What is "spacetime" really?

[modest]

I largely agree with what you have said, but I don't agree that the zero spacetime interval is comparable to the speed of light. The speed of light being the same in every frame of reference is an axiom of special and general relativity. It is assumed to be true, and the Lorentz transformation and the spacetime metrics follow directly from that assumption.

 

Right, I think we're saying pretty much the same thing. The postulates of special relativity are the same as Minkowski spacetime:

• invariance of the speed of light
  
• laws of physics hold in all inertial frames
  
• space is isotropic and homogeneous

These are equivalent to saying laws of physics are invariant between any two arbitrary inertial frames with respect to Lorentz transformations which is the underlying principle of Minkowski spactime. It’s a framework on which laws of physics work given special relativity.

 

The zero spacetime interval is a conclusion from the spacetime metrics.

 

And the spacetime metric is a conclusion from the postulates of SR. In that respect, it describes something that we know is real (note: I’m not saying it’s a real “entity” :rolleyes:, I’m saying it really happens). Consider someone who is traveling at half the speed of light compared to you. Following the equation:

[math]\Delta s^2 = c^2 \Delta t^2 - \Delta x^2[/math]

Their path length (or spacetime interval) is going to be less than your own. This doesn’t make intuitive sense because they are traveling some great distance implying any path length should be large. Also, following a normal Pythagorean relationship of adding the squares should make their path length larger than your own.

 

But, considering time dilation and length contraction in special relativity this is not true. The clock belonging to the person traveling half the speed of light is slowed and the distance they are traveling is shorter by length contraction. The closer they get to the speed of light, the greater the time dilation and length contraction. Their path length actually gets smaller. Their spacetime interval becomes less.

 

In my understanding, the spactime interval is taken to be the "distance" in spacetime between two events. But a zero spacetime interval does not mean that the events happen at the same place (co-located), nor that they happen at the same time (simultaneous). So what does a zero spacetime interval mean?

 

Two events that are collocated and simultaneous do have an interval of zero. Similarly, something that travels one unit of length per one unit of time should also have an interval of zero.

 

Time dilation and length contraction approach infinite as something approaches the speed of light. Light attains this speed. The physical interpretation of this is quite clear. Light does not experience the passage of time [6191]. Nor does light "see" the universe as having any length at all. It travels (from it’s perspective) zero distance in zero time to get from where it was to where it’s going. Special relativity implies this.

 

It is then not surprising that the spacetime interval of light is zero.

 

Anyway, I don't see that any of this answers my question.

 

I still don’t think I’ve answered your question. Why do physical equations work in an invariant way when time and space are subtracted rather than added? Philosophically, I don’t know. I think the question is fundamentally the same as asking why special relativity works, or why the postulates of special relativity are true. :shrug:

 

By the way, the spacetime interval is only one of many quantities that is invariant when space and time are combined in a Minkowski way. Energy and momentum combine when treated like the spacetime interval (a subtracted from of the Pythagorean theorem) to yield its invariant mass. Depending on the reference frame of the observer, the values of momentum and energy will change, but they combine yielding the invariant mass.

 

Also, charge and current (as related to electricity) are the space and time parts of a 4-dimensional invariant. In other words, some observers can see more of a static charge while others see more of an electric current, but if each observer combines the two in the subtracted Pythagorean way then they each get a value they can both agree on.

 

What all this means philosophically... I think is over my head. At the very least, I guess it implies (perhaps strongly) that we live in a 4-dimensional universe and that time is somewhat different from the other 3 dimensions. If it were just another space-like dimension (even if all but the 'present moment' were hidden from us) then we could use the Pythagorean theorem on it as well. But, we know we can't, so it's a very interesting question.

 

Very good question, Jedaisoul.

 

~modest

[watcher]

Yes, I carefully avoided your question of "what is an act?"

 

I assume it is the interaction of the so-called "higher" dimensions with each other, which then generates what we perceive as paticles/energy.In my view those "higher" dimensions are the basics of existence,

the truely lowest dimensions (without spacetime); and their conformation determines the placement of matter/energy in a new, emergent dimension (ours); which then generates spacetime as described previously.

 

i also think of spacetime as emergent out of something else. in emergent system, the whole is greater than the sum of its parts or we can say that the stuff from which spacetime emerges could be totally different from the properties/substance of spacetime itself. so to ask what is spacetime could also mean to inquire to its source or origin.

