I suspect that the basis for many disagreements regarding SR and GR are based in a misunderstanding of the notion of space-time.

I'd like people to post their impression of what space-time is, and what it implies, as they think Einstein and his followers intend it.

To me, I see space time in the typical rubber sheet analogy.. but it's hard to envision the time aspect of the rubber sheet, and the implications as to whether or not two events are simultaneous.

It's my primitive understanding that two events, or X events can only be simultaneous to a separate observer.... that Frame A and Frame B can only really be thought of as simultaneous from a third frame "C" or "the observer". I see it as "any frame sees any other combination of frames as simultaneous", but they are not necessarily so at all.

That's all I'll say about that for now.

*Please, this is not intended to be a discussion of other theories.. but how SRT and GR use the notion of Space-time... and it seems to me that it's a nefarious concept that people struggle to comprehend.. and that perhaps it's this difficulty that impedes communication amongst those in the ongoing arguments on the topic.

Please post your thoughts, clarifications, etc. on the topic.

Spacetime is a four-dimensional manifold possessing a metric. Three of the dimensions are regarded as "spatial," while the other is considered "time." There is no distinction between space and time dimensions mathematically -- the only difference appears in the metric, wherein the time dimension has a sign opposite that of the spatial dimensions.

- Warren

There is no time. Time is nonexistent. Time is just a way of measuring and comparing the rate of change of physical entities with respect to other physical entities. You cannot measure time without the existence of mass and this therefore supports the idea that time is intrinsically linked to our physical surroundings. When you measure time, you are not measuring time but instead measuring a comparison between one set of arbitrary physical oscillations with some other set of events. It's as simple as that, in MY opinion. Time dilation is not TIME dilation, but a PHYSICAL dilation of matter itself.

This idea renders the "time" bit in spacetime nonsense. What you're left with is "space" and that's where the problem begins.

Oh, and don't listen to dav57, he has no idea what he's talking about.

- Warren

Spacetime is a four-dimensional manifold possessing a metric. Three of the dimensions are regarded as "spatial," while the other is considered "time." There is no distinction between space and time dimensions mathematically - WarrenAgreed, but let's flesh it out a little. The 4-manifold "space-time" is a man-made construction to help us better understand our world. We could choose any N-manifold to do the same job, but parsimony does not require this.
The idea that coordinate systems are somehow intrinsic to "reality" is a misconception, as looking out your window would show readily enough (huh? see any coordinates out there?).... the only difference appears in the metric, wherein the time dimension has a sign opposite that of the spatial dimensions.Which is, of course, only true if you are using the Minkowski metric - other metrics don't impose that restriction.

EDIT IN: Hmm, is that right? Scrabbling for buried textbooks....Oh, and don't listen to dav57, he has no idea what he's talking about.C'mon, this is sciforums, man!!

There is no time........ supports the idea that time is intrinsically linked to our physical surroundings. Which is merely to reiterate what we all know - space and time cannot be meaningfully seperated.
This idea renders the "time" bit in spacetime nonsense. What you're left with is "space" and that's where the problem begins.Oh no! That's where it ends. What, you think "space", as in spatial coordinates, is a thing, like in Star-Trek maybe?

I agree totally with chroot...here is an easy way to "see" that time "exists" as a dimension within our universe.

In a 2 dimensional graph : y=2 as a definition of an "entity" is what? A line at y=2 that extends the length of the x axis.

Add a dimension (axis z), y=2 is a plane that extends the length of the x axis AND z axis.

Same example but define x and y as 2 (x=2,y=2); now you have a line that is at (2,2) that extends the length of the z axis.


Now add another dimension (axis t) and define x,y,z each as 2 (2,2,2). We now have a point that exists at (2,2,2) and extends the length of axis t.
It is hard to picture is not it? Here is a practical application: Define a meeting for you and a date. You would say something to the effect of corner of 22nd St and 14th Ave. You have now defined a point on a two dimensional coordinate system. It is understood that you are going to meet your date on the ground so y=0. So now we have a coordinates defined along 3 axis, in a 3 dimensional representation of the world? Is there anything missing? If you HAVE to stay there (at that exact location) until you meet your date, do not you think it might be a good idea to specify a location on axis t as well? Otherwise, as in my last example above you only have 3 dimensions of a 4 dimensional system defined, the result of this is a line that extends infinitely along the t axis (4th axis). Therefore, the only way you can be sure to meet your date is to stand there until they come...this may be today, tommorow, next week, or 2000 years from now.

To accurately describe any event in our universe you need 4 pieces of relavent information: distance along axis x,y,z, and t relative to a reference point.

- KitNyx

Spacetime is a four-dimensional manifold possessing a metric. Three of the dimensions are regarded as "spatial," while the other is considered "time." There is no distinction between space and time dimensions mathematically -- the only difference appears in the metric, wherein the time dimension has a sign opposite that of the spatial dimensions.

- Warren

Excellent, concise. Thank you.

If you don't mind, please tell me if the following is consistent with your explanation:

"It's my primitive understanding that two events, or X events can only be simultaneous to a separate observer.... that Frame A and Frame B can only really be thought of as simultaneous from a third frame "C" or "the observer". I see it as "any frame sees any other combination of frames as simultaneous", but they are not necessarily so at all."

?

To accurately describe any event in our universe you need 4 pieces of relavent information: distance along axis x,y,z, and t relative to a reference point.

- KitNyxTo all your your post, yes (apart from split infinitives). A point in space-time is an event. Let's say this this.
Any two or more coincident points in space-time are the same event, or different manifestations of it.

To all your your post, yes (apart from split infinitives). A point in space-time is an event. Let's say this this.
Any two or more coincident points in space-time are the same event, or different manifestations of it.

Can you give an example of two or more coindident points in space-time?

If it is the same event as you say, then would you not be dicussing different features rather than manifestations?

I almost said manifistation. Got sort of a sick chuckle out of that.

see here,
the trouble with formalism -mathematical logics- is that the scope of information is beyond the breadth of conscious/simultaneous apprehension... yeah, i said it. so that even if you know a LOT, you're still missing some pieces, and this encourages us to accept
logical trajectories that don't take comprehensive account of possible explanations...
so, regarding the minkowskian/gaussian 'block universe' explanation suggested (not unexpectedly) by chroot, kitnyx... i dunno. This is a vestige of a post-Einsteinian orthodoxy. But it don't seem right... too conclusive, ain't it? -the implication seems to be that all events, past, present and future, exist as this static mathematical entity... and a fairly simple one... but universe doesn't seem to work that way... does it?
i mean, not to deny the fairly obvious logic of kitnyx's explanation, or the explanatory power of the minkowskian 4space, ...just that certain phenomena remain unexplained thereby, and that our spacetime conceptions may underly these limitations. like, what we know can sometimes prevent us from finding what we're looking for... yeah?

=b

p.s. for an example of an interesting 'hole' in the s/g rt paradigm, check out the thread on 'the death of relativity'
~

To accurately describe any event in our universe you need 4 pieces of relavent information: distance along axis x,y,z, and t relative to a reference point.

- KitNyx

Hmm.

So in order to accept or intelligently discuss this model, you have to accept (for the purposes of the discussion) that time is relative?

So without a clock specific to x, y, z... , t has no basis?

To me, this seems to be the root of the objection given by anti SRT folks.

It seems more of a failure of comprehension of the concepts involved, or a flat rejection of the possibility than anything else.

Maybe I just fail to comprehend them.

Can you give an example of two or more coindident points in space-time?

If it is the same event as you say, then would you not be dicussing different features rather than manifestations?

Thunder and lightning, different manifestations of the same event.
Coincident in space-time. Not necessarily so for an observer, say, at 10km distance.
You want to say there's some profound difference between "features" and manifestations"?

from quark:
"A point in space-time is an event."
yeah. but an event has to have a duration, so there wouldn't really be any such thing as
"points" in spacetime, would there? we'd be talking trajectories... which way are they going? -deterministic 'future already exists' scenario results. what of it?

Well it seems to me to be that way, yes. A feature is part of something. A manifestation (at least to me) is apart from the original thing... It's what was the thing, manifested into whatever, but it's not the thing. So yeah that seems like a big difference to me. I just want to be sure I understand the intended meaning as clearly as possible.

