Block Time vs Q. Indeterminacy

[nitsuj]

(-+++) is called a metric "signature"; it means the sign of timelike squared intervals is negative and the sign of spacelike squared intervals is positive. For a diagonal metric, this means the "t-t" metric coefficient is negative and the "x-x", "y-y", and "z-z" metric coefficients are positive. (A null squared interval is always zero.)

The fact that it is possible to have negative, zero, and positive squared intervals means that the metric is *not* positive definite; a positive definite metric only has positive squared intervals (except in the limiting case where we are evaluating the "interval" from a point to itself, which is zero).

Thanks for the reply.

One more thing, can an interval be the distance between two mirrors of a light clock?

Where an observer traveling very fast with the clock sees it as say one second, and an at rest observer sees it as taking more time? Is the spacetime? between the mirrors (events) of the same interval for the two observers? is that right?

 

[PeterDonis]

can an interval be the distance between two mirrors of a light clock?

Yes, it would be a spacelike interval (assuming the mirrors themselves were moving on timelike worldlines, as they would have to be to be part of a light clock). However, this interval is not the same as the interval between the events of light striking one mirror and then the other; the latter is a null interval. See below.

Where an observer traveling very fast with the clock sees it as say one second, and an at rest observer sees it as taking more time? Is the spacetime? between the mirrors (events) of the same interval for the two observers? is that right?

The spacetime interval between two given events is always the same for all observers. That's the basic foundation of SR. (Strictly speaking, this is only true when spacetime is flat, so SR is valid globally. We'll ignore the complications introduced by GR here.) However, how that interval is split up into "space" and "time" parts is observer-dependent. A pair of events that occur at the same point in space as seen by an observer at rest relative to the mirrors (say, successive bounces of the light beam off one of the mirrors) will *not* occur at the same point in space as seen by an observer to whom the mirrors are moving. So the latter observer will see a *larger* time separation between the two events, but will also see a space separation, and the interval, t^{2} - x^{2}, will be the same for both observers.

 

[nitsuj]

Yes, it would be a spacelike interval (assuming the mirrors themselves were moving on timelike worldlines, as they would have to be to be part of a light clock). However, this interval is not the same as the interval between the events of light striking one mirror and then the other; the latter is a null interval. See below.



The spacetime interval between two given events is always the same for all observers. That's the basic foundation of SR. (Strictly speaking, this is only true when spacetime is flat, so SR is valid globally. We'll ignore the complications introduced by GR here.) However, how that interval is split up into "space" and "time" parts is observer-dependent. A pair of events that occur at the same point in space as seen by an observer at rest relative to the mirrors (say, successive bounces of the light beam off one of the mirrors) will *not* occur at the same point in space as seen by an observer to whom the mirrors are moving. So the latter observer will see a *larger* time separation between the two events, but will also see a space separation, and the interval, t^{2} - x^{2}, will be the same for both observers.

Awesome, thanks Peter! I'm gunna re-read that when I get home (null interval specificaly, cause t=0? for things at c?). It looks like it's gunna help me understand spacetime diagrams / terminology better.

 

[PeterDonis]

(null interval specificaly, cause t=0? for things at c?)

Careful; a null interval is an interval whose squared length is zero, but that does *not* mean that "t = 0". It means that t^2 - x^2 = 0, where t, x are coordinates in some inertial frame, i.e., as assigned by some observer moving on a timelike worldline; which means that t = +/- x, i.e., null intervals are intervals along lines that are sloped at 45 degrees on a standard spacetime diagram. Such lines are not timelike; they are not possible worldlines for any timelike observer. So it's not a good idea to use the word "time" or anything that could be interpreted as "time" (such as "t") to refer to intervals along such lines. Null lines *are* worldlines of massless objects, such as light rays; but again, since those worldlines are not timelike, saying that "time stops" or "t = 0" for objects moving on such worldlines is not a good idea because it invites a lot of erroneous inferences.

There's a whole other thread that is largely about this issue, in which I've posted a number of times:

https://www.physicsforums.com/showthread.php?t=552175

(There are other threads running that touch on this too.)