 

in string theory there is a great force present in the folding and compression of space as the dimensions goes higher which i think will eventually goes in the direction of singularity . whether the folding space creates the tension force or the force creates space as it exerts/unfolds depends on the perspective of the space or the perspective of the force. but if our spactime is the generated result, then the former should be the case. ie. the stuff from which spacetime emerged is a force that generates spacetime. (perhaps at a rate of 186,000 miles of space per second?) hehe

 

in Qm, spacetime imo seemed to emerge from a field of potential. simply put, spacetime exists because it can. what is this field of potential, dunno, perhaps , it's just that, but we need to assume that if only to answer the question why is there something instead of nothing.

[Essay]

...is a force that generates spacetime. (perhaps at a rate of 186,000 miles of space per second?) hehe ...

I'll have to reread the rest of your post a few times, but for now....

 

Hey! Two years ago I was posting about how "c" should be called the speed of space (...or spacetime).

I wrote something like, "c" was the speed at which space cracks to admit the transfer of energy from one thing to another (change in conformation in one place, causing a change in conformation of something else); the emission and subsequent absorption of a photon.

 

...although that space "crack" or fissure metaphor is sort of reversed from the image of "generated" space;

:) ...but it's just a matter of perspective.

 

~ ;)

[jedaisoul]

Right, I think we're saying pretty much the same thing. The postulates of special relativity are the same as Minkowski spacetime:

• invariance of the speed of light
  
• laws of physics hold in all inertial frames
  
• space is isotropic and homogeneous

[snip]

And the spacetime metric is a conclusion from the postulates of SR.

I agree with everything you are saying here. I was just pointing out, as you have agreed, that the invariance of the speed of light is a postulate (axiom) whereas the spacetime metric is a conclusion. A postulate and a conclusion are very different things.

 

The clock belonging to the person traveling half the speed of light is slowed and the distance they are traveling is shorter by length contraction. The closer they get to the speed of light, the greater the time dilation and length contraction. Their path length actually gets smaller. Their spacetime interval becomes less.

I disagree with this interpretation. The time dilation and distance contraction are alternative descriptions of the same phenomenon. If time dilates, you get there quicker because less time is taken to travel the original distance. If the distance contracts, you get there quicker at the same rate of time flow, because the distance is shorter. These are two ways of saying the same thing. However, if you claim that time dilates and the distance contracts you are counting the effect twice, and will come to incorrect conclusions.

 

Two events that are collocated and simultaneous do have an interval of zero. Similarly, something that travels one unit of length per one unit of time should also have an interval of zero.

This is false logic. Two events that are co-located and simultaneous have an spacetime interval of zero because the spatial distance and time interval are both zero. This has nothing to do with when the spatial distance and time interval are not zero. You appear to be generalising from the fact that, in a spacetime diagram, the light cones meet the plane representing space at the point where the spatial distance and time interval are both zero. This is the only place where the light cones meet the plane representing space. So you cannot generalise from the circumstances that are true at that point.

 

Light does not experience the passage of time (photons have no time). Nor does light "see" the universe as having any length at all. It travels (from it’s perspective) zero distance in zero time to get from where it was to where it’s going. Special relativity implies this. It is then not surprising that the spacetime interval of light is zero.

Exactly. I could not have put it better. But what does this mean? I would suggest that it means that specetime describes the universe as experienced by a photon. But material objects cannot travel at the speed of light, so material objects do not travel zero distance in zero time to arrive at a destination.

 

Putting it another way. You seem to agree with my point that, when the spatial distance and time interval are non-zero, a zero spacetime interval does not imply that the events are co-located nor simultaneous. They are only co-located and simultaneous for light. So if what is true for light is not true for material objects when the spacetime interval is zero, why should we assume that it is true for material objects when the spacetime interval is non-zero? That simply does not make sense.

 

This suggests that spacetime mechanics rests on a false premise, that what is true for light is true for material objects as well. We have shown that not to be true at one point (zero space-time interval), so it would seem logical to assume that it is equally false at all other points.

[Michael Mooney]

First, I can hardly navigate this site without the full-screen "Chase" ad/spam locking in and making me exit and then re-enter the website (twice now.)

If this is a permanent hurdle, I haven't the patience for it and will no longer be participating here. There must be a limit to users' tolerance of pop-up ads to pay the site bills.

I will now go to the "homework" link given me by Modest in his last post to me.