"Thunder and lightning, different manifestations of the same event."

nicely put quark, but lemme ask; thunder results from lightning, don;t it?
i mean, doesn't the constancy of lightspeed suggest that light is somehow the ultimate driver of physical changes?

How can people think time is something "real".

Imagine that there were no humans, no creatures, no life, no observer in the universe... what would time mean? What would be the "difference" between a second and a thousand years... between a thousand years and infinity? Time would simply not exist without us... without consciousness... mind... (as a matter of fact... nothing would exist without an observer) Time and matter are just illusions in our consciousness.

There are several problems with the rubber sheet model:

1) It's 2-D. Spacetime is 4-D. Even a 3-D model of spacetime would be closer. The rubber sheet only shows spatial distortion in one direction--down.

2) It assumes a force of gravity below the sheet. For an object placed on the rubber sheet to create a dent in the sheet, a larger force of gravity must exist below the sheet--bypassing the theory of gravity as a function of spacetime in the first place.

3) There is no way to model energy on a rubber sheet, except as a line. Also, travel is limited to two dimensions in this model.

A better model for spacetime might be a gel consistency, where objects that spin drag the spacetime with it, heavy objects bend the gel towards the object, and objects traveling through the gel will bend as the gel towards the larger objects. Spacetime is often described as a sheet or fabric, but these words connotate something so radically limited that it cannot viably describe the concept.

Spacetime in of itself is what the universe is molded from. To describe precisely what it "is" in terms of matter and energy is impossible. Since all of our experiences deal with matter and energy, ascribing a metaphor to it is equally difficult. The best metaphor for spacetime is probably the foundation of existence, where matter and energy are spawned and interact. It is rather like the arena where the events take place.

from quark:
"A point in space-time is an event."
yeah. but an event has to have a duration, so there wouldn't really be any such thing as
"points" in spacetime, would there? Be careful, here! In space-time you have to be careful what you say.
I said a point in space-time is an event. You then start talking about "duration".
A point in space-time has a unique set of coordinates which, by the definition, has zero extension along any of the four coordinates. In other words, an "event" is a zero-dimensional point in space-time. And hence unique.

but check it out:
"A point in space-time has a unique set of coordinates which, by the definition, has zero extension along any of the four coordinates. In other words, an "event" is a zero-dimensional point in space-time. And hence unique." -q

-you can't have an event with zero dimensions. for any 'event' to occupy spacetime, it must have values at 4 (by convention) ds. these values are, by definition, 'durational' (assuming nominal equivalency of space/time dimensions). zero-point existence is in fact nonexistence.
therefore, i would suggest, (quoting 12night) that a view of spacetime as "the arena where events take place" is incomplete, and misleading.
events don't 'take place' within an imaginary lattice of determinate dimensions... rather, the spacetime i envision emerges as a result of the interactions among events, and is comprised of an accumulate of possible event-trajectories resulting from the interaction.
the theory of relativity is poised for expansion to encompass these interactions, in my opinion. Q&A?
=b

Oh, and don't listen to dav57, he has no idea what he's talking about.

- Warren

Ok, then show me how time exists without the existence of matter.

If you can't, then doesn't that go to show that time IS directly related to the existence of matter?

Just because you've digested and preach textbook physics to a relatively high level doesn't mean to say that what YOU preach is correct. Many great scientists have experienced ultimate embarrassments previously! I'm sure you're not entirely exempt from this possibility.

Now, please explain how time exists without mass?

Ok, then show me how time exists without the existence of matter.

Now, please explain how time exists without mass?Which do you want? They are not the same thing.
Time is one of 4 coordinates invented by Man to help describe the world and events in it. Time exists, in this sense, because we say it does, whether matter is present or not.

It's perfectly true, that there is a natural sense of "before" and "after", but it's a big logical step to say "therefore time is natural".

It's also perfectly valid, but completely trivial, to say that, as time is a man-made invention, in the absence of men there is no time.

In fact, I defy you to say what you mean by "time exists" with any degree of precision, other than the sense that I gave.

I suspect that the basis for many disagreements regarding SR and GR are based in a misunderstanding of the notion of space-time.

I'd like people to post their impression of what space-time is, and what it implies, as they think Einstein and his followers intend it.

To me, I see space time in the typical rubber sheet analogy.. but it's hard to envision the time aspect of the rubber sheet, and the implications as to whether or not two events are simultaneous.

It's my primitive understanding that two events, or X events can only be simultaneous to a separate observer.... that Frame A and Frame B can only really be thought of as simultaneous from a third frame "C" or "the observer". I see it as "any frame sees any other combination of frames as simultaneous", but they are not necessarily so at all.

That's all I'll say about that for now.

*Please, this is not intended to be a discussion of other theories.. but how SRT and GR use the notion of Space-time... and it seems to me that it's a nefarious concept that people struggle to comprehend.. and that perhaps it's this difficulty that impedes communication amongst those in the ongoing arguments on the topic.

Please post your thoughts, clarifications, etc. on the topic.
Wes Morris,
I have trouble with the "space' as many see it. "rubbery" or otherwise. when space is given attributes of "expansion" then to me this signals space as having some substance, "masslike" even, if you allow the stretch. However, these attributes are invariabley offered to explain away other theoretical findings or conclusions.
Geistkiesel

Which do you want? They are not the same thing.
Time is one of 4 coordinates invented by Man to help describe the world and events in it. Time exists, in this sense, because we say it does, whether matter is present or not.

It's perfectly true, that there is a natural sense of "before" and "after", but it's a big logical step to say "therefore time is natural".

It's also perfectly valid, but completely trivial, to say that, as time is a man-made invention, in the absence of men there is no time.

In fact, I defy you to say what you mean by "time exists" with any degree of precision, other than the sense that I gave.

Time is like length - it only exists in terms of MEASURING something which already exists. Same with WEIGHT, there aint no weight without matter.

Do you thing length exists?

Do you think weight exists?

Of course not, it's just a way of describing different effects on universal matter relative to other bits of matter.

Time is no different. Time does not exist where there are no "events" occurring. There is nothing to measure with, and nothing to compare it to. I can extrapolate this argument further by suggesting that, like length, time simply measures one physical entity against another, nothing more. Therefore, time is just a mathematical tool to distinguish varying sequences of events with reference to a chosen set of arbritary physical oscillations. There is nothing "trivial" in what I am saying and this argument is perfectly valid.

If time does not exist, as I am suggesting, then there is a causal link with our current ideas of time dilation. How can something that doesn't exist dilate? How can you prove the phenomenon NOT to be attributed to the "physical" mechanisms of space?

Clocks are proven to slow down as they speed around the Earth or penetrate a gravitational field. So what? This "proves" nothing!

>> Time is like length - it only exists in terms of MEASURING something…

Hum, that sounds like one hand clapping…
A tree in a forest does not exist unless someone see it.

Well maybe.

But, as what is conceived today, the properties of space time and matter do seem to be finely tuned so that they are reliant on each other and changing the value on one is meaningless.
In short a universe without matter could not expand, Too much and it would collapse before it could inflate.

And likewise it is supposed that removing the splitting process (symmetry breaking) to create , say, the temporal dimension, would lead to a still born universe that would quickly collapse.

The creation of matter/antimatter, and the small CPT violation, in our Universe, is a consequence of the formation of the 3 dimensions and time. Imho.

Though it may be hypothesised that a unverse may be created without , say , matter, then that does not lead us on to say that a temporal or special dimension cannot exist.

Time is like length - it only exists in terms of MEASURING something which already exists. Same with WEIGHT, there aint no weight without matter.Yep, that's what I said. Time is a man-made coordinate - a measuring device, if you prefer. With the qualification I gave earlier, that there is a definite reality that corresponds to the order of events.



Time is no different. Time does not exist where there are no "events" occurring. There is nothing to measure with, and nothing to compare it to. Time "exists" whenever we say it does, as it's man-made. My ruler exists, even when I have nothing to measure, as does the concept of length, or distance. What's your problem?
There is nothing "trivial" in what I am saying and this argument is perfectly valid.Fair enough. Just that I would regard the statement that "if there are no clocks and no men, time can't be measured" as trivial. You apparently don't.

OK by me.

If time does not exist, as I am suggesting, Explain what you mean by "time existing", or the converse, and maybe we can agree on that too.