 

[nitsuj]

Careful; a null interval is an interval whose squared length is zero, but that does *not* mean that "t = 0". It means that t^2 - x^2 = 0, where t, x are coordinates in some inertial frame, i.e., as assigned by some observer moving on a timelike worldline; which means that t = +/- x, i.e., null intervals are intervals along lines that are sloped at 45 degrees on a standard spacetime diagram. Such lines are not timelike; they are not possible worldlines for any timelike observer. So it's not a good idea to use the word "time" or anything that could be interpreted as "time" (such as "t") to refer to intervals along such lines. Null lines *are* worldlines of massless objects, such as light rays; but again, since those worldlines are not timelike, saying that "time stops" or "t = 0" for objects moving on such worldlines is not a good idea because it invites a lot of erroneous inferences.

There's a whole other thread that is largely about this issue, in which I've posted a number of times:

https://www.physicsforums.com/showthread.php?t=552175

(There are other threads running that touch on this too.)

I think that clarifies space time diagrams for me.

Simply put, st diagrams are done where ct= x and 1ct= 1x slope is the 45 degree line that represents c. This is a null line. Two events along this path ( like light passing something) one second apart is a null interval. One side of the 45 is time like(2ct = 1x), the other space like (1ct = 2x), the line itself null (1ct = 1x).

Am I getting that right?

 

[bobc2]

The way you have written your equations, it seems like X1, X4 (regardless of color) are numbers, i.e., lengths along the lines along which they're marked. That means they can't be coordinates on the same chart; blue X1, X4 are coordinates on the blue chart, and red X1, X4 are coordinates on the red chart. Are you saying that you do not intend your X1, X4 of various colors to be numbers, but that each of them are 4-tuples giving the coordinates of the points you have labeled (presumably in the black coordinate chart)?

If you are thinking of them as 4-tuples, then I see why you are saying they are "coordinates on the same chart"; but you should recognize that you are squaring these 4-tuples, so they function in your equations exactly the same as if they are numbers taken from the chart of the appropriate color, because the "square" of a 4-tuple can only be its squared length, which is equivalent to a single number giving the corresponding coordinate from the chart of the given color--i.e., the squared length of the 4-tuple "blue X1" is the *coordinate* "blue X1", i.e., the X1-component of the 4-tuple from the blue coordinate chart that describes the indicated point. So both ways of talking about your X1, X4 of various colors are equivalent in this sense.

Also, none of this is relevant to the objections I've been making, which center around the fact that the metric of spacetime is not positive definite. See further comments below.

You can freely choose the coordinates, yes. But once you choose the coordinates, you can't freely choose the metric. The metric is determined by the actual, physical intervals between points, so the metric coefficients in your chosen coordinate system are fully determined once you have chosen your coordinates.

This is where you keep missing my point. The metric of the black "rest frame" is *NOT* positive definite. Squared intervals on the underlying spacetime can be positive, negative, or zero, and the metric has to capture that. The underlying spacetime, as a *metric space*, is *not* Euclidean.

An affine space doesn't have a metric; it doesn't "know" anything about lengths. You can define basis vectors, but since there is no metric, there is no way to assign squared lengths to the basis vectors, so you can't even express the concept of a "spatial" vector as opposed to some other kind, because you can't express the concept of a "squared length", let alone its sign.

As an *affine space*, yes, you can call R4 "Euclidean", as long as you remember that that *only* refers to the *affine* properties of Euclidean space, *not* its metrical properties.

Good job, Peter. Yes, you caught me red handed trying to pass the Affine space off as a metric. I'm too use to looking at a distance and calling it a metric (or not considering the mixed use of the term, "distance"). The usual treatment to get the metric is to use Minkowski's ict for X4, then get the ++++ signature. Einstein referred to that as a Euclidean space. But, I've never liked the ict treatment. I guess it works for mathematicians (and obviously for many physicists).

Anyway, you are a great asset for this physics forum. Thanks.

 

[PeterDonis]

The usual treatment to get the metric is to use Minkowski's ict for X4, then get the ++++ signature.

Yes, this treatment appears in many textbooks and many physicists seem to like it. Hawking, for example, uses it in his "no-boundary" proposal for quantum cosmology. (I'll see if I can dig up a reference.) And it's used a lot in quantum field theory in general, where it goes by the name "Wick rotation" to confuse lay people. But not all; IIRC, Misner, Thorne & Wheeler spend a page or so explaining why they *don't* use it. I'm not a great fan of it either, since it seems to me to obscure the physical distinction between time and space.

Anyway, you are a great asset for this physics forum. Thanks.

You're welcome. Thanks for the kudos!