At present my perspective is as follows: I have studied general and special relativity as a non-mathematician and still question the jump from Euclidean to non-Euclidian space as a basis for relativity. Specifically I question "curved space" and "dilated time" etc. The mathematics is beyond my expertise, as I have said, so throwing more math at me is like saying, "science has gone beyond Euclidean cosmology, and the math proves it... catch up to speed or shut up."

 

So maybe it is good that this thread is in the "philosophy of science" section rather than in a section focused on mathematical proofs of relativity and its basic axioms that space is something that bends, expands, etc., and time is something that speeds up and slows down... (not just contents/objects in space with curved trajectories or clocks responding to changes in inertial force.)

 

As a base line reality check, I asked for an explanations of how the shortest distance between two points is no longer a straight line, and how it is that parallel lines can converge... both tenants of non-Euclidean "space"... which used to be simply "emptiness." Then the argument continues that there *must be something* between objects besides empty space to convey the force of gravitation... and "spacetime" is born, though it is just an "metaphorical" device as an aid to understanding the math of relativity... which is, for sure a better predictor of certain gravitational phenomena than the less sophisticated math of pure Newtonian/Euclidean space and cosmology.

 

Still, there has been nothing offered against the assumption that electromagnetic, gravitational, and micro-scale quantum forces can traverse empty space without an intermediate agent of some kind "filling space."

 

I will now go to my homework and return later. (I do not expect to become a mathematician meanwhile and offer rebuttal in that realm.)

[modest]

I disagree with this interpretation. The time dilation and distance contraction are alternative descriptions of the same phenomenon. If time dilates, you get there quicker because less time is taken to travel the original distance. If the distance contracts, you get there quicker at the same rate of time flow, because the distance is shorter. These are two ways of saying the same thing. However, if you claim that time dilates and the distance contracts you are counting the effect twice, and will come to incorrect conclusions.

 

I’m not being very clear. The clock (on the rocket) slows from our perspective. The distance the rocket travels becomes less from the rocket’s perspective. Both are necessary for the description to work and they are not the same thing.

 

It’s the rocket’s clock that measures the spacetime interval (so long as the motion between the two is inertial). That interval needs to be invariant. We can’t have different people in different reference frames thinking a young astronaut landed at his destination while others think an old astronaut landed. That would reek havoc on reality.

 

Two events that are collocated and simultaneous do have an interval of zero. Similarly, something that travels one unit of length per one unit of time should also have an interval of zero.

This is false logic. Two events that are co-located and simultaneous have an spacetime interval of zero because the spatial distance and time interval are both zero. This has nothing to do with when the spatial distance and time interval are not zero.

You’re reading too much into the word “similarly” and you’re not considering what you asked me. I’m responding to this:

But a zero spacetime interval does not mean that the events happen at the same place (co-located), nor that they happen at the same time (simultaneous). So what does a zero spacetime interval mean?

When, indeed, a null spacetime interval can very well mean two events are collocated and simultaneous. There’s nothing false about that logic, I’m simply correcting you.

 

As far as why I’d use the word “similarly”. There’s 2 reasons. First, when dt = dx, any value you choose for dt or dx will give you a zero spacetime interval. It works for 3, for 2, for 1, and, similarly, it works for 0. You seem to have decided that collocated and simultaneous events must have zero spacetime separation because dt and dx are both zero. That’s fine, but all that really matters is that they’re both the same (zero or not).

 

Second reason: a clock that moves through neither space nor time (as viewed from any frame) does not advance. [math]\Delta S[/math] is zero. Similarly, a clock that moves at the speed of light (as viewed from any frame) does not advance. [math]\Delta S[/math] is zero. The physical interpretation is the same in both cases.

 

Putting it another way. You seem to agree with my point that, when the spatial distance and time interval are non-zero, a zero spacetime interval does not imply that the events are co-located nor simultaneous. They are only co-located and simultaneous for light.

 

That's the idea. For any inertial reference frame, light travels one light-year per year. Light's proper time and its interval are therefore zero. All observers can agree on this.

 

But, the case of light's proper time being zero is kind of an odd case to consider. It works just as well (and more intuitively) when an inertial frame is being considered. The spacetime interval between two events which can be causally connected is the proper time between the events as measured by an inertial observer who intersects them both.

 

This value is invariant using the equation [math]ds^2 = c^2dt^2 - dx^2...[/math]

 

This suggests that spacetime mechanics rests on a false premise, that what is true for light is true for material objects as well. We have shown that not to be true at one point (zero space-time interval), so it would seem logical to assume that it is equally false at all other points.