-you can't have an event with zero dimensions. Why not, in space-time? Zero dimensional in space-time simply means "instantaneous and not moving through space". What's wrong with that? . rather, the spacetime i envision emerges as a result of the interactions among events, and is comprised of an accumulate of possible event-trajectories resulting from the interaction. Envision away, I have no idea what this means.
the theory of relativity is poised for expansion to encompass these interactions, in my opinion.
=bGo ahead and do it, then. The rest of us will watch TV while you do

This is the common arguement..."Time is a man made way to measure"...measure what? Guess what, meters, feet, inches, nanometers, aus, etc...are all ways that we use to measure distance...are you going to argue that the spacial dimensions do not exist? Seconds, minutes, hours, days, years are ways we measure time...no difference.

wesmorris - you believe it odd that dimension t requires the existance of the x, y, z axis? First, it was never stated that this was the case, but you are right IN OUR universe. Of course, the existance of the x axis is dependant on the existance of the y, z, t axis, just as the existance of the y axis is dependant on the existance of the x, z, t axis...etc. Can you imagine living in a world without height (flatworld)...

If time is not a fundamental "dimension" in our universe then it is absurd to use it in the math describing it...try it...good luck.

- KitNyx

...are you going to argue that the spacial dimensions do not exist? Can you prove that they do? Certainly, in our every-day experience, we have an intuition about flatness and height, that's why we invented Cartesian coordinates, but invent them we did.

Of course, the existance of the x axis is dependant on the existance of the y, z, t axis, just as the existance of the y axis is dependant on the existance of the x, z, t axis...etc. This is, of course, incorrect. The axes you are talking about are, by the definition, linearly independent (i.e. orthogonal) so of course they don't depend on each other.

If time is not a fundamental "dimension" in our universe then it is absurd to use it in the math describing it...try it...good luck.

- KitNyxWhy? There are plenty of things that mathematics uses that are not "fundamental in our universe" (i.e. not cooked up by anybody other than us - think of √-1). We use them either because they simplify life or because they allow us to get on and do want we really want to do.

wesmorris - you believe it odd that dimension t requires the existance of the x, y, z axis?

No not at all, I was just asking if you saw that as a proper conclusion from your assertions. If you really want to know how I hypothesize time fits into the big picture, I offer the following cool graphic:

http://jwesmorris.home.mchsi.com/time.jpg

I include a fourth component of "a human's being" because I allow a dimension for the abstract component of existence... though I realize this is highly unconventional.

I think the reason I do so is perhaps incredibly sophomoric, in that "imaginary time" seems to me to be the perfect realm of the abstract. It's my understanding that in "imaginary time" one would not be constricted by the arrow of time as in "time" as we see it. That one can do that same abstractly in one's mind seems to be something of importance to me... but I'm no physicist and should probably shut up about that now.

First, it was never stated that this was the case,

Yes I know, hence the question.

but you are right IN OUR universe.

Well I was really asking if I was correct in terms of what you said. I sometimes over-confirm from other's perspective that communication is what I think it is.

Of course, the existance of the x axis is dependant on the existance of the y, z, t axis, just as the existance of the y axis is dependant on the existance of the x, z, t axis...etc. Can you imagine living in a world without height (flatworld)...

I can imagine it, but I can't relate to it.

If time is not a fundamental "dimension" in our universe then it is absurd to use it in the math describing it...try it...good luck.

Agreed.

If time is not a fundamental "dimension" in our universe then it is absurd to use it in the math describing it...try it...good luck.

- KitNyx

Nothing is "absurd" about the calculations and nothing is wrong with introducing the "concept" of time into the calculations. The problem lies with what we understand (or lack of understanding) about what we are ACTUALLY measuring here.

When we measure times between events and record our findings, we are only using something physical, convenient, accurate, consistent (in our frame) and happens to oscillate quite nicely. We then compare our equipment with the events to determine some arbitrary measurement. Everything throughout the entire process is physical.

When atomic clocks are seen to dilate due to speed it is quite obviously due to their internal mechanism having been physically affected by either the speed or acceleration through space, or its proximity to a gravitational field.

So if this is true, and I can't seem to find anyone to disprove it as yet, we are left with the notion that spacetime is not quite what we thought it was.

Time is not real, time does not affect anything in our universe and neither does it dilate. When are you people going to get to grips with this?

Time is not real, time does not affect anything in our universe and neither does it dilate. When are you people going to get to grips with this?

If time is not real, changes also are not real, gravitational field (gravitywave) is also not real, nothing is real.. hahaha.. dav57..you..you..crazy

If time is not real, changes also are not real, gravitational field (gravitywave) is also not real, nothing is real.. hahaha.. dav57..you..you..crazy


Yes, me crazy - you stupid.

How can you stand there and say " changes are not real"? That is dumb to say the least. We have substantial evidence to prove that things change continuously throughout our universe. Gravity is also very real and observable.

Time, however, is nothing more than a way of showing a relationship between intervals of physical events. Just because you "think" you are aware of time passing doesn't mean to say that time exists in any form whatsoever. All you are witnessing is your brain having the capacity (well, perhaps not yours) to record snapshots of your surrounding world - it's all physical and there's nothing clever about it. The clever bit is to realise that our perception of time is not really a perception of time but rather an observation and measurement of physical events. Like I said before, time is like length and weight, they don't exist. Get it?

Like I said before, time is like length and weight, they don't exist. Get it?
haha.. you are getting crazier.. length and weight are measurements of distance and gravitational force on massive objects.. length and weight are as much real as distance and force.. you need time to measure duration of/between events if they are not happening at same place at same moment.. you seem to believe that 'time' means an absolute time that is running for the entire universe like processor clock controlling the operations of chips/peripherals of a computer system!!! hahahahaha....

you seem to believe that 'time' means an absolute time that is running for the entire universe like processor clock controlling the operations of chips/peripherals of a computer system!!! hahahahaha....

No, I don't mean that and I have never said that. Don't put words into my mouth and read what I am saying before you interpret it incorrectly!

Without substance or physical presence, length, distance, weight, force and time are non-existent. All these elements are is an expression of the relative behaviour of physical events.

Tell me, Everneo, explain to me what you think you are actually measuring / recording when you use your stopwatch to time how long it takes to , say, play a chess match from start to finish? What is your stopwatch ACTUALLY doing?

Stopwatch shows how much units of time passed between start and stop. Why?

Stopwatch shows how much units of time passed between start and stop. Why?

No it doesn't. It's simply making a comparison between the number of physical oscillations of a quartz crystal and the events of the chess match.

The Chess match didn't take any TIME. All that happened was a set of ordered physical happenings began to unfold. The stopwatch simply compared its own ordered, physical happenings with some other comparable events.

The only reason our brains seem to get "a feel" for time is that evolution has given us the capacity to store and allocate memories of our physical surroundings. We are able to compare what order events took place and distinguish between continual snapshots of our lifetime. My guess is that our brains operate at a given arbitrary frequency which helps us differentiate events and "apparent" passage of time.

Time is a nonsense, but a very convenient part of relativity equations which balances the incorrect notion that speed of light is constant to all observers, when we clearly don't have the apparatus or communication speed to correctly measure the ACTUAL speed of light in a moving frame.

You may laugh, Everneo, but one day you might get egg on your face. Anything is possible.

One day i may get an egg on my face as some skin-care process unfolds (which i doubt), not for the reason you indicate.

btw, unfolding requires time, what you say?

One day i may get an egg on my face as some skin-care process unfolds (which i doubt), not for the reason you indicate.

btw, unfolding requires time, what you say?

Look, lets take length as an analogy. If you have two measurable objects which are moving apart at some measurable rate and you attempt to measure their distance at different moments, think for a moment about what is actually changing relative to what. Is length changing? Of course not because "length" doesn't exist. Instead, think about it as being a relationship between the changes in your measuring stick and the objects you are measuring. The objects are simply moving their coordinates in space. They don't NEED time to do this. It just requires a physical process. Length is relative and arbitrary and non-existant, relying wholly on physical change before it measurably alters relative to something else.

Look, lets take length as an analogy. If you have two measurable objects which are moving apart at some measurable rate and you attempt to measure their distance at different moments, think for a moment about what is actually changing relative to what. Is length changing? Of course not because "length" doesn't exist. Instead, think about it as being a relationship between the changes in your measuring stick and the objects you are measuring. The objects are simply moving their coordinates in space. They don't NEED time to do this. It just requires a physical process. Length is relative and arbitrary and non-existant, relying wholly on physical change before it measurably alters relative to something else.