 

[ghwellsjr]

Good job, Peter. Yes, you caught me red handed trying to pass the Affine space off as a metric. I'm too use to looking at a distance and calling it a metric (or not considering the mixed use of the term, "distance"). The usual treatment to get the metric is to use Minkowski's ict for X4, then get the ++++ signature. Einstein referred to that as a Euclidean space. But, I've never liked the ict treatment. I guess it works for mathematicians (and obviously for many physicists).

Anyway, you are a great asset for this physics forum. Thanks.

Does this mean you are persuaded that the 4th dimension is not spatial but temporal?

 

[stglyde]

So bobc2 and PeterDonis and others,

Do you deny or accept the possibility the future already exist but contained domain of high probability and low probability... like Barbour's mist of Platonia in Hilbert Space. Watch this interview with Max Tegmark:

http://discovermagazine.com/2008/jul/16-is-the-universe-actually-made-of-math/article_view?b_start:int=3&-C=

"So the mathematical structure that is the theory of relativity has a piece that explicitly describes time or, better yet, is time. But the integers don’t have anything similar.

Yes, and the important thing to remember is that Einstein’s theory taken as a whole represents the bird’s perspective. In relativity all of time already exists. All events, including your entire life, already exist as the mathematical structure called space-time. In space-time, nothing happens or changes because it contains all time at once. From the frog’s perspective it appears that time is flowing, but that is just an illusion. The frog looks out and sees the moon in space, orbiting around Earth. But from the bird’s perspective, the moon’s orbit is a static spiral in space-time.

The frog feels time pass, but from the bird’s perspective it’s all just one eternal, unalterable mathematical structure.

That is it. If the history of our universe were a movie, the mathematical structure would correspond not to a single frame but to the entire DVD. That explains how change can be an illusion.

Of course, quantum mechanics with its notorious uncertainty principle and its Schrödinger equation will have to be part of the theory of everything.

Right. Things are more complicated than just relativity. If Einstein’s theory described all of physics, then all events would be predetermined. But thanks to quantum mechanics, it’s more interesting.

Plausible conclusion: Although the future may already exist. They are not definite. There are Hilbert Platonia mist where Obama would be reelected. Another probability mist where Arnold Schwarzenegger would become president. And everything is not yet set in stone.. but the probabilities are not unlimited but set within certain boundaries or limits of possibilities.

No problem about this aspect of the future. But the more question now. Does the past still exist? If not, why do many physicists explore time travel to a past that still exist? I'm partly interested in this because I'd like to go 5 years prior to do some things right.

 

[PeterDonis]

Simply put, st diagrams are done where ct= x and 1ct= 1x slope is the 45 degree line that represents c. This is a null line.

Yes.

Two events along this path ( like light passing something) one second apart is a null interval.

Yes; to clarify a bit, two events along the worldline of a light ray which are separated by one second in time will also be separated by one second in space--i.e., one light-second, or 3 x 10^8 meters. The same for any other time separation.

One side of the 45 is time like(2ct = 1x), the other space like (1ct = 2x), the line itself null (1ct = 1x).

Yes; again, to clarify a bit, the timelike "side" of the 45-degree line (which is usually called the "light cone" because it looks like a cone if we put back in one more spatial dimension) contains worldlines where t > x, i.e., the "slope" dt/dx is greater than 1 (it doesn't have to be 2, it can be any value > 1). This is more usually stated as the velocity, dx/dt, being less than 1 (i.e., less than the speed of light, which is 1 in the units usually used in SR, where c = 1).

The spacelike "side" of the light cone, OTOH, contains curves (which aren't called "worldlines" because they're not possible paths for any real object) for which t < x, i.e., the "slope" dt/dt is less than 1.

 

[PeterDonis]

Plausible conclusion: Although the future may already exist. They are not definite. There are Hilbert Platonia mist where Obama would be reelected. Another probability mist where Arnold Schwarzenegger would become president. And everything is not yet set in stone.. but the probabilities are not unlimited but set within certain boundaries or limits of possibilities.

This seems like a reasonable summary of what you quoted from the Tegmark interview.

No problem about this aspect of the future. But the more question now. Does the past still exist? If not, why do many physicists explore time travel to a past that still exist? I'm partly interested in this because I'd like to go 5 years prior to do some things right.

