Can Lorentz transformations determine the middle point on a photon's worldline?

[tomprice]

Ok so I begin by entertaining the following hypothesis:

Suppose a number of points are selected in spacetime, and then all relationships between these points that are invariant under Lorentz transformations are recorded. Given only these relationships, one should be able to reproduce the original set of points. (or a translated/lorenz transformed/reflected version of the original set of points).

Now, suppose that we choose three distinct points on one o' them there worldlines of a photon. If we were to specify the Lorentz interval between each pair of points, we would get 0, 0, and 0. This would not tell you which point is in the middle.
I'm pretty sure that, even though you can move these points all over the place with Lorentz transformations, it is not actually possible to change which one is in the middle.

So my question is, which of the following is true:
A) My original hypothesis is wrong.
B) There is an invariant that will tell you which point is in the middle (if this is the case, please describe said invariant)
C) My assumption about changing the middle point through Lorentz transformations is wrong.

Thank you very much for your thoughts.

 

[genneth]

I'm thinking A

 

[my_wan]

I believe what you are talking about is the space-time interval. It is never 0 unless the points are in the same place, i.e., the same point. Your intuition is more or less correct wrt the middle but requires some caveats. Your thinking about Lorentz transformations is somewhat distorted (I think).

First you are using points instead of objects. This can lead to troublesome misconceptions. It implies an absolute point that doesn't really move, like an absolute frame of reference. I'll therefore assume these points are actually objects that can move. It is always bad to assume the coordinates you chose to define something has any objective reality outside your mind, even if your an etherist. Even thinking of it that way can lead to chasing ghosts when trying to understand relativistic effects.

Since you did describe them as points I'll assume these objects are at rest wrt each other. If this is the case then the set of space-time intervals between all pairs of objects defines their position wrt each other, but not wrt any other observers or the galaxy itself. If one can be defined as in the middle then, given the lack of motion wrt each other, the set of space-time intervals will say which one, always.

It gets slightly more complicated if the objects are moving at very high velocity wrt each other. If one object is in the middle for a short time then the other objects may not agree on when it was in the middle. To a lesser extent this high velocity may also mean that they disagree on whether the three objects are lined up in a straight line. As long as you defined "middle" as simply the closest to one to the middle they should still always agree on which one.

Another issue when the objects are in motion is that the space-time can change with time in some circumstances even when none of the objects accelerated any.

 

[pmb_phy]

I believe what you are talking about is the space-time interval. It is never 0 unless the points are in the same place, i.e., the same point.

That is incorrect. If the magnitude of the spacetime displacement 4-vector is zero then either the 4-vector is zero or the displacement is lightlike, i.e. the two points are on the worldline of a null geodesic such as that of the worldline of a photon. In that case cdt = dL where dL is the spatial displacement and cdt is the temporal displacement.

Pete

 

[robphy]

Let P,Q,R be sequential events on a lightray [in Minkowski spacetime].
Although the square-intervals PQ,QR,PR are all zero, you still have directional information, like
$$\vec{PQ}+\vec{QR}=\vec{PR}$$ . If each vector is future-lightlike [i.e. have positive dot-products with a future-timelike 4-velocity], then Q is between P and R. These relationships don't change under a Lorentz transformation.

Pictorially, Q is on the intersection of the future-lightcone of P and the past-lightcone of R. So, that can be used to define Q as being between P and R. That won't change under a [proper] Lorentz Transformation.

There are other ways to capture this idea.

 

[Ich]

[i.e. have positive dot-products with a future-timelike 4-velocity]

Ah, now I see what your signature is good for.
Thanks, I was puzzling about this question since tomprice asked. So we have to include another vector to solve it.

Believe it or not, it took me 3 1/2 years to get this pun.

 

[robphy]

Ah, now I see what your signature is good for.
Thanks, I was puzzling about this question since tomprice asked. So we have to include another vector to solve it.

Yeah, I like to think positively.

 

[my_wan]

That is incorrect. If the magnitude of the spacetime displacement 4-vector is zero then either the 4-vector is zero or the displacement is lightlike, i.e. the two points are on the worldline of a null geodesic such as that of the worldline of a photon. In that case cdt = dL where dL is the spatial displacement and cdt is the temporal displacement.

Pete

Yes a technicality. robphy's method was fine.

More intuitively,
In defining the interval the space and time components must be defined such that we know $$\Delta t_2$$ and $$\Delta r_2$$ such that $$c_2\Delta t_2 = \Delta r_2$$. So technically yes, $$s^2 = c^2\Delta t^2 - \Delta r^2 = 0$$ yet the definition of the specific interval contains the non-zero separations $$\Delta r$$.

 

[robphy]

Yes a technicality. robphy's method was fine.

More intuitively,
In defining the interval the space and time components must be defined such that we know $$\Delta t_2$$ and $$\Delta r_2$$ such that $$c_2\Delta t_2 = \Delta r_2$$. So technically yes, $$s^2 = c^2\Delta t^2 - \Delta r^2 = 0$$ yet the definition of the specific interval contains the non-zero separations $$\Delta r$$.

Actually, the "technicality" is one of the distinctive features of the Minkowskian geometry of special relativity... and it's a reminder that you have to be careful with trying to apply your Euclidean intuition in this spacetime geometry.

 

[tomprice]

Alright thanks again robphy that answers it pretty well.

 

[pmb_phy]

Yes a technicality. robphy's method was fine.

What you call a "technicality" is known as an indefinite metric.

More intuitively,
In defining the interval the space and time components must be defined such that we know $$\Delta t_2$$ and $$\Delta r_2$$ such that $$c_2\Delta t_2 = \Delta r_2$$. So technically yes, $$s^2 = c^2\Delta t^2 - \Delta r^2 = 0$$ yet the definition of the specific interval contains the non-zero separations $$\Delta r$$.

Non-zero sace time seperations can occur along a null geodesic and still have a zero magnitude. In fact that's why its called a null geodesic. But for null worldlines the interval is zero and as such $$\Delta \tau$$ = 0. Yet spacetime displacements which are parallel to null worldlines need not vanish.

Pete