I see time all over there. may be i am stupid.. all i can suggest is don't neglect time as non-exiting.. you will have a lot of problems.

I see time all over there. may be i am stupid.. all i can suggest is don't neglect time as non-exiting.. you will have a lot of problems.

Well, you don't need "length" for a particular distance to increase relative to another fixed, arbitrary distance. You only need a physical change to take place. Likewise, you don't need time to be able to measure the relative nature of how matter interacts with other matter. It just happens because it CAN happen and it does. Just because these processes take place one after the other and can be measured relative to other processes is by no means proof of the existence of time.

Relativity doesn't need time to dilate. It is based on the fact that the physical processes of matter vary depending on where the measuring device is placed and how fast it is travelling. It is this which affects the particles, not time.

Relativity doesn't need time to dilate. It is based on the fact that the physical processes of matter vary depending on where the measuring device is placed and how fast it is travelling. It is this which affects the particles, not time.

Unfortunately,
Relativity does need a temporal dimension.

Though it may be possible to add or remove spatial dimensions.

The idea that something is `travelling`, or `changing`, by its very nature, includes the concept of a temporal dimension.

Unfortunately,
Relativity does need a temporal dimension.

Just because relativity requires a temporal dimension could mean that relativity is wrong :D


Come on, think about it guys. Think about what time really is and how and what we are REALLY measuring. Think about the very real possibility of atomic clocks being affected by their physical surroundings and hence slowing due to physical reasons rather than the notion that time slows.

Do you relativists not think this as at all plausible?

Come on, think about it guys. Think about what time really is and how and what we are REALLY measuring. Think about the very real possibility of atomic clocks being affected by their physical surroundings and hence slowing due to physical reasons rather than the notion that time slows.

you are maybe right, we can't measure time directly we measure energy. Time can be defined most clearly as 'the trivial changing', it is a matter of convinience to define time as axis in 4-dimensional manifold, we can obtain relativity laws and without this. I think that time-axis is not correctly to be put together in a common manifold with space axis. The reason is that the result is not a numeric space at all. It is not rotational invariant and so on... But maybe if time axis is imaginery , i.e if we multiply it by i - the imaginery unit, then space-time become complex dual space and maybe it corresponds more to reality.

you are maybe right, we can't measure time directly we measure energy. Time can be defined most clearly as 'the trivial changing', it is a matter of convinience to define time as axis in 4-dimensional manifold, we can obtain relativity laws and without this. I think that time-axis is not correctly to be put together in a common manifold with space axis. The reason is that the result is not a numeric space at all. It is not rotational invariant and so on... But maybe if time axis is imaginery , i.e if we multiply it by i - the imaginery unit, then space-time become complex dual space and maybe it corresponds more to reality.

I'd like to make it clear that time, like length, is not imaginary but a useful way of making a comparative analysis of something changing or different to something else.

Before the universe came into existence, there was no length and no time. There was nothing to compare anything to so time was not relevant.

The universe didn't require time before it began to unravel itself - it just started because something physical happened to take place which began the change. The universe hasn't aged but rather changed from one snapshot to the next. Our perception of time is founded by the fact that we have the ability to make comparisons of events using other convenient and arbitrary events. Time is a way of describing how one physical thing changes with respect to something else.

So when relativity introduces a time dilation factor into the equations, you need to think about it in a different way, which happens to have the same outcome. That is, your clocks are slowing down due to the physical processes slowing rather than time slowing. This is very fundamental and crucial to how we explain spacetime.

Remember, time is not responsible for the universe evolving but rather the billions of particle / energy interactions which have taken place one after the other since the universe began.

But maybe if time axis is imaginery , i.e if we multiply it by i - the imaginery unit, then space-time become complex dual space and maybe it corresponds more to reality.Huh? First a dual space V* is a set of linear functionals f_v that associates a scalar to every w in V (a vector space) such that f_v(w) = (v·w), the inner product, for all v,w in V. So where is your vector space?
Second, i(t) implies that under certain circunstances time can be reversed, which hardly "corresponds more to reality".

i(t) implies that under certain circunstances time can be reversed, which hardly "corresponds more to reality".

Well,
complex transformations of spatial dimensions are now being applied to current string theories.
The three dimensions (length x Breath x height) and their imaginary transformations (length_i x Breath_i x height_i) may be more familiar to those who follow `<i>twistor theory</i>` but they seem to encompass all the variations and freedoms that are required by the current string theories 10/26 dimensions.

As for <b>time</b>, it too, may have a complex transformation, (ie i propose time is a spatial dimension), and yes, as it stands, time can be reversed.
(All particles can and do move forwards and backwards in the time dimension.)

<b>But</b>, unfortunately, if you include the `boundary conditions` of the universe (<i>one side is closed and the other open</i>) then destructive interference of propagating waves from those time travelling particles automatically gives a direction, or arrow to time.
Therefore we do not usually see particles moving backwards in time, or `cause and effect` violations.

Well,
complex transformations of spatial dimensions are now being applied to current string theories.Yes. I am very well aware that quantum physics prefers to work over the complex field. But, as I understood the discussion, we were talking about the macroscopic world.
<b>But</b>, unfortunately, if you include the `boundary conditions` of the universe (<i>one side is closed and the other open</i>) then destructive interference of propagating waves from those time travelling particles automatically gives a direction, or arrow to time.
Therefore we do not usually see particles moving backwards in time, or `cause and effect` violations.Sorry, but I have no idea what you are getting at here. You can choose to beleive the universe is bounded if you like, but what do you mean by boundary conditions? And what are time travelling particles?
I'm sitting in my office, tapping away, and I'm time travelling. I have no choice, sadly. My arrow of time, as you call it, is irreversible. What has that got to do with destructive interference of propogating waves?

Sorry, but I have no idea what you are getting at here.

Whoops, my mistake, forgot to add the bit about the ` Wheeler-Feynman Absorber Theory`; which is unimportant here, as the main point is that complex transformations and the mathematics can be made to fit with what we observe.
Links:
http://arxiv.org/abs/physics/0104066
http://www.npl.washington.edu/npl/int_rep/dtime/node2.html

<hr>
Abstract:<b>
Quantum Theory Looks at Time Travel
Authors: Daniel M. Greenberger, Karl Svozil
</b>
We introduce a quantum mechanical model of time travel which includes two figurative beam splitters in order to induce feedback to earlier times. This leads to a unique solution to the paradox where one could kill one's grandfather in that once the future has unfolded, it cannot change the past, and so the past becomes deterministic.
On the other hand, looking forwards towards the future is completely probabilistic. This resolves the classical paradox in a philosophically satisfying manner.
http://arxiv.org/pdf/quant-ph/0506027 (PDF)

<hr>
Abstract::

<b>Spontaneous Inflation and the Origin of the Arrow of Time</b>
Microscopic laws of physics are essentially time-reversal invariant, but macroscopic thermodynamics exhibits a profound time-asymmetry; entropy typically increases in closed systems. This intriguing feature of the real world has a cosmological origin: the entropy of the early universe was fantastically small.
After a century of effort, it has been difficult to explain this arrow of time without assuming time-asymmetric boundary conditions.

Jennifer Chen and Sean Carroll have suggested a simple scenario in which increasing entropy is natural, based on the idea that the entropy can increase without bound
(there is no equilibrium state) and that the way entropy increases is by creating universes like our own.
In their picture, any generic state first evolves to an empty de Sitter phase; the small temperature of de Sitter allows for fluctuations into a proto-inflationary configuration, which grows and reheats into a conventional Big-Bang space-time.

The same thing happens in the far past, but with a reversed arrow of time.