From our perspective (the frog's perspective, as it's described in the Tegmark quote), no, the past doesn't "still" exist. "Still" is a concept that's only applicable from the frog's perspective. The frog may *remember* past events, but those memories are not in the past, they're in the present; they are part of the frog's present state. A week from now, the frog may remember events that are happening now, but those events will not "still exist" a week from now; only their traces in the frog's memory (or other records that they leave) will exist then.

However, this does not, in itself, prevent time travel to the past; from the frog's perspective, traveling to a point in time five years ago would be just like traveling to tomorrow; you would experience time "flowing" forward as usual, but your experience would happen to pass through events that it had passed through before. One of the best descriptions of what this could be like that I've read, at least in fiction, is the classic Heinlein story By His Bootstraps:

http://en.wikipedia.org/wiki/By_His_Bootstraps

From the bird's perspective, the word "still" does not apply. The entire 4-dimensional spacetime is a single "thing" that is just there; it does not "flow" or "evolve" or anything like that. If this single "thing" happens to contain closed timelike curves, i.e., timelike curves (possible worldlines that "frogs" could follow) that pass through the same event more than once, then time travel to the past is part of the 4-dimensional thing.

 

[stglyde]

This seems like a reasonable summary of what you quoted from the Tegmark interview.



From our perspective (the frog's perspective, as it's described in the Tegmark quote), no, the past doesn't "still" exist. "Still" is a concept that's only applicable from the frog's perspective. The frog may *remember* past events, but those memories are not in the past, they're in the present; they are part of the frog's present state. A week from now, the frog may remember events that are happening now, but those events will not "still exist" a week from now; only their traces in the frog's memory (or other records that they leave) will exist then.

However, this does not, in itself, prevent time travel to the past; from the frog's perspective, traveling to a point in time five years ago would be just like traveling to tomorrow; you would experience time "flowing" forward as usual, but your experience would happen to pass through events that it had passed through before. One of the best descriptions of what this could be like that I've read, at least in fiction, is the classic Heinlein story By His Bootstraps:

http://en.wikipedia.org/wiki/By_His_Bootstraps

From the bird's perspective, the word "still" does not apply. The entire 4-dimensional spacetime is a single "thing" that is just there; it does not "flow" or "evolve" or anything like that. If this single "thing" happens to contain closed timelike curves, i.e., timelike curves (possible worldlines that "frogs" could follow) that pass through the same event more than once, then time travel to the past is part of the 4-dimensional thing.

But if Many Worlds were true. Things may not be that simple. If quantum choices can split worlds, so can choices in the future or past be their own worlds. According to David Deutch and Michael Lockwood in their article "The Quantum Physics of Time Travel" (saw complete article link in 2009 archive https://www.physicsforums.com/showthread.php?t=360188 ):

"What, then, does quantum mechanics, by Everett's interpretation, say about time travel paradoxes? Well, the grandfather paradox, for one, simply does not arise. Suppose that Sonia embarks on a paradoxical project that, if completed, would prevent her own conception. What happens? If the classical space-time contains CTCs, then, according to quantum mechanics, the
universes in the multiverse must be linked up in an unusual way. Instead of having many disjoint, parallel universes, each containing CTCs, we have in effect a single, convoluted space-time consisting of many connected universes. The links force Sonia to travel to a universe that is identical, up to the instant of her arrival, with the one she left, but that is thereafter different because of her presence."

Can you refute it or give penetrating arguments why Many Worlds can't be true in the Present, Past or Future Timelines?

 

[atyy]

But if Many Worlds were true. Things may not be that simple. If quantum choices can split worlds, so can choices in the future or past be their own worlds. According to David Deutch and Michael Lockwood in their article "The Quantum Physics of Time Travel" (saw complete article link in 2009 archive https://www.physicsforums.com/showthread.php?t=360188 ):

"What, then, does quantum mechanics, by Everett's interpretation, say about time travel paradoxes? Well, the grandfather paradox, for one, simply does not arise. Suppose that Sonia embarks on a paradoxical project that, if completed, would prevent her own conception. What happens? If the classical space-time contains CTCs, then, according to quantum mechanics, the
universes in the multiverse must be linked up in an unusual way. Instead of having many disjoint, parallel universes, each containing CTCs, we have in effect a single, convoluted space-time consisting of many connected universes. The links force Sonia to travel to a universe that is identical, up to the instant of her arrival, with the one she left, but that is thereafter different because of her presence."

Can you refute it or give penetrating arguments why Many Worlds can't be true in the Present, Past or Future Timelines?