 

Let's say a rocket travels from one planet to another. Different people observe the event from different reference frames moving at different speeds relative to the two planets and rocket.

 

These different observers will disagree on the distance the rocket travels (as measured from their frame) and they will disagree on how much time it took the rocket to make the trip (as measured from their frame). But, when those two values are squared and subtracted then they all get a value they can agree on.

 

It's the same value that the guy in the rocket agrees to. He travels zero distance (relative to himself) and experiences t time. His proper time is then his spacetime interval and everyone agrees.

 

~modest

[modest]

Still, there has been nothing offered against the assumption that electromagnetic, gravitational, and micro-scale quantum forces can traverse empty space without an intermediate agent of some kind "filling space."

 

You, yourself, have said there are fields (electromagnetic and gravitational) traversing space. A field is *by definition* an intermediate agent. I think, you don't realize the contradictions you introduce in order to validate your view.

 

How can a Newtonian vector field of unknown origin be any more satisfactory than an Einstein tensor field of unknown origin? It makes no sense to say one fills up space while the other does not.

 

~modest

[Michael Mooney]

You, yourself, have said there are fields (electromagnetic and gravitational) traversing space. A field is *by definition* an intermediate agent. I think, you don't realize the contradictions you introduce in order to validate your view.

 

How can a Newtonian vector field of unknown origin be any more satisfactory than an Einstein tensor field of unknown origin? It makes no sense to say one fills up space while the other does not.

 

~modest

 

I have not yet checked out the link you offered as my homework... My life is way too busy. So, first I will responding to the above. If math per se is your whole argument, let me remind you that math must have actual observational referents in the real world. Models by themselves are not empirical science, and theory, including the math, must be verified by evidence upon which reasonable , objective people can agree. Don't count me out just yet from the latter group just because I haven't your math expertise.

 

Is "by definition" is your argument that space must be *filled* with "an intermediate agent," some-thing to accommodate the fact that forces traverse (perhaps empty) space? I think that such fields as above do not need something like "spacetime" to convey their force, one mass upon another.

A related consideration:

Do you have an explanation for the info exchange between two "entangled particles?" (ref: QM experiments) So far everyone is mystified. Perhaps you can clear up the mystery with a more reasonable explanation of the "medium" spacetime than a methphorical graphical aid to understanding.

 

I think you don't realize the extent of your a-priori assumptions, philosophically speaking, regarding how substantive "space" must be to allow forces to traverse it.

 

Finally, since all agree that "spacetime" is not an actual malleable medium (yet *it* bends and expands and such!??), and that, ala Einstein, there would be no spacetime without the mass, etc which the equations describe, (it is not a medium in and of itself.)

 

If it is just a graphic visual aid... well established in the scientific visual cortex by computer-generated graphics... let us be honest about "what it is."

 

Finally, you claim to have answered my "six pointed questions* (minus the infinity of space... no possible boundary) in your subsequent post. Your answers appealed only to math jargon and post-Euclidean science, as if math itself has proven Euclidean space/cosmology wrong. I will first review your post above and then your next one and its link.

 

(Hope I have time for all this sooner rather than several days later!) Lots of demands on my time, and some must have priority, for reasons I will not here explain. (Not interested in sharing details of my personal life.)

I'll be back when I can.

Michael

[modest]

Finally, you claim to have answered my "six pointed questions* (minus the infinity of space... no possible boundary) in your subsequent post. Your answers appealed only to math jargon and post-Euclidean science, as if math itself has proven Euclidean space/cosmology wrong.

As I said, your questions were answered "at length in post #37 right after you asked them". There was no math in post 37.

Don't count me out just yet from the latter group just because I haven't your math expertise.

I honestly am neither expert in math nor physics.

"The worst thing that can ever happen for philosophy, and for science, is that people are so overawed by the conventional wisdom in areas where they feel inadequate (like math) that they are actually afraid to ask questions that may imply criticism, skepticism, or, heaven help them, ignorance."

 

-Dr. Kelly Ross

Is "by definition" is your argument that space must be *filled* with "an intermediate agent," some-thing to accommodate the fact that forces traverse (perhaps empty) space?