On ultra-large scales, therefore, entropy is growing without bound in the asymptotic future and past.
In the absence of inflation, systems coupled to gravity usually evolve asymptotically to the vacuum, which is the only natural state in a thermodynamic sense. In the presence of a small positive vacuum energy and an appropriate inflation field, the de Sitter vacuum is unstable to the spontaneous onset of inflation at a higher energy scale.
Starting from de Sitter, inflation can increase the total entropy of the universe without bound, creating universes similar to ours in the process.
An important consequence of this picture is that inflation occurs asymptotically both forwards and backwards in time, implying a universe that is (statistically) time-symmetric on ultra-large scales.

http://arxiv.org/PS_cache/hep-th/pdf/0410/0410270.pdf

blobrana: Two interesting refs, which I have only had time to skim. For the first, it seems to me the essence is in this statement of theirs:
" No matter how unlikely the events are that could have led
to your present circumstances, once they have actually occurred, they cannot be changed." They nowhere seem to take seriously the notion that time can meaningfully be reversed, although, as they are in the quantum world, they might just have well have done!

As to the second, they seem to be doing some wierd things with the second and third laws of Thermodynamics. Again, I geuss they are entitled to because of the regime in which they are modelling, and because these laws are axiomatic in nature anyway.

My real problem is with their unquestioning adherence to Eddington's definition of the "arrow" of time being determined by &Delta;S, the change in entropy. This is, to my mind, an unrealistic definition.

Nevertheless, interesting stuff, thanks.

There is no distinction between space and time dimensions mathematically -- the only difference appears in the metric, wherein the time dimension has a sign opposite that of the spatial dimensions.

- Warren

The time dimension itself doesn't have an oppsite sign. In the metric, the negative sign appears along the squares (either of space, or of time). This, in fact, suggests that the coordinates of space, or of time, should carry a factor of 'i', where i is the square root of -1. This doesn't some to make much sense at first sight, because they might be implying the existence of imaginary numbers in reality, but further thought is needed before saying anythig of this kind. But suppose that the coordinates are these: ix, iy, iz, t. This is mathematically correct. It also raises another question.

Most beginners in relativity seem to think that going faster than ligh deposit us in the past. But actually, putting v>c in the transformations gives us time and length with an 'i' attached to them. If we choose the coordinates as I've shown, this might imply that the roles of time and space reverse themselves when we travel faster than light. It's not very worthwhile constructing fantasies of this kind, but there might be some substance in this.

They nowhere seem to take seriously the notion that time can meaningfully be reversed


They begin with the notion that time <b>can</b> be reversed.
The feedback mechanism that cancels out all the incompatible pathways backwards in time is very similar to the Wheeler-Feynman Absorber Theory , in that they <b>require</b> movement backwards in time.

In the case of the old <b>Absorber Theory</b> that is simply electromagnetic radiation (which simply radiates outwards into space, and <i>forwards and backwards into time</i>) the radiation will interact with other particles that produces more `radiation` (retarded & advanced waves that propagate through a temporal dimension); the waves that move backwards eventually get back to the <b>big bang singularity</b> and get <i>reflected</i> forwards - - but they are now out of phase; the forward waves just keep going as that end (<i>the boundary condition</i>) of the universe is open and they don’t get reflected...

The bottom line is that only the retarded waves survive the destructive interference, thus there is a direction to time.



My real problem is with their unquestioning adherence to Eddington's definition of the "arrow" of time being determined by &Delta;S, the change in entropy. This is, to my mind, an unrealistic definition.


Yes I too find it an unrealistic definition.
i suppose most ppl find it difficult to separate entropy and time.
It could be likened to primitive sailing ships that get blow along by the wind.
Their direction (&Delta;S) basically <i>dictates</i> which way the wind blows (hehe)

But tomorrow the wind will blow in the opposite direction. (<i>ie, We may find that once this universe has ended and another one born in its place the entropy in that universe may well go from High to low….but the direction of time is still `the same` (forwards)</i>)


the roles of time and space reverse themselves when we travel faster than light.


Hum,
interesting idea....

The time dimension itself doesn't have an oppsite sign.
I never said it did. Dimensions don't have signs.
In the metric, the negative sign appears along the squares (either of space, or of time). This, in fact, suggests that the coordinates of space, or of time, should carry a factor of 'i', where i is the square root of -1.
I gather that you have no idea what a metric is.

- Warren

This, in fact, suggests that the coordinates of space, or of time, should carry a factor of 'i', where i is the square root of -1.Sorry chum, but you can't move 3 of 4 dimensions onto the complex field - it's all or none.
Let's stay in Minowski space. The metric is
&eta; = diag{1,1,1,-1}
If you impose an imaginary coefficient on each of the derivatives, all you do is is reverse signs. Which many authors do. It's no big deal.

It's no big deal.

Hum,
I found it quite invigorating the concept the <font color=red>“roles of time and space reverse themselves when we travel faster than light.</font>”

A sort of self-imposed speed limit, ©.

It maybe quite a big deal….

Spacetime is a four-dimensional manifold possessing a metric. Three of the dimensions are regarded as "spatial," while the other is considered "time." There is no distinction between space and time dimensions mathematically -- the only difference appears in the metric, wherein the time dimension has a sign opposite that of the spatial dimensions.

- Warren

Nicely said.
2 questions:
First, why do the 4 dimensions require a sign?
Second, why is it that Time has the opposite sign (and what does this opposition imply)?

Thanks in advance.

Nicely said.
2 questions:
First, why do the 4 dimensions require a sign?
Second, why is it that Time has the opposite sign (and what does this opposition imply)?

Thanks in advance.I take it that this question was addressed to Chroot, but I can't resist jumping in.
First, assume we are in a Euclidean 4-dimensional space E⁴ i.e. one where the shortest distance between two points is what we ordinarily think of as being a straight line. (Actually in 4-D it's called Minkowski space). We now establish the Pythagorean relation

dx<sup>2</sup >+ dy² + dz² = d(ct)²
(ct rather than t is used to equalise dimensions)

and dx² + dy² + dz² - d(ct)² = 0

define a "line element" ds such that
dx² + dy² + dz² - d(ct)² =ds²

( the ds is nothing more then a zero space-time point, i.e. a stationary instantaneous "event".)

Collapse this, with no loss, to

ds² = &sum;_ab dx^adx^b and now ask what are the coefficients on the (dxⁱ)². Here they are {1,1,1,-1}. This is the so-called Minkowski metric. But assuming the coefficients may take any value, we need something (bit like a data file, I geuss) to tell us what these values should be. This is called the metric, and is denoted by g (usually). It's used like this;
ds² = &sum;_abg_abdx^adx^b.

And when we move out of E⁴ things start to get ineteresting!

define a "line element" ds such that
dx^2 + dy^2 + dz^2 - d(ct)^2 =ds^2


As I see it, this metrics can be obtained only if we have a 3 real ortogonal axis for the spacial dimensions, and one imaginery for time, then because i*i = -1 the aboved equality will be obtained.

I never said it did. Dimensions don't have signs.

I gather that you have no idea what a metric is.

- Warren
Sorry, I didn't mean the metric, I meant the line element. As to QuarkHead, please see the above post from XGen. He's understood what I was saying. You might appreciate my statement a bit more if, instead of just playing around with the metric, you tried to use it and derive actual coordinate transformations which give rise to the metric. This is what XGen has done.

You might appreciate my statement a bit more if, instead of just playing around with the metric, you tried to use it and derive actual coordinate transformations which give rise to the metric. This is what XGen has done.What do you mean? Coordinate transformations in no sense "give rise to the metric". And I see no coordinate transformations from Xgen, merely bald assertions.

As I see it, this metrics can be obtained only if we have a 3 real ortogonal axis for the spacial dimensions, and one imaginery for time, then because i*i = -1 the aboved equality will be obtained.Did you read my post? Did you understand how the time coordinate got it's negative sign? I hope so, as it was the whole point of the post.

I know that you can get the metric directly from the fact of invariance of the relativistic space-time interval, that is, by using Lorentz transformations. Let me ask you. How do you get the metric tensor for say, a sphere embedded in space? You take the equations of transformation from the Euclidean 3D space, to that of the curved 2D space (surface) of the sphere. Then you find all the dervitives, etc., and use them to get the metric tensor. Suppose you have the metric tensor, as in our case. Have you tried obtaining the transformations previously mentioned from the metric already given?

How you 'get' the negative sign is another matter. It's somewhat like using calculus of variations to obtain the path of a particle, instead of using the force law directly. But once you do have the path, you may use it to search for a force law which satisfies it. I know I didn't make myself very clear, but think about it and try it out.