I'm not sure such a theory of quantum gravity is known. We do have a working theory of quantum gravity as a low energy effective theory. It assumes that the topology of spacetime is boring.

 

[PeterDonis]

If the classical space-time contains CTCs, then, according to quantum mechanics, the universes in the multiverse must be linked up in an unusual way. Instead of having many disjoint, parallel universes, each containing CTCs, we have in effect a single, convoluted space-time consisting of many connected universes. The links force Sonia to travel to a universe that is identical, up to the instant of her arrival, with the one she left, but that is thereafter different because of her presence."

Can you refute it or give penetrating arguments why Many Worlds can't be true in the Present, Past or Future Timelines?

I don't see that I would have to refute it. What you say looks OK to me, and I don't see any inconsistency with what I said. Considering Sonia as a "frog", she still experiences "flowing" along her worldline normally; when she arrives at the event that, classically, would complete a CTC, she simply experiences the alternate version in which she is present at that event (where, supposing she passed through the same event before, she would remember experiencing the "original" version where she was not present). So the frog's eye view still works for Sonia.

From the bird's-eye perspective, as the quoted passage says, spacetime is now a convoluted, multiply-connected thing, instead of a simply-connected thing that happens to contain CTCs. But it's still a single thing that is just "there", and the statement "the past still exists" is still not applicable from this point of view. From Sonia's point of view, the event at which she arrives back in her own past can be considered a "splitting point", where a decision she makes causes the past to "change"; but from the bird's-eye perspective, her "decision" is simply an event on her convoluted worldline in the convoluted multiply connected spacetime, which does not change; it is all there "at once" as part of the whole spacetime.

 

[stglyde]

I don't see that I would have to refute it. What you say looks OK to me, and I don't see any inconsistency with what I said. Considering Sonia as a "frog", she still experiences "flowing" along her worldline normally; when she arrives at the event that, classically, would complete a CTC, she simply experiences the alternate version in which she is present at that event (where, supposing she passed through the same event before, she would remember experiencing the "original" version where she was not present). So the frog's eye view still works for Sonia.

From the bird's-eye perspective, as the quoted passage says, spacetime is now a convoluted, multiply-connected thing, instead of a simply-connected thing that happens to contain CTCs. But it's still a single thing that is just "there", and the statement "the past still exists" is still not applicable from this point of view. From Sonia's point of view, the event at which she arrives back in her own past can be considered a "splitting point", where a decision she makes causes the past to "change"; but from the bird's-eye perspective, her "decision" is simply an event on her convoluted worldline in the convoluted multiply connected spacetime, which does not change; it is all there "at once" as part of the whole spacetime.

Do you agree that there are two kinds of time. Real time experienced and Pime (Parameter Time) in Spacetime equations? And distinguising them can illuminate many areas in physics? It's according to Demystifier who wrote:

http://fqxi.org/data/essay-contest-files/Nikolic_FQXi_time.pdf

Do you or do you not agree with it and why?

 

[PeterDonis]

Do you agree that there are two kinds of time. Real time experienced and Pime (Parameter Time) in Spacetime equations? And distinguising them can illuminate many areas in physics? It's according to Demystifier who wrote:

http://fqxi.org/data/essay-contest-files/Nikolic_FQXi_time.pdf

Do you or do you not agree with it and why?

I'll have to read the paper you linked to before I can really respond to your questions, since I'm not sure what specifically the issue is that you're concerned about. From a quick skim of the abstract, it looks like the distinction between "Real time" and "Pime" is more an issue of theories of consciousness than theories of physics.

 

[nitsuj]

Yes.
Yes; to clarify a bit, two events along the worldline of a light ray which are separated by one second in time will also be separated by one second in space--i.e., one light-second, or 3 x 10^8 meters. The same for any other time separation.
Yes; again, to clarify a bit, the timelike "side" of the 45-degree line (which is usually called the "light cone" because it looks like a cone if we put back in one more spatial dimension) contains worldlines where t > x, i.e., the "slope" dt/dx is greater than 1 (it doesn't have to be 2, it can be any value > 1). This is more usually stated as the velocity, dx/dt, being less than 1 (i.e., less than the speed of light, which is 1 in the units usually used in SR, where c = 1).

The spacelike "side" of the light cone, OTOH, contains curves (which aren't called "worldlines" because they're not possible paths for any real object) for which t < x, i.e., the "slope" dt/dt is less than 1.