 

I'll quote Faraday:

The Sun, for instance, can be said to create a gravitational field, which spreads outward through space, its intensity diminishing as the inverse square of the distance from the Sun. Earth “feels” this gravitational field locally—right where Earth is—and reacts to it by accelerating toward the Sun. The Sun, according to this description, sends its attractive message to Earth via a field rather than reaching out to influence Earth at a distance through empty space. Earth doesn’t have to “know” that there is a sun out there, 93 million miles distant. It only “knows” that there is a gravitational field at its location. The field, though nearly as ethereal as the ether itself, can be said to have physical reality. It occupies space. It contains energy. Its presence eliminates a true vacuum. We must then be content to define the vacuum of everyday discourse as a region free of matter, but not free of field.

It's interesting, considering what you were saying about math, that this quote can be found in "Great Physicists by William H. Cropper" which goes on to explain that Faraday's ideas on this topic were not well-received and gives reasons why:

Faraday’s theories were opposed because they were revolutionary, always sufficient reason to stir opposition, and also because Faraday did not speak the sophisticated mathematical language his fellow theorists expected to hear. Beyond rudimentary arithmetic, Faraday had no mathematics; his mathematical methods were about the same as those of Galileo. In Faraday’s time, that may actually have been an advantage for creativity. The field concept was the product of “a highly original mind, a mind which never got stuck on formulas,” wrote a great twentieth-century field theorist, Albert Einstein.

Of course, what Faraday says in the quote above is now conventional wisdom. Part of the quote can be found in the third paragraph of wikipedia's page on Fields in physics.

 

I think that such fields as above do not need something like "spacetime" to convey their force, one mass upon another.

I just can't see how you're ruling-out spacetime while ruling-in any type of other unknown field.

 

The point is, there should be something mediating gravitational effects, and whatever it is, it follows the equations of General Relativity as accurately as we've ever measured.

 

~modest

[jedaisoul]

That's the idea. For any inertial reference frame, light travels one light-year per year. Light's proper time and its interval are therefore zero. All observers can agree on this.

This is true. but seems to miss the point I'm making. The zero interval expresses spacetime as light experiences it, not as we experience it. I'm concerned with the meaning of the zero interval as we experience the relationships.

 

This suggests that spacetime mechanics rests on a false premise, that what is true for light is true for material objects as well. We have shown that not to be true at one point (zero space-time interval), so it would seem logical to assume that it is equally false at all other points.

Let's say a rocket travels from one planet to another. Different people observe the event from different reference frames moving at different speeds relative to the two planets and rocket.

 

These different observers will disagree on the distance the rocket travels (as measured from their frame) and they will disagree on how much time it took the rocket to make the trip (as measured from their frame). But, when those two values are squared and subtracted then they all get a value they can agree on.

 

It's the same value that the guy in the rocket agrees to. He travels zero distance (relative to himself) and experiences t time. His proper time is then his spacetime interval and everyone agrees.

You do not seem to be "getting" the point I'm making. If we can identify one circumstance (zero intervals) where spacetime does not reflect reality as material objects experience it, then the assumption that spacetime reflects reality as material objects experience it fails. It is as simple as that. It seems to me that nothing you have said addresses that philosophic argument.

[modest]

I understand what you’re saying Jedaisoul. I’m trying to explain, it’s supposed to be zero in the case of light given what we measure.

[math]s^2 = c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2[/math]

This equation has a left and right side. The left side is supposed to represent what the object we’re describing (rocket, light, planet, etc.) experiences. The right side is supposed to represent what we (as observers) measure. It is from our perspective. This makes perfect sense with light as well as every other inertial frame.

 

We see (on the right side of the equation) light go one lightyear in one year. The right side is supposed to represent what we measure. It is then 1 - 1. The left side of the equation is what light experiences. It is zero. The equation is then satisfied: 0 = 1 - 1. This is not wrong. This is exactly what we see happen in nature. Light is not a counter-example. Light does exactly what we’d expect a null spacetime interval to do, just as much as a time-like interval does what it's supposed to do.

 

~modest

[jedaisoul]

I understand what you’re saying Jedaisoul. I’m trying to explain, it’s supposed to be zero in the case of light given what we measure.

[snip]

We see (on the right side of the equation) light go one lightyear in one year. The right side is supposed to represent what we measure. It is then 1 - 1. The left side of the equation is what light experiences. It is zero. The equation is then satisfied: 0 = 1 - 1. This is not wrong. This is exactly what we see happen in nature. Light is not a counter-example. Light does exactly what we’d expect a null spacetime interval to do, just as much as a time-like interval does what it's supposed to do.

