really wouldn't happen because…XYZ. I didn't think I was slipping in any FTL behavior, but if it turns out that I was, then that would be the answer I'm looking for.

The problem is that I haven't come up with a detailed enough example, as WannabeNewton rightly points out. I am sure that pervect is correct to say that "the basic answer is that it doesn't happen", as is DaleSpam. I don't want to feel like I'm just taking their assertions on faith, but the onus is on me to produce an alleged paradox in enough detail to be refuted.
)

Now PAllen seems to have given exactly what I was looking for…if I were smart enough to understand how the "boundaries of state transition move >c" because "the transition space of events is a spacelike hypersurface." Can you explain further or perhaps suggest what kind of textbook would cover event transition spaces?

 

[JVNY]

Your description is still very vague. How is the heat generated between the points? . . .

Perhaps a flame on one end of the object and a fuse on the other. The two are not close enough in the rest frame of the pole to light the fuse. But, get the flame close enough to the fuse and it lights. In the spirit of the original post, one might think that length contracting the pole will bring the flame close enough to the fuse to light it.

. . . Don't use heat, I guarantee you it will become overly complicated-the Lorentz transformations for heat flow are highly non-trivial. . .

Good thing I didn't try

gabeeisentei, I turn it back over to you. Is there a simple example you are thinking of? One can't stop after just calculating the length contraction. One has to do the Lorentz transformations for the other parts of the scenario, such as the heat flow in my example, and these transforms will show why the parts of the length contracted pole do not interact with each other any differently than they do in the rest frame of the pole.

Sorry: the others posted before I did, so this is obsolete.

 

[gabeeisenstei]

JVNY thanks for your input too, but I wasn't thinking about length contraction, just mixed states in skewed simultaneity slices.

Another off-the-wall idea I had was a pole made of alternately matter and antimatter atoms. There are probably many things wrong with that idea.

 

[Dale]

the "boundaries of state transition move >c" because "the transition space of events is a spacelike hypersurface." Can you explain further or perhaps suggest what kind of textbook would cover event transition spaces?

In your OP you described a pole which alternates color in its rest frame such that it is always a single color. So, let's make this a little more concrete. Let's say that the pole is covered in little LED lights that can either be red or green.

Now, let's consider two LED's on opposite ends of the pole, say $x=x_0$ and $x=x_1$. Now, suppose that we have the LED's turn from red to green at $t=t_0$ and $t=t_1$ respectively.

We can calculate the "spacetime interval" separating those two events as follows:
$\Delta s^2=-c^2\Delta t^2 + \Delta x^2 = -c^2(t_1-t_0)^2+(x_1-x_0)^2$

If the spacetime interval, $\Delta s^2$, is negative then we call the two events "timelike" separated, and a massive particle can go from one to the other at v<c. If $\Delta s^2$ is positive then we call the two events "spacelike" separated, and not even light can go from one to the other. If $\Delta s^2$ is 0 then we call the two events "lightlike" or "null" separated, and light will be able to make it from one to the other at c.

The comment you were asking about means that all of the events of the LED lights changing would be spacelike separated from each other.

 

[WannabeNewton]

Now PAllen seems to have given exactly what I was looking for…if I were smart enough to understand how the "boundaries of state transition move >c" because "the transition space of events is a spacelike hypersurface." Can you explain further or perhaps suggest what kind of textbook would cover event transition spaces?

PAllen has given an excellent description of the complementary nature of simultaneity and causality. It will show up implicitly in all SR texts. Here's a concrete but very simple example. We place two identical light sources at rest at different points in an inertial frame. The light sources periodically alternate between emission of red light and blue light and do so simultaneously in this inertial frame; a measuring device capable of simultaneously recording multiple wavelengths is placed at rest equidistant from the two light sources in this inertial frame. If the device simultaneously records wavelengths corresponding to blue light and red light then it is rigged to explode. Clearly in this inertial frame it will never explode. But we can find an inertial frame moving relative to this one in which one light source emits red light and the other light source emits blue light simultaneously.

Does this mean that in this frame the measuring device will explode? No because in this frame these two light signals won't reach the measuring device simultaneously-the measuring device is catching up to one of the light signals whereas it is moving away from the other light signal.

The point is that in order for the measuring device to explode, there has to be some sort of communication between it and the two light sources and this communication has to respect causality. This means that just because in the moving inertial frame the two light sources simultaneously emit light signals of different colors doesn't mean the measuring device will explode since each light signal takes an associated time to propagate to the measuring device. If the information about the wavelengths of the emitting signals was to say instantaneously be transmitted to the measuring device then the story would be very different but this overtly breaks causality.

I see DaleSpam beat me to it!

 

[FactChecker]

My question is about physical processes, like the explosion in my example, generated by the contiguity of features in the moving frame that are not contiguous in the rest frame.

The correct physical processes are as though the inertial reference frame of that process is not moving. Any other reference frame that observes those processes will have to take the relative velocity into account to understand why things happened in an unexpected way (In your example, chemicals might appear to mix but really did not). In other words, the result of an experiment in a fast moving reference frame would have to agree with the changes in time, space, and simultaneity for that reference frame.

 

[JVNY]

The latest responses utilize lights placed at a distance from each other. But what about the suggestion of atoms next to each other:

. . . Another off-the-wall idea I had was a pole made of alternately matter and antimatter atoms. . .

If I understand this, the idea is that instead of lights changing colors, the pole is made of atoms that cycle between being matter and antimatter in the pole's frame. In that frame, all of the atoms that make up the pole are either matter or antimatter at any given time. But in another reference frame moving relative to the pole, the pole would consist of alternating matter and antimatter atoms abutting each other along the pole's length. That would seem to create a problem.

I suspect that the atoms cannot interact instantaneously, so that there is only a difference in degree, not kind, between atoms along the length of the pole and lights separated along the pole.

 

[WannabeNewton]

gabeeisenstei, here's another very simple scenario that you should try working out yourself as an exercise on the complementarity of simultaneity and causality except this time it doesn't involve light signals. It's a standard textbook variation of the usual ladder-Barn paradox, taken from "A First Course in General Relativity"-Schutz. The key point again is subluminal propagation speed of information.

https://www.physicsforums.com/attachment.php?attachmentid=64255&d=1385320084

 

[WannabeNewton]

If I understand this, the idea is that instead of lights changing colors, the pole is made of atoms that cycle between being matter and antimatter in the pole's frame.

It's a physically impossible scenario that breaks all kinds of conservation laws so don't focus too much on it.

 

[gabeeisenstei]

Thanks very much, I think I've got it. DaleSpam & WbN's posts were both helpful and, I think, complementary. The example of signals sent to the middle of the pole was a good reminder.
I suspected all along that the working principle would be to analyze processes in their rest frame, and that there would be "unexpected" phenomena in the moving frame. (Events that seem, from the moving frame, to be close enough to generate interaction are really not close enough in spacetime). Here it's tempting to say that the rest frame determines which events are "really" simultaneous, whereas in the moving frame they only seem to be simultaneous. But I guess it's better to say, like WbN, that simultaneity and causality are complementary concepts.