Negative squares using the space time interval invariance

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Discussion Overview

The discussion revolves around the concept of spacetime interval invariance in special relativity (SR), particularly focusing on the implications of negative squares in spacelike intervals. Participants explore the relationships between events in different reference frames, the effects of Lorentz contraction, and the interpretation of time differences as perceived by a moving observer.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a scenario involving two points A and B emitting light flashes simultaneously in their rest frame, leading to a spacelike interval represented as -(AB)².
  • Another participant suggests that the negative square indicates a mistake in the calculations, proposing to ignore the sign when taking the square root.
  • Some participants clarify that spacelike intervals are indeed negative under the (+,-,-,-) signature convention and express curiosity about the implications of this in calculations.
  • There is a discussion about the change in simultaneity for a moving observer, with references to Lorentz transformations and their effects on time coordinates.
  • One participant emphasizes that their goal is to understand why a negative square arises rather than to find a numerical answer.
  • Another participant questions the application of length contraction to events that are not causally connected, raising doubts about the validity of certain mathematical expressions.
  • Several participants engage in correcting each other's mathematical expressions and assumptions, particularly regarding the relationship between spacetime intervals and Lorentz contraction.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and confusion regarding the implications of negative squares in spacelike intervals. While some agree on the mathematical framework, others challenge the assumptions and interpretations, indicating that multiple competing views remain without a clear consensus.

Contextual Notes

Limitations include unresolved mathematical steps and dependencies on signature conventions. The discussion reflects uncertainty about the application of Lorentz transformations and the interpretation of simultaneity in different frames.

ersteller
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Hallo
I'm new to this (wonderful) forum, and to SR too...
I've a general question about the space time interval invariance.
Say we have two points A and B, at rest each other, at distance AB.
Now A and B simultaneously in their reference frame emit a flash of light.
The space time interval between these two events is spacelike and equals -(AB)2.
Now a fast spaceship C travels from A toward B and is reached by the two flashes at the middle point between A and B. For C the two flashes are not emitted simultaneously, and I want to use the invariance of the space time interval to compute T, that is the difference in time (measured by C) between the two flashes. The distance between A and B measured by C is shortened by the Lorentz contraction: ABc < AB. The space time interval measured by C should be:
(CT)2 - (ABc)2
and this number should equal -(AB)2. It results
(CT)2 = (ABc)2 - (AB)2 < 0.

So, how can I find T, as a square root of a negative number?

Sorry if the question is stupid...
er
 
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This is a space-time diagram of your scenario as I understand it. The first is from A and B's rest frame, the second is from the ship frame. I think you can read off the time between flashes on the diagram. The ship is traveling at 0.42c in the A,B frame.

I think you must have made a mistake, if you're trying to take the sqrt of a negative number. With this signature (+,-,-,-) a timelike interval is negative, so just ignore the sign, take the square root, the put the sign back.
 

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Thank you

With this signature (+,-,-,-) a SPACElike interval is negative, and this is our case.

And for sure (CT)2 = (ABC)2 - (AB)2 IS negative.

What I find very interesting in your answer is when you say: "just ignore the sign, take the square root, the put the sign back"... Really so simple?

Thank you very much!
er
 
ersteller said:
Thank you

With this signature (+,-,-,-) a SPACElike interval is negative, and this is our case.

And for sure (CT)2 = (ABC)2 - (AB)2 IS negative.

What I find very interesting in your answer is when you say: "just ignore the sign, take the square root, the put the sign back"... Really so simple?

Thank you very much!
er

I don't know if space-like intervals are invariant ( they are, see below) so ignore my advice.

The flashes are at the same time in the A,B frame (tA-tB=0) but clearly not when the diagram is boosted by -0.412c. We're talking about a change in simultaneity here, and I'm not sure how proper length comes into it. I'll try and work it out later.

[later]

Spacelike intervals are invariant, so it seems there's not much point in using one to explain a change in simultaneity. In my diagrams, the interval is sqrt(-100) in both frames. The quantity you want is one side of the triangle that defines the interval.

If two events ( tA,xA), (tB,xB) are boosted by \gamma the transformed time coords are (Y is gamma, b is beta)

t'A = YtA+YbxA
t'B = YtB+YbxB

so in your scenario
t'A - t'B = Yb(xA - xB)

which shows that the change in simultaneity depends on the spatial separation, and the relative velocity.
 
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Thank you.

My point (now) is not to *find* a number, but only to understand: By using the invariance of the spacetime interval, all quantities in SR kinematics can be found, as I've understood. So my question, that perhaps you have already answered, is: Why have I a negative square in this case?
You said: "just ignore the sign, take the square root, the put the sign back" and perhaps this is the simple answer (but I don't really understand WHY...)

Thanks again
er
 
ersteller said:
Thank you.

My point (now) is not to *find* a number, but only to understand: By using the invariance of the spacetime interval, all quantities in SR kinematics can be found, as I've understood. So my question, that perhaps you have already answered, is: Why have I a negative square in this case?
You said: "just ignore the sign, take the square root, then put the sign back" and perhaps this is the simple answer (but I don't really understand WHY...)

Thanks again
er

The sign of a spacelike interval depends on the signature convention, so there's no harm in changing the sign and then taking the square root and putting sign back.

I still can't see how to get the t'A - t'B in my last post without doing a Lorentz transformation. I'll have another try later.
 
Mentz114 said:
The sign of a spacelike interval depends on the signature convention, so there's no harm in changing the sign and then taking the square root and putting sign back.

Very good! Now I can do my simple calculation without harm!
Thank you!

I still can't see how to get the t'A - t'B in my last post without doing a Lorentz transformation. I'll have another try later.

Here perhaps I can help you: From the invariance of the spacetime interval

(CT)2 - (ABc)2 = (AB)2

(and without problems arising from the 'negative square'...) one can immediately find T... or not?

Ciao
er
 
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ersteller said:
Very good! Now I can do my simple calculation without harm!
Thank you!
Here perhaps I can help you: From the invariance of the spacetime interval

(CT)2 - (ABc)2 = (AB)2

(and without problems arising from the 'negative square'...) one can immediately find T... or not?

Ciao
er
I think that should be

(CT)2 - (ABc)2 = -(AB)2

so

(CT)2= (AB)2( 1/(1-b2) - 1 ) = b2Y2(AB)2

as above. But by using a Lorentz contraction, have you not assumed that an LT connects the frames ?

(it's late so don't assume I've done the math correctly).
 
You're right, I forgot a minus sign...

(CT)2 - (ABc)2 = - (AB)2

that is

(CT)2 = (AB)2 (\frac{1}{\gamma^{2}} - 1) (<0 but no more a problem...)
But by using a Lorentz contraction, have you not assumed that an LT connects the frames ?
Yes, I must admit!
er
 
  • #10
Now I am confused ( nothing new there ). This is wrong

[STRIKE](CT)2= (AB)2( 1/(1-b2) - 1 ) = b2Y2(AB)2[/STRIKE]

but is this ?

t'A = YtA+YbxA
t'B = YtB+YbxB

so in your scenario
t'A - t'B = Yb(xA - xB)

Now I have doubts the validity of (ABC)=1/Y(AB). This sort of length contraction is because the moving frames perception of simultaneity when measuring the ends of a timelike interval. These events are not causally connected so can length contraction be applied ?
 
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  • #11
Mentz114 said:
These events are not causally connected so can length contraction be applied ?

I guess YES, but is perhaps better to wait for an explanation...

Now I've less doubts, thank you...
er
 
  • #12

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