I agree with everything you say, but none of it changes what I said. You are looking at the zero interval dynamically as "what will be seen by an object moving at that velocity". From that perspective, what you say is true. But that is not the whole story.

 

For example: I am stationary with respect to another observer who is 600,000 kilometers (two light seconds) away from me. At time zero I clap my hands. Two seconds later the other observer claps his hands. According to special relativity, there is a zero spacetime interval between these two events.

 

It is true to say that for a photon travelling from me to the other observer these events happen at the same place and time. So a zero spacetime interval makes sense for light. That is true.

 

But it is not true from my and the other observer's point of view. The events are neither simultaneous nor co-located from the perspective of either observer, nor indeed for any material object. It's only true for light. Hence the claim that spacetime reflects the universe as experienced by material objects fails. Pointing out that it is true for light is irrelevant.

[modest]

I don't know how to make it any clearer.

 

Your objection is that the left side of the equation isn't what we see (or, as I say: measure). But, the left side is not what we're supposed to measure—so this is in no way a problem.

 

For example: I am stationary with respect to another observer who is 600,000 kilometers (two light seconds) away from me. At time zero I clap my hands. Two seconds later the other observer claps his hands. According to special relativity, there is a zero spacetime interval between these two events.

 

I've drawn a diagram for this.

Instead of 2 planets, I made 3. The three observers on the 3 planets are 1, 2, and 3. Their world lines are shown parallel with the time axis. The light emitted at event a travels to event b then event c. It is the 45° line.

 

According to the spacetime separation equation, all three observers (1, 2, and 3) should measure 2 light seconds distance between a and b and likewise between b and c. It also says there should be two seconds duration between events a and b and again between b and c. This is exactly what we expect to measure/see, and it's what we see/measure.

 

You have it in your mind that the spatial distance of a null interval should be zero for us (who are not in the frame of the null interval). You expect this as if the metric implies it. The metric doesn't imply that.

 

If the interval is 6, it doesn't mean your ruler or my ruler should measure 6 meters. If the interval is 4, it doesn't mean your clock or my clock should measure 4 seconds. It's all relative. The only person that gets to do the proper measuring of the proper time is the one on the proper path.

 

~modest

[jedaisoul]

According to the spacetime separation equation, all three observers (1, 2, and 3) should measure 2 light seconds distance between a and b and likewise between b and c. It also says there should be two seconds duration between events a and b and again between b and c. This is exactly what we expect to measure/see, and it's what we see/measure.

 

You have it in your mind that the spatial distance of a null interval should be zero for us (who are not in the frame of the null interval). You expect this as if the metric implies it. The metric doesn't imply that.

I must sincerely thank you for your efforts at elucidating the matter, and I agree everything you have said. The observers should measure 2 light seconds distance between a and b (and a delay of 2 seconds in time between the events). I agree that.

 

My question is:

What is the spacetime interval between these events from my perspective "stationary" at a?

 

You see, as far as I'm aware, because the spacetime interval is invariant, and (we agree) that it is zero for light travelling from a to b, surely it must also be zero for me? If so, and I'm pretty sure that is so, we have what appears to be a logical fallacy that the spacetime interval is zero, but the events are neither co-located nor simultaneous (to me). And the same applies to any other material entities.

 

Hence I suggets that, logically, the spacetime interval does not reflect how material objects experience the universe. I can't see a logical flaw in that argument.

[modest]

My question is:

What is the spacetime interval between these events from my perspective "stationary" at a?

 

You can’t be stationary with an event. An event is a point in space and time. The observers 1, 2, and 3 are moving up in the diagram.

 

Nevertheless, the distance you measure is 2 and the duration you measure is 2. The interval is then 0 = 2 - 2. The interval is zero.

 

This is the same interval that anyone else viewing the above situation should find. Somebody else (with different relative motion) might measure the distance between the planets as 1.2 and the duration as 1.2, but the interval is still zero.

 

If so, and I'm pretty sure that is so,

 

It is.

 

we have what appears to be a logical fallacy that the spacetime interval is zero, but the events are neither co-located nor simultaneous (to me).

 

Why would they be collocated and simultaneous to you?

