Spacetime interval - alternative view - maybe?

[whatif]

With regard to special relativity…

Whenever, I come across the spacetime interval, written like this, say, (Δs)² = (Δt)² – (Δx)² – (Δy)² – (Δz)² , it is as if it has to be that way. However, it seems to me it is this way by definition and does not have to be so. Sometimes, it seems to be referred to as the modified Pythagoras theorem.

What if we apply the usual Pythagoras theorem so that (Δs)² = (Δt)² + (Δx)² + (Δy)² + (Δz)² and apply the following analysis. For simplicity, say the coordinates are arranged such that Δy = 0, and Δz = 0, so that (Δs)² = (Δt)² + (Δx) ². Then, for timelike events:Consider 3 events, where E₁ and E₂ are timelike, and E₁ and E₃ are lightlike:

(Δs₁₂)² = (Δt₁₂)² + (Δx₁₂)²
(Δs₁₃)² = (Δt₁₃)² + (Δx₁₃)²)

(Δs₁₂)² - (Δs₁₃²)
= {(Δt₁₂)² + (Δx₁₂)²} - {(Δt₁₃)² + (Δx₁₃)²}
= {(Δt₁₂)² + (Δx₁₂)²} - {(Δx₁₃)² + (Δx₁₃)²}
= {(Δt₁₂)² + (Δx₁₂)²} - {2(Δx₁₃)²}
= {(Δt₁₂)² + (Δx₁₂)²} - {2(Δx₁₂)²}
= (Δt₁₂)² + (Δx₁₂)² - 2(Δx₁₂)²
= (Δt₁₂)² - (Δx₁₂)²
= the same invariant as when the “modified Pythagoras theorem” is used

In words, for timelike events E₁ and E₂ ...
... the square of the invariant, is equal to the square of the spacetime interval between events E₁ and E₂, minus the square of the spacetime interval for light to get from E₁ to the spatial location of E₂.

A similar case can be made for spacelike and lightlike events.

Lightlike events in different inertial frames would have different spacetime intervals between the same 2 events, but would not have the oddity of no spacetime interval between separate events.

?

 

[Dale]

What if we apply the usual Pythagoras theorem so that (Δs)2 = (Δt)2 + (Δx)2 + (Δy)2 + (Δz)2

Then Δs is not invariant.

Since we are always interested in invariants we would still need a symbol and a term to refer to the invariant quantity. That symbol and term may as well remain what it currently is.

 

[whatif]

Then Δs is not invariant.

Since we are always interested in invariants we would still need a symbol and a term to refer to the invariant quantity.

That is all very well. However, I did provide an invariant which is the same as the existing invariant. A symbol can always be created for the invariant as I expressed it. It seems to me that the interpretation of that symbol could stand for something consistent with the common understanding of an interval.

That symbol and term may as well remain what it currently is.

So we should be content that different timespace events that are lightlike are not separated by a timespace interval?Anyway, from my limited perspective it seems that my different interpretation would work, but my comments are contingent on that being the case. Whether an expert agrees is the kind of response I appreciate, so thank you Dale.

 

[Ibix]

?

Stripped of the algebra, you've simply defined a quantity that is the interval squared between events $E_1$ and $E_3$ plus twice the spatial distance squared (in some frame) between them. This is clearly not an invariant, so doesn't function in any way analogously to either length or interval. In fact, its transformation law is probably horrible since it isn't a Lorentz scalar. It doesn't appear to have a simple relationship to any obvious coordinate system (although it could be one quarter of a rather bizarre one). It doesn't seem to relate simply to measurements.

Note also that the invariant you define is just the interval. You've simply defined $\Delta s_{whatif}^2=\Delta s^2+2\Delta x^2$ and then found a very long winded way of saying $\Delta s^2+2\Delta x^2-2\Delta x^2=\Delta s^2$.

In short, what's the point? All you seem to be doing is layering algebraic obfuscation on top of an already conceptually challenging theory.

 

[Dale]

However, I did provide an invariant which is the same as the existing invariant. A symbol can always be created for the invariant as I expressed it.

I would recommend using the symbol $\Delta s^2$ and the term "spacetime interval".

My point is that we still need the existing concept. So it makes no sense at all to take an existing term, change the meaning of the existing term, then create a new term to refer to the previously existing term. That makes people unlearn an existing term, relearn the new definition of the existing term, and learn a new term. Plus from then on you have to worry if a document you are reading is using the old meaning or the new meaning.

It would be far better to invent a new term for your new concept. Maybe something like the Euclidean spacetime distance. That way people only need to learn your new concept, and not go through all of the mess of changing an established term. If your new concept has any value then it will be adopted on its own with its own name and no need to take an established name.

So we should be content that different timespace events that are lightlike are not separated by a timespace interval?

Sure, why not? It follows directly from the second postulate:

In any inertial coordinates: $c^2 \Delta t^2 = \Delta x^2 + \Delta y^2 + \Delta z^2$ describes a sphere whose radius is expanding at c, so $0 = - c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$ is clearly an invariant due directly to the second postulate. Then it takes very little imagination to guess that $\Delta s^2= - c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$ might also be an invariant for things that are not necessarily on the light cone.

 

[SiennaTheGr8]

Lightlike events in different inertial frames would have different spacetime intervals between the same 2 events, but would not have the oddity of no spacetime interval between separate events.

So we should be content that different timespace events that are lightlike are not separated by a timespace interval?