We propose a new framework for solving the hierarchy problem which does not rely on either supersymmetry or technicolor. In this framework, the gravitational and gauge interactions become united at the weak scale, which we take as the only fundamental short distance scale in nature. The observed weakness of gravity on distances 1 mm is due to the existence of new compact spatial dimensions large compared to the weak scale. The Planck scale is not a fundamental scale; its enormity is simply a consequence of the large size of the new dimensions.

While gravitons can freely propagate in the new dimensions, at sub-weak energies the Standard Model (SM) fields must be localized to a 4-dimensional manifold of weak scale "thickness" in the extra dimensions. This picture leads to a number of striking signals for accelerator and laboratory experiments. For the case of new dimensions, planned sub-millimeter measurements of gravity may observe the transition from Newtonian gravitation. For any number of new dimensions, the LHC and NLC could observe strong quantum gravitational interactions.


Furthermore, SM particles can be kicked off our 4 dimensional manifold into the new dimensions, carrying away energy, and leading to an abrupt decrease in events with high transverse momentum TeV. For certain compact manifolds, such particles will keep circling in the extra dimensions, periodically returning, colliding with and depositing energy to our four dimensional vacuum with frequencies of Hz or larger. As a concrete illustration, we construct a model with SM fields localized on the 4-dimensional throat of a vortex in 6 dimensions, with a Pati-Salam gauge symmetry in the bulk.

Did you understand the purpose of this thread. I don't think we're discussing a theory of gravity, or a unified field theory. This thread is solely about space-time.

Indeed. I hypothesize that those on this site who doggedly reject GR and SR, actually just reject the notion of space-time. As it seems to me, they do not argue about GR and SR within its context, but from some absolutist model of the universe. They claim it isn't consistent without actually addressing why it isn't, but rather somewhat unwittingly hammer away at the foundation of the model, which for the purposes of the calculations of SR and GR, are assumed (and thus not subject to criticism unless the argument is specifically about the boundary conditions of the theories).

Great confusion is going on here..

i*i = -1 and all is OK, but how it is linked to -ve metric of time coordinate in space-time is beyond my understanding. So because of this coincidence it was said that time is imaginary? You people are living in an imaginary world, i say.

when c is taken as 1, ct = t ; time is treated as spacelike (imaginary space).

OR you treate spatial coordinates as timelike by having -ve metric to spatial dimensions and +ve metric to time; that is spatial dimensions are treated as timelike (imaginary time).

Spacetime is built upon the dimensions interrelating to one another, if you don't understand that please reread my paragraph once again to get a better understanding.

I know that you can get the metric directly from the fact of invariance of the relativistic space-time interval, that is, by using Lorentz transformations. No. Invariance of the line element is imposed after the metric is derived, where, as in the present case, the question was "why does the time dimension have an opposite sign to that of space". Lorentz transformation applies after you have declared one coordinate system to be in uniform motion relative to another. Let me ask you. How do you get the metric tensor for say, a sphere embedded in space? You take the equations of transformation from the Euclidean 3D space, to that of the curved 2D space (surface) of the sphere. Then you find all the dervitives, etc., and use them to get the metric tensor. Suppose you have the metric tensor, as in our case. Have you tried obtaining the transformations previously mentioned from the metric already given?
Yes, but that wasn't what was being asked. You are saying that when notions of length and angle are defined
in Euclidean space, then in order for these to be inherited by curvilinear space one needs a coordinate transformation protocol. Of course.

WHAT IS SPACE?

Space is the feeling of distance that zero needs to cancel in order to reach the infinity it rejects. This means that this distance can never be cancelled. Space is consciousness ("light") itself.

WHERE DOES SPACE COME FROM AND WHEN?

Space comes from the rejection of infinity-principle by the zero-principle in order to be and to remain nil. This takes place here and now.

WHAT IS TIME?

Time is the feeling of memory of what zero has managed to merge with, compared to the future of what it has not yet managed to merge with. Time is love itself.

WHERE DOES TIME COME FROM?

Time comes from the necessity of unity (of the two principles of Nothingness), that consciousness feels. What must be (nothingness) is and remains an AIM, since consciousness eternally refuses infinity (so that the zero-principle remains actual). So, the infiniteness of nothingness remains an unreachable goal giving the center of infinity a feeling of eternity. This is why mind is eternal, and why we'll never fall into nothingness.

Well, that's it folks. The "infiniteness of nothingness" says it all. Thank goodness, we can all sleep easy tonight.

Well, that's it folks. The "infiniteness of nothingness" says it all. Thank goodness, we can all sleep easy tonight. Did you possibly think your consciousness could exist without it?

And yet events are subject-centerd by definition. That is, it requires a subject or observer to define an "event" out of an undefined background matrix in which there are no event. Only pure duration or "duree," as Henri Bergson would say. Reason being that events are only those things that are significant for some reason to an observer, either on a physical or a cognitive basis. There are an infinite number of events that could be defined in history, for example - so many that it is not separate events but a single unified field without subjectivity. It is subjects who decide that events x and y exist because x and y are relevant to the very structure and existence of the observer.

Quarkhead, take some time to think about this. Don't think about the math alone.
<Br>
Okay. you're given the (2X2) metric tensor for the surface of a sphere embedded in space. From this, you can get three partial differential equations, and solving these gives you the transformations between the Euclidean 3-space, and non-Euclidean 2-space. If you take the coordinates of spere-surface to be x and y, and square their distance and add up, you won't get the length of an arc on the sphere. Of course, that's where the metric comes into action. Because of this property, an inhabitant of the surface can conclude that he isn't living in a Euclidean sapce.
<Br>
Now, if you take the coordinates of our space-time to be x,y,z,t, you don't get the length of an arc in space-time if you simply take these coordinates and add up their squares. Thats why you've to use the metric. Hence, we can conclude that we're <I>not</I> living in a Euclidean space-time. Okay, you never said that we did. That's not what I'm saying. There's more to come. If we take the coordinates to be x,y,z, <I>i</I>t, then it will do to add up the squares of <I>these</I> coordinates.
<Br>
I'll put this in another way which appeals to your <I>physical</I> sense. An inhabitant of the surface of a sphere can find out what the space he lives in is like, by analyzing the properties of the space from within the space itself. That, I think, is what intrinsic geometry is about. Suppose we want to find out what our space-time is like. What space-time <I>looks</I> like. We can't visualize 4D space-time or even 3D space-time from an outside perspective. Agreed. So we take a 2D space-time. The metric is {1,-1}. You can obtain the tranformations which give rise to this metric. Tell me, what geometric figure <I>is</I> 2D space time?

Rosnet: I think we should talk some about manifolds, but today I don't really have the time. But let me quickly ask you this? If you are on the surface of the 2-sphere, how do you populate the metric? You need something to measure curvature. Alternatively, you reference to a set of coordinate nieghbourhoods, work out the transformation rules between them, in which case you don't need the metric.
With regard to your desire to make the time coordinate imaginary, you cannot mix the fields over which the manifold is defined. It has to be all real or all complex.

Finally, the geometry of any 2D object embedded in 3-space can be planar, parabolic, spherical, ellipsoid or toroid, whether the coordinates are spatial or spatio-temporal.

With regard to your desire to make the time coordinate imaginary, you cannot mix the fields over which the manifold is defined. It has to be all real or all complex.

Why? And I'm assuming you meant all real, or all imaginary. Because a real number is specialisation of a complex number.

Spacetime is built upon the dimensions interrelating to one another, if you don't understand that please reread my paragraph once again to get a better understanding.

Are you replying to my post?

BTW, you may cite the reference from where you lifted the paragraph.
- http://arxiv.org/PS_cache/hep-ph/pdf/9803/9803315.pdf

Anyway thanks for the link.

What is "velocity"? Can this "property" of "mass" exist without time?

(originally only "property" was in inverted commas, until noticing that the other terms are also 'so called'. They don't have a definition beyond that which is not understood. Or if they do the definition is circular.

What is "velocity"? Can this "property" of "mass" exist without time? By definition, velocity is simply a rate of change of spacial coordinates with time. So the answer to your second question is no since the definition of velocity inherently includes time.

(originally only "property" was in inverted commas, until noticing that the other terms are also 'so called'. They don't have a definition beyond that which is not understood. Or if they do the definition is circular. Well then, it would seem every definition is circular - this is nothing spectacular.

For example;

speed = distance / time , or number of distances covered per second.

We don't know what a distance is (space?) or what time is (change?)