 

That's not how the metric works, so it obviously shouldn't be interpreted that way. In relativity; time, distance, and simultaneity are all relative. The Minkowski metric is no different. With a time-like interval (a positive interval) the only person whose clock should measure the value of the interval is the person with inertial motion between the events. All other frames will measure something different and use the interval equation to find the interval. With a space-like interval the only person who has a ruler that measures the value of the interval is the person for whom the two events are simultaneous. All other frames use the interval equation to find the interval.

 

It is then blatantly obvious how we should interpret a light-like (or zero) interval. The only frame where a clock and ruler give the value of the interval is one that intersects both events and judges them both to be simultaneous—which belongs to light. Massive particles cannot move between the events nor can we be simultaneous with them. We (you and I) are massive. We then should not expect that we would measure collocation or simultaneity of the two events.

 

You seem to have the idea that the spacetime interval is what you, personally, measure. But, reality is relative. In the metric, for a null interval, you are supposed to measure one unit space for one unit time—and you do. The spacetime interval is not what you, personally, are supposed to see.

 

~modest

[jedaisoul]

That's not how the metric works, so it obviously shouldn't be interpreted that way. In relativity; time, distance, and simultaneity are all relative. The Minkowski metric is no different. With a time-like interval (a positive interval) the only person whose clock should measure the value of the interval is the person with inertial motion between the events. All other frames will measure something different and use the interval equation to find the interval. With a space-like interval the only person who has a ruler that measures the value of the interval is the person for whom the two events are simultaneous. All other frames use the interval equation to find the interval.

Agreed.

 

You seem to have the idea that the spacetime interval is what you, personally, measure. But, reality is relative. In the metric, for a null interval, you are supposed to measure one unit space for one unit time—and you do. The spacetime interval is not what you, personally, are supposed to see.

I'm merely trying to attach a meaning to the spacetime interval (and hence to spacetime).

 

It is then blatantly obvious how we should interpret a light-like (or zero) interval.

It may be blatantly obvious to you, but my point is that the meaning of the spacetime interval is not blatantly obvious to many people. (Hence this thread?)

 

The only frame where a clock and ruler give the value of the interval is one that intersects both events and judges them both to be simultaneous—which belongs to light. Massive particles cannot move between the events nor can we be simultaneous with them. We (you and I) are massive. We then should not expect that we would measure collocation or simultaneity of the two events.

Agreed. Hence my comment that the zero spacetime interval is meaningful as the view light has of spacetime. The events are co-located and simultaneous to light. But (as we seem to agree) that meaning does not apply to what any massive object experiences.

 

The spacetime interval has been described (not by you) as "the distance in spacetime between two events". I'm trying to show that that interpretation is misleading, and you seem to agree. Does this matter? Yes, I think it does, because even some very clever people are not mathematically literate. They have little interest in the maths, but they may be interested in the philosophic meaning of spacetime. And that brings us back to the question this thread started with "what is spacetime"?

 

My answer would be:

Spacetime is a mathematical model of the universe. It is based on the spacetime interval, which is an unique description of the "distance" in space and time between two events. However, the spacetime interval is not what an observer will necessarily see as the distance in space, or interval in time, between the events. That varies. It is a "magic number" from which the distance in space, and the time interval, between two events can be computed for any observer.

 

The trouble is, I don't think that anyone not familiar with the mechanics of spacetime would find that satisfactory. Is spacetime really just a mathematical model? If not, what is it?

[jedaisoul]

I've just had another thought. Perhaps the following would be a more meaningful definition of spacetime:

Spacetime is a description of the universe. It is composed of three "spacelike" dimensions and one "timelike" dimension. A distance in spacetime is composed of two intervals:

• A "spacelike" interval. This is the distance in space between two events.
  
• A "timelike" interval. This is the time interval between two events.

The two intervals are combined into two components:

• A "lightlike" component. The "lightlike" component is the amount of spacetime where the "spacelike" interval matches the "timelike" interval between the events.
  
• A "spacelike" or a "timelike" component. This is the remainder (if any) of the "spacelike" or "timelike" interval.

The intervals are combined in this manner because the spatial distance and time interval between two events that will be observed by an observer depends upon the frame of reference in which it is observed.

This definition makes the point that spacetime is not just a mathematical model. It also stresses the "lightlike" component of the spacetime interval. Mathematically this is zero (because the spacelike and timelike intervals cancel), so is ignored mathematically, but is very important conceptually. I think it is the omission of the "lightlike" component in descriptions of the spacetime interval that leads to many misunderstandings.

 

Anyway, what do you (anyone) think?