It seems you're uncomfortable with the fact that the spacetime interval between lightlike-separated events is zero, and it seems that you're trying to overcome your discomfort by inventing some other related quantity and calling it "the interval" instead.

You may be committing the "nominal fallacy," confusing the map for the territory. Don't let yourself get sidetracked by words! We care about the quantity $(\Delta t)^2 - (\Delta x)^2$ not because it's called "the interval," but because it is Lorentz-invariant. A rose by any other name...

So if you're uncomfortable with $\Delta s$ being zero for lightlike-separated events, the solution is not to dub some other frame-dependent quantity "the interval." That's semantics, not physics. Rather, you should crack open a textbook on relativistic kinematics and solve some problems involving the interval. That's how you'll come to appreciate the usefulness of the quantity $(\Delta t)^2 - (\Delta x)^2$. I recommend the first few chapters of Taylor and Wheeler's Spacetime Physics.

 

[PeterDonis]

I did provide an invariant which is the same as the existing invariant

More precisely, you rearranged the math for the standard spacetime interval in a way that, to you, seems more intuitive. But you haven't changed any physics. See below.

Lightlike events in different inertial frames would have different spacetime intervals between the same 2 events

No, they wouldn't. Physically, the "spacetime interval" is still what you've described here:

the square of the invariant, is equal to the square of the spacetime interval between events E1 and E2, minus the square of the spacetime interval for light to get from E1 to the spatial location of E2.

Which obviously means the square of the invariant is zero when you're talking about a light ray, since the events E1 and E2 are the events light passes through when going from E1 to the spatial location of E2.

I know you'll say that you defined your terms so that what I just quoted above only applies to timelike intervals, not lightlike intervals. But nature doesn't care how you define your terms. The physical spacetime interval between a pair of lightlike separated events is zero.

In other words, as above, you've just relabeled the math in a way that, to you, seems more intuitive. But you haven't changed any physics. The thing you are calling "the spacetime interval" does not describe anything physical; it's just your idiosyncratic label for a particular artifact of the math as you've expressed it. You can't change physics by rearranging the math.

 

[whatif]

Stripped of the algebra, you've simply defined a quantity that is the interval squared between events E1E1E_1 and E3E3E_3 plus twice the spatial distance squared (in some frame) between them.

All my calculations are based on using the same frame.

You've simply defined

... I defined Δs² = Δt² + Δx². Then I showed that the same invariant can be arrived at with this definition by reference to a lightlike pair of events which shares one of the timelike pair of events.

 

[whatif]

In short, what's the point?

This is my point:

Whenever, I come across the spacetime interval, written like this, say, (Δs)2 = (Δt)2 – (Δx)2 – (Δy)2 – (Δz)2 , it is as if it has to be that way. However, it seems to me it is this way by definition and does not have to be so. Sometimes, it seems to be referred to as the modified Pythagoras theorem.

The spacetime interval, as currently expressed, is not the same as Pythagorean theorem. It is typically implied as a necessary consequence of the geometry and that is misleading. It is a consequence of the expression chosen to define the interval (an expression that is inconsistent with what an interval is generally understood to be).

 

[whatif]

That's how you'll come to appreciate the usefulness of the quantity (Δt)2−(Δx)2(Δt)2−(Δx)2(\Delta t)^2 - (\Delta x)^2. I recommend the first few chapters of Taylor and Wheeler's Spacetime Physics.

I have read Taylor and Wheeler's Spacetime Physics, I have solved many of the problems and I do appreciate the usefulness of the quantity (Δ t)² − ( Δ x )²

 

[Dale]

It is typically implied as a necessary consequence of the geometry and that is misleading.

It is a necessary and very direct result of the second postulate as I showed above

 

[whatif]

No, they wouldn't.

A pair of lightlike events would have different spacetime intervals in different frames if the spacetime interval is defined as Δs² = Δt² + Δx².

Which obviously means the square of the invariant is zero when you're talking about a light ray, since the events E1 and E2 are the events light passes through when going from E1 to the spatial location of E2.

Correct, the invariant would be zero but the timespace interval would not be zero. It just happens that, in lightlike cases, the time it takes light to travel the distance would be equal to the timespace interval.

 

[whatif]

It is a necessary and very direct result of the second postulate

The invariant is a consequence. The definition of the timespace interval in not a consequence.

 

[Dale]

The definition of the timespace interval in not a consequence.

That is a vacuous statement. No definition is a consequence of anything. It is just a definition and doesn’t need to be a consequence of anything.

I think we are about done here. The spacetime interval is very useful and the terminology is well accepted. Attempting to change the terminology is not something this community is likely to support.

I certainly don’t, particularly not for such a weak justification and a bad solution. You don’t like that the spacetime interval is degenerate but your proposed fix is to make it not invariant. That is truly a solution worse than the problem.

 

[PeterDonis]

It is a consequence of the expression chosen to define the interval (an expression that is inconsistent with what an interval is generally understood to be)

You've got it backwards. You don't start from a mathematical expression like the Pythagorean theorem and then assume that something in physics must match it. You start with the physics--the actual, physical relationships between events--and then you find a mathematical model that correctly describes those relationships. That is why the spacetime interval expression is what it is in standard SR. And no amount of mathematical manipulation will change it, because you can't change physics by changing the math.

the invariant would be zero but the timespace interval would not be zero.

The thing that you are calling the "timespace interval" would not be zero. But that's because the thing you are calling the "timespace interval" has no physical meaning, while the standard SR spacetime interval does. Once again, you can't change physics by changing the math.

And with that, this thread is closed.