Would you agree that only the present "time" exists and that the past no longer exists and the future doesn't exist either, since it is not the present? It would seem then that the present has zero duration....

Agree/disagree?

Well then, it would seem every definition is circular - this is nothing spectacular.

Do you think physics will just keep adding infinite circular definitions about reality?

For example;

speed = distance / time , or number of distances covered per second.

We don't know what a distance is (space?) or what time is (change?)

Would you agree that only the present "time" exists and that the past no longer exists and the future doesn't exist either, since it is not the present? It would seem then that the present has zero duration....

Agree/disagree? I challenge you to define present.

Does the "present" exist? It would seem that it does but how do you "measure" it? Let's take for instance your sight. What you conceive of sight in your mind is actually a blurr of several images stored in your mind over a very short interval of time. This is why when you look at a fan, you see the blades in mutliple places at the same instant - but we know that is not actually so. So this definition of "present" can not agree with what you seem to want to call present as it includes times before (read: past).

Do you think physics will just keep adding infinite circular definitions about reality? I don't think your understanding of what exactly physics is or what it is used for is correct which makes it impossible to explain anything to you until you have a better understanding.

Why? And I'm assuming you meant all real, or all imaginary. Because a real number is specialisation of a complex number.Yes I'm aware of that.
But look, I've now realised we are at cross purpose. At work today I sneaked a quick look at Lovelock & Rund, and I think I now see what you were saying about the derivation of the metric. Hopefully, I'll back later. (This could be fun!)

Well, you never came back. And just when things were starting to get interesting

I challenge you to define present.

Though I have no idea of the context excepting "time", I can't help but give it a crack.

The "present" is an infinitessimal slice of time right between what is about to happen, and what just happened (locally).

Though I have no idea of the context excepting "time", I can't help but give it a crack.

The "present" is an infinitessimal slice of time right between what is about to happen, and what just happened (locally). This definition seems adequate - but I'll note, the challenge was for Trilobyte. Also, by your definition, you are rejeting what Trilobyte said - which is why I gave him the challenge in the first place.

This definition seems adequate - but I'll note, the challenge was for Trilobyte. Also, by your definition, you are rejeting what Trilobyte said - which is why I gave him the challenge in the first place.

Like I said, I didn't look up the context of the challenge. I just don't get to use the term "infinitessimal" very often so I went for it. ;)

The "present" is an infinitessimal slice of time right between what is about to happen, and what just happened (locally).That's nice. I prefer: It's an infinitessimal slice of space-time perpendicular to an observer's world-line.

Too abstract Aer?

Well, you never come back. And just when things were starting to get interestingHa! Not my fault the forum went on holiday. What was the question? Derivation of the metric, or something? Lemme check (been on the road all day).

EDIT: Got it now. I was coming to that in my monologue on tensors.

But I'm thinking of discontinuing that, as there seems to be no real interest.

That's nice. I prefer: It's an infinitessimal slice of space-time perpendicular to an observer's world-line.

Too abstract Aer? Nope - if you want abstract, try to read science by Trilobyte :D

Well, you never come back. And just when things were starting to get interestingOK, Ros, looks like I'd better do it here. It won't be as easy to do (or follow) as a stand-alone, but here goes!

Consider a rank one affine tensor Aⁱ defined on a rectangular coordinate system xⁱ. We want the transformation rule for transistion to another rectanguler system y^j. (i and j = 1,....n)We see it must be of the form y¹ = f¹(x¹,.....xⁿ).......yⁿ = fⁿ(x¹,.....xⁿ).

Let's write this as y^j = f^j(xⁱ). Now we know that the value of each y^j is given by the f^j so we may rewrite this as
y^j = y^j(xⁱ), with the derivative such that the transformation of any Aⁱ is

A'^j = &sum;_i(&part;y^j/&part;xⁱ)Aⁱ.

Now, we insist that such a transformation leaves all metric properies of the tensor unchanged, so we rigidly enforce the condition

dA'^j = &sum;_i(&part;A'^j/&part;xⁱ)Aⁱdy^j = 0, for any abitrary displacement dyⁱ. In general, we will say that the relation

A'^jB'^m = &sum;_i&sum;_n(&part;y^j/&part;xⁱ)(&part;y^m/&part;xⁿ)AⁱBⁿ
holds. By index contraction, after setting j = m, we get

&sum;_jA'^jB'^j = &sum;_j&sum;_i&sum;_n(&part;y^j/&part;xⁱ)(&part;y^j/&part;xⁿ)AⁱBⁿ.

And if we now choose to consider this as the inner product between the vectors A and B, we may consider the quantity
&sum;_j(&part;y^j/&part;xⁱ)(&part;y^j/&part;xⁿ)

to be a measure of this, and we will call it a metric on the vector space containing A and B and write it g_mn

If you see any errors here, they are probably typos.

NOTE: This now amended as per Rosnet's suggestion (see below)

I have a small doubt, but I'll post it in the Tensor thread. However, our original problem was that since we know this method of derivation, and given the Minkowski metric, what would be the coordinate transformations from which we can derive the Minkowski metric in the above given manner? I've already done this, and that's why I originally talked about complex axes.

You can't directly, I think. You can certainly show that it is a condition of orthogonality that the g_ij = &delta;_ij (= 1 if i = j, zero otherwise). It's then a matter of the definition of Minkwoski space that g_ij = diag[1,1,1,-1].

But listen, Ros. Why are you making me do all the work? Let's see what you've got!

I've already done this, and that's why I originally talked about complex axes.OK, I was waiting for you to see it yourelf, having shown you two ways of looking at the metric. Without getting into a lengthy detour of spaces and fields, let's say this.

Many objects that describe the physical world can be scaled, that is multiplied by a scalar to give a new object of the same type. Scalars, vectors, tensors, matrices etc. are among these. Whenever we want to do this, we are entitled ask "is the scalar real or complex?"

As you should have seen, the metric is a rank 2 tensor on a Reimannian space-time manifold, so any scalar applied to it (even the unit scalar) must be either real or complex. You can't have both!

Digression.
There's a small mistake in your post.

∑<Sub>j</Sub>(∂y<Sup>j</Sup>/∂x<Sup>j</Sup>)(∂y<Sup>m</Sup>/∂x<Sup>n</Sup>)

Instead of this, the correct one is

∑<Sub>j</Sub>(∂y<Sup>j</Sup>/∂x<Sup>i</Sup>)(∂y<Sup>j</Sup>/∂x<Sup>n</Sup>)

since the contraction was m=j

Ooops! Quite right, thanks, and it ain't so small! (It's because it's all in-line). You do see, don't you, this naturally leads to the "length" |A| of the vector A?

As I demonstrated earlier, if the coordinates are chosen to be t, ix, iy, iz, then the line element will be dt^2 + d(ix)^2 + d(iy)^2 + d(iz)^2
= dt^2 - dx^2 - dy^2 - dz^2.

First, you have to use ct rather than t, otherwise dimensions don't match. Second, you cannot mix scalar fields like that, it has to to real or complex, as I've told you before.
The metric follows naturally from dx² + dy² + dz² = d(ct)²

dx² + dy² + dz² - d(ct)² = ds²

It is usual to choose such units that ct=t (that is, c=1). For instance, time in Years, and distance in Lightyears.
Why shouldn't they be mixed (since it obviously works fine when we mix them)?

Secondly, do you know anything about Quaternions? I've become aware of their existence recently, after reading an old post here in SF.

Quarkhead:
http://home.pcisys.net/~bestwork.1/specrel.html
http://home.pcisys.net/~bestwork.1/index.html
http://home.pcisys.net/~bestwork.1/2Dempl.html#2DExample

Dammit to hell, I think you're right after all about using ict in the Minkowski metric. Although I was adamant you were wrong about setting the x⁴ to ict I still continued to think about it. (And that's not based on your web links which I haven't looked at - I never do in situations like this).
But the reason you're right is rather subtle, although the arithmetic isn't hard, and it's extremely good fun. I don't have the energy tonight to go through it, so maybe you could give me the full reason why you originally thought it was OK, and see if we come to the same conclusions.

Am I making any sense? I've done a 12h shift in a bar today. Yuk!