Yea a cone is 2 spatial dimensions, a sphere for 3 i guess. The diagram helps me ignore time as being the period between events and more on it being simply a "cause effect" thing. Said differently the null line illustrates causality (I can't picture where it goes with a sphere). Oh cool, the moon is 3d, the distance between it and me is the 4th lol (i pictured causality as a sphere, neat-o)

I wonder if it's a shared point; having to measure the two way speed of light to calculate c and causality. So in a big stretch, "reality" falls onto an observer at c from all around. ANY direction you move in changes things accordingly, via the Lorentz transformations and the constancy of c, or causality in this example. I just got all Greene there

Time is easily identified as a dimension from this perspective; time and distance in meters. However, block universe really doesn't pass the smell test now.

 

[bobc2]

From the bird's-eye perspective, as the quoted passage says, spacetime is now a convoluted, multiply-connected thing, instead of a simply-connected thing that happens to contain CTCs. But it's still a single thing that is just "there", and the statement "the past still exists" is still not applicable from this point of view. From Sonia's point of view, the event at which she arrives back in her own past can be considered a "splitting point", where a decision she makes causes the past to "change"; but from the bird's-eye perspective, her "decision" is simply an event on her convoluted worldline in the convoluted multiply connected spacetime, which does not change; it is all there "at once" as part of the whole spacetime.

Good way to size it up. Peter gets it right and puts things in the proper perspective, as usual.

 

[ghwellsjr]

Good way to size it up. Peter gets it right and puts things in the proper perspective, as usual.

As I asked earlier:

Does this mean you are persuaded that the 4th dimension is not spatial but temporal?

 

[PeterDonis]

Time is easily identified as a dimension from this perspective; time and distance in meters. However, block universe really doesn't pass the smell test now.

Why not?

 

[nitsuj]

Why not?

Simply because I can't picture it.

I could from the point of veiw of there being only 2 spatial dimensions and one time. Said differently that the sum of individual "Now Slices" make up 3D (and only because time "plays out" / distance).

Now that I can picture the Spacetime diagram from 1 space dimension through to the 3 there are, I cannot see how the block universe [STRIKE]"plays out" in time[/STRIKE] has three spatial dimensions.

The "present" moment surrounds me as a sphere. There is no "this direction future" that "direction past". Perhaps from a conscious perspective it could be thought of as future is out there (outside what ever one would define as the present moment sphere) and the past is merely the memory. Ah, the future "comes in" / "falls onto" the observer from all directions, there's no "room" for the past lol.

I may be wrong with my understanding of block universe, among a number of other things :)

EDIT: I guess I'm not buying the block universe concept moslty because of the spatial dimensions. I don't see three spatial ones in the block universe. I am going to read some more about it to try and see how it is 3 spatial dimensions and a time one, but I don't think it's there. Just can't see how.

 

[PeterDonis]

Simply because I can't picture it.

Obviously you can't picture a 4-dimensional thing with a mind that's only able to visualize 3 dimensions. What does that have to do with the physics?

EDIT: I guess I'm not buying the block universe concept moslty because of the spatial dimensions. I don't see three spatial ones in the block universe. I am going to read some more about it to try and see how it is 3 spatial dimensions and a time one, but I don't think it's there. Just can't see how.

One of the reasons we invented mathematics was to be able to reason in areas where we can't trust our intuitive capacities, such as "visualizing" things. Mathematically, 4-D spacetime is just 3-D spacetime (2 spatial plus 1 time), which you have said is perfectly OK, with one more spatial dimension added. Adding the one more spatial dimension creates no problems at all, mathematically; the model is still perfectly consistent and the tools for dealing with it still work perfectly well.

To make things as simple as possible, try starting with plain flat Minkowski spacetime, the full 4-D ("3+1") version. The metric is:

d\tau^{2} = dt^{2} - dx^{2} - dy^{2} - dz^{2}

The corresponding "2+1" version is:

d\tau^{2} = dt^{2} - dx^{2} - dy^{2}

What's the problem with adding the - dz^{2}?

 

[nitsuj]

Obviously you can't picture a 4-dimensional thing with a mind that's only able to visualize 3 dimensions. What does that have to do with the physics?

Nice tone peter. In that sense, what does physics have to do with reality then?

When you say we can't visualize a 4 dimensional thing 'cause we're only able to visualize 3 dimensions, leads to the point of confusion over time as a dimension. I don't think a dimension (specifically time) is necessarily spacial. So with that being said...

I can easily visualize the 3 spatial dimensions and another for the interval between me and whatever else I see. Or that whatever I see, it will always be cause first, then effect no matter how fast I move to try and exceed my present moment.


My comments are regarding intupreting the block universe visualy as being a 4D spacetime continuum. I can't see it. I dislike the concept more then before. But I still want to read more about it. But not from PF threads, too heated/biased.

 

[PeterDonis]

My comments are regarding intupreting the block universe visualy as being a 4D spacetime continuum. I can't.

Yes, I can't visualize all four dimensions at once either. If that's all you are saying then I'm confused about why you said the block universe concept "doesn't pass the smell test", since that implies that you think it's not valid physics, not just that you can't visualize it.

I can easily visualize the 3 spatial dimensions and an other for the interval between me and whatever else I see.

If by "interval between me and whatever else I see" you mean "interval between the 3 dimensional slice I call 'now' and some other 3 dimensional slice from which the light I am seeing 'now' was emitted", then I don't think this is any different from the 4-D "block universe", except insofar as some people who talk about the "block universe" talk about it as though any concept of 4-D spacetime required complete determinism, which it doesn't.

 

[nitsuj]

Yes, I can't visualize all four dimensions at once either. If that's all you are saying then I'm confused about why you said the block universe concept "doesn't pass the smell test", since that implies that you think it's not valid physics, not just that you can't visualize it.



If by "interval between me and whatever else I see" you mean "interval between the 3 dimensional slice I call 'now' and some other 3 dimensional slice from which the light I am seeing 'now' was emitted", then I don't think this is any different from the 4-D "block universe", except insofar as some people who talk about the "block universe" talk about it as though any concept of 4-D spacetime required complete determinism, which it doesn't.

Oh I see now, my comment about it not passing the smell test is far from an interpretation of the block universe from the perspective of a seasoned physicist. Is it valid physics? In this sense I haven't the slightest clue how to determine what is valid physics and what isn't.

Seems too strange to be able to say litteraly, the start of the universe is on one side of me, the future of it, on the other side of me. Any hoo, ima read more about block universe, since I don't doubt your belief in the block universe.

 

[stglyde]

Nice tone peter. In that sense, what does physics have to do with reality then?

When you say we can't visualize a 4 dimensional thing 'cause we're only able to visualize 3 dimensions, leads to the point of confusion over time as a dimension. I don't think a dimension (specifically time) is necessarily spacial. So with that being said...

I can easily visualize the 3 spatial dimensions and another for the interval between me and whatever else I see. Or that whatever I see, it will always be cause first, then effect no matter how fast I move to try and exceed my present moment.


My comments are regarding intupreting the block universe visualy as being a 4D spacetime continuum. I can't see it. I dislike the concept more then before. But I still want to read more about it. But not from PF threads, too heated/biased.

Just imagine 4 dimensional space and time as absolute. Only time and space are relative. If you are stationary in space, all energy is allocated for time so your time moves fast. If you now move in space, some of the energy for time is allocated for space so time slows down. Now block time is simply cutting the absolute 4D spacetime in different angles. This is one good quick way to visualize spacetime.

 

[nitsuj]

Just imagine 4 dimensional space and time as absolute. Only time and space are relative. If you are stationary in space, all energy is allocated for time so your time moves fast. If you now move in space, some of the energy for time is allocated for space so time slows down. Now block time is simply cutting the absolute 4D spacetime in different angles. This is one good quick way to visualize spacetime.

Is that all that's meant by block universe? I completely agree with that description. Seems to be the way it is. (leaving out things like "energy is allocated for time..." idk what that means, but am aware of the term 4 velocity, and leaving out 4D space, i think it's only 3D and spacetime 4D)

I thought block universe also implied that the future and past are equally as "real" as the present. And from there things like traveling to the past come up, and that's why I [had] disliked the concept.

When I pictured a light cone as a sphere, it really made it clear only the future is out "there" lol, so that's why i said block universe doesn't pass the smell test.

Out of future, past, present & time, the past seems to be the only thing that's an "illusion". I guess for me that's what really muddied the water trying to understand what time is. There's future and present / cause and effect, but no "past". (the past spacetime is still there of course)

Oh lol, that's why it's called a continuum right? cause-effect-cause-effect...

 

[PeterDonis]

I thought block universe also implied that the future and past are equally as "real" as the present. And from there things like traveling to the past come up, and that's why I [had] disliked the concept.