OK, let's see what to Lorentz transforms if we use spacetime coordinates {x,y,z,ict}.
Let's first restrict ourselves to 2-dimensional spacetime i.e. bodies in motion along, say the x-axis. We consider a coordinaste system of this sort {x,ct} in which a body A is at rest. If we now introduce a second body B, in uniform motion relative to A, we will say that this motion is inertial if there is a coordinate transformation {x',ct'} relative to which B can now be considered to be at rest.

We see straight away that this is a rotation of {x,ct} to {x',ct'}. Lets look in our high-school texts and write this down in matrix form:

cos&psi;........sin&psi;

-sin&psi;.........cos&psi;

We now change gear, and ask how else can we describe rotations. Lets think of the world lines of A and B as vectors in 2-dimensional vector space. If we multiply any vector by a real number, we think of this as merely extending its length, right? But if we multiply by a complex number, say z = a + ib, we do that but we also induce a rotation.
(This is simply because the complex number line is defined on the 2-dimensional complex plane, with real and imaginary axes. Specifically, iz is a &pi;/2 anticlockwise rotation in the complex plane). Note here I am not distinguishing between coordinate transformations and vector transformation, which sloppy but not wrong, as they are equivalent.


So just to keep it relevent, we'll set z = re^i&psi;, with r and &psi; real. To be simple we'll additionally set r = 1, i.e. multiplying any vector by z so defined is merely a rotation.

Now let's look at Euler's definitions of circular trig functions:

sinx = 1/2(e^ix - e^-ix)
cosx = 1/2(e^ix + e^-ix)

and the hyperbolics

sinhx = 1/2(e^x - e^-x) etc.
So we see we can get somewhere with these guys.

But first let's write down the standard transformations: we'll say x' = &gamma;(x - vt), t' = &gamma;(t - xv/c²), and to save myself a lot of "supping" later, I'll define &lambda; = v/c
and &gamma; = (1 - &lambda;²)^-1/2.

I've just seen a major error in my next step, so I'll have to break off here and sort that out.

QH,

it seems your orginal postion seems reasonable.

Consider the minskowski space-time.

The world line of light is 45 deg with both time (say y) and space (say x) co-ordinates. (c=1)

Any real world line, at any small space-time interval, should have more than 45 degs with x co-ordinate, and of course, less than 45 deg with y coordinate (velocities less than c).

That is, take any 2 small segments d1 and d2 in a real world line. let d1 and d2 has equal space-time interval. if d1's space interval is bigger than d2's space interval then inevitably d1's time interval must be less than that of d2. increase in time interval is at the cost of space interval hence -ve metric. vice versa.

everneo: Thanks for that, I'll back to it later. Let me first continue with my thought-train, as I've fixed the bug (I think!).
Remembering our algebraic Lorentz transforms, and that "time" is ict, we'll write the Lorentz transform matrix as

&gamma;...........i&gamma;&lambda;

-i&gamma;&lambda;........&gamma;

where the entries here are the coefficients, under transformation, on the x and ict coordinates, top right is x'(ict), bottom left is x(ict)'. I think that's right (it had better be as the following depends on it).

Let's stick with the hyperbolic trig functions (you will have noticed from the Euler definitions that the imaginary unit i is "understood" in these cases).

Let's find a &beta; such that cosh&beta; = &gamma;.
Looking in our elementary text again, we see that cosh²&beta; - sinh²&beta; = 1. then
sinh&beta; = (cosh²&beta; - 1)^-1/2, which I get to be &lambda;(1 - &lambda;²)^-1/2 = &lambda;&gamma;

We can now write the Lorentz transformation matrix in 2-dimensions as

cosh&beta;..........i(sinh&beta;)

-i(sinh&beta;)..........cosh&beta;

and - bingo! - we have our rotations in the complex plane, as promised.

Note that the matrix is orthogonal (it had better be!), is non-singular, and hence invertible, in other words all that we would expect of a linear transformation on rectilinear coordinates.

Now whether this still works when we consider spacetime to be a manifold, I harbour many doubts, as I've repeatedly said that a single scalar field must be applied, all real or all complex. Nevertheless, the above rings my bell, somewhat.

Great work!
I had done most of this earlier. Because when I was learning GR, I wanted to see what the Christoffel Symbols were. So I had to know what the coordinates were, in order to differentiate the metric tensor. That's what set me off, initially.

Right now, I'm reading up on hyperbolic geometry, and quaternions. Have a look at them. For quaternions, you may look at the links I gave in my earlier post.

And also, do you know that the transformations can be seen as a skewing of the axes? So that they won't be orthogonal anymore.

Great work!Thanks! In fact it was a flight of fantasy, but it looked OK, and I had fun.

And also, do you know that the transformations can be seen as a skewing of the axes? So that they won't be orthogonal anymore.Hmm? Explain what you mean here. Are you talking about transferring to curvilinear coordinates, or something else? This could be fun, so please elucidate.

Oh. You might like this one, if you really want to get into the quaternions.

http://world.std.com/%7Esweetser/quaternions/qindex/qindex.html

I already saw that (the link). There are some good links in the site. But I don't know about the rest of it.

About my other remark, I'll try to explain quickly.

Consider the transformation

x' = ax + by
y' = cx + dy

The equations of the x and y axes are y=0 and x=0, resp. Right? The equations of the x' and y' axes are y'=0 and x'=0.

If we substitute the values of x' and y' in terms of x and y, we get two linear equations. Plote these, and you will get the x' and y' axes as seen from the XY coordinate system. If the transformations are not orthogonal, then you get skewed axes. In order to see whether they're orthogonal. I think it is enough ot check whether the Jacobean matrix of the transformation is orthogonal or not. I'm not sure about that part.

About my other remark, I'll try to explain quickly.

If the transformations are not orthogonal, then you get skewed axes. In order to see whether they're orthogonal. I think it is enough ot check whether the Jacobean matrix of the transformation is orthogonal or not. I'm not sure about that part.Sorry, but I really don't know whatyou're getting at here. Earlier you said:

And also, do you know that the transformations can be seen as a skewing of the axes? So that they won't be orthogonal anymore.Now you seem to be saying we can, if we choose, transfer to skewed axes. Of course we can, but why do that? Usually one tries to go the other way i.e. if axes are not orthogonal, let's make them so. Certainly, in the context of a vector space, we know how to orthogonalize the basis, courtesy of Gramm-Schmidt.

Regarding the Jacobean - you are half right. A condition for a transformation to be invertible is that the Jacobean be non-singular i.e. the functional determinant is non-zero. And, of course, it is a condition of orthogonality that the matrix representation of a transformation is invertible.

No, you misunderstood. We have no choice here. The transformations (Lorentz) result in skewed axes.

I'm sorry, but this wrong. Look - draw X,Y axes, label Y "time", X "space. Draws the worldlines of A stationary wrt these axes, and that of B in motion wrt them. We'll say that the motion of B relative to A is inertial if there is a coordinate transformation that will bring B to rest wrt new axes.

This is a rotation, right, a linear orthogonal transformation. We agreed this before. But the axes are still orthogonal, not skewed.

Aha! I think I've got it. By "skewed axes you mean B's are "skewed" relative to A's?

Hmm. First, that's not what is usually meant by skewed axes, normally it means not orthogonal.

Second, the "skewing" of B's axes relative to A's is an artefact of combining all spatial coordinates into a single dimension. Do the same thing in E⁴ and you will see it's a rotation, as I keep saying.
Anyway, as you said in another thread, it's not important.

Well, they aren't othogonal.
I did mean, skewed w.r.t A's axes, and I mentioned this too, but only in my first post.
I didn't get that last part. How is it a rotation? Or do you mean that skewing can be seen (in this case) as a rotation through complex angles?

Well, they aren't othogonal.
I did mean, skewed w.r.t A's axes, and I mentioned this too, but only in my first post.
We're starting to develope a real communication problem here. A's axes are orthogonal if we say they are, likewise B's. Just to be clear - orthogonal means x&middot;y = 0, i.e. the projection of y on x is zero.
If you're trying to say A's axes aren't "orthogonal" to B's, don't, as they are different spaces. Specifically, we can think of them as basis vectors spanning different vector spaces. Orthogonality between vector spaces has no meaning.
I didn't get that last part. How is it a rotation? Or do you mean that skewing can be seen (in this case) as a rotation through complex angles?I'd rather you didn't keep calling it skewing, normally that means something else. It's a rotation in the sense