That's what I was trying to clarify, what you thought didn't pass the "smell test".

I think of it this way: a "block universe" is a mathematical model that describes a *possible* 4-D spacetime. But to correlate this model with the real world, we have to divide it, conceptually, into three portions.

(1) A portion of this 4-D spacetime is already known to correspond with the real world; that's the part we call the "past". This is the portion that's in the past light cone of the point in the model that corresponds to where we are here and now. This portion of the model cannot change, in the sense that it specifies a single set of conditions that must be fulfilled by *any* possible 4-D model of the complete spacetime. But what's in our past light cone is not sufficient to pick out a single unique 4-D model for the entire spacetime, so we have to consider a number of "possible" models.

(2) Another portion of this 4-D spacetime is the part that we can affect, causally, from here and now. This is the part we call the "future", and decisions we make here and now can "change" it, so we think of it as not being fixed. But in order to "change" it, we have to change the conditions here and now, and that means the model changes; we have to re-compute what the future will be from the changed conditions, and that gives us a *different* 4-D spacetime, which is now our "best estimate" of what block universe is the "real" one (out of all the "possible" ones that are consistent with our knowledge up to now).

(3) The third portion of this 4-D spacetime is not known by us, here and now, to correspond with the actual world, because it's outside of our past light cone; but we also can't affect it causally, because it's outside of our future light cone. The only thing we can do is to compute, based on what we know of our past light cone, what this portion "should" look like; but at any time, we might have new information come to us, as more of this region comes within our past light cone, that forces us to re-compute. And again, every time we re-compute, that yields a *new* 4-D spacetime that becomes our new best estimate of what block universe is the real one.

So each block universe, considered as a single 4-D model of a possible spacetime, is fixed; but we don't know exactly which possible 4-D model corresponds to reality; the best we can do is to estimate it based on the information we have and the decisions we make. A portion of each model we compute is fixed--the part that corresponds to our past light cone. But the rest of it can vary based on our decisions and on information coming in from regions that were spacelike separated from us until just now.

 

[bobc2]

As I asked earlier:

I wish. I would really like to see a 4-dimensional spatial universe dispelled. Then, we would be left with just deep mystery--something I could handle. It would be more comfortable to have just the mystery as compared to the spectre of implications that come along with the 4-dimensional space-space and still have mystery lingering.

No, I can't yet shake the 4-D space-space. But, PeterDonis did a credible job of flagging my blunder of identifying the Affine Space as a metric space.

Now, I just need PeterDonis to build the mathematical machinery properly, beginning with a R4 manifold and positive definite metric that allows me to freely choose coordinates (without resorting to Minkowski's imaginary ict). We then establish a 4-D orthonomal vector space consistent with defining Pythagorean Theorem distances in the coordinate system.

I would then try to visualize a simple example universe having two straight line 4-D objects that each begin at the orgin of the orthonormal coordinate system, tracing out world lines angled symmetrically about the 4th dimension of the orthonormal vector space. One object is rotated counter-clockwise 22.5 degrees, and the other object is rotated clockwise by 22.5degrees. PeterDonis would have to give us the field theory that allows a correct mathematical accounting of the 4-D objects.

At this point we are saying nothing about time and nothing about observers. There are no affine X1 coordinates established--and no charts--unless PeterDonis insists that we cannot have two 4-D objects in our universe without a chart. In that case we would have to let the two objects define the chart.

Now, I would want to choose another set of orthonormal coordinates, X1' and X4', rotated counter-clockwise by 22.5 degrees relative to the original orthonormal coordinates (X4' is rotated 22.5 degrees counter-clockwise to X4, and X1' is rotated 22.5 degrees counter-clockwise to X1).

We now pick a point on the clockwise rotated 4-D object and note that the distance from the orgin to the point is invariant--it can be determined equally well with either X1 and X4 or X1' and X4'.

If PeterDonis O.K.'s things so far, we can continue the pursuit of the affine space mapped onto the orthonormal cartesian space. (we still don't have any observers in our universe)

 

[PeterDonis]

Now, I just need PeterDonis to build the mathematical machinery properly, beginning with a R4 manifold and positive definite metric

Why do you keep insisting on a positive definite metric? The whole point of Minkowski spacetime is that the metric is *not* positive definite. That's what you need to describe a *spacetime* as opposed to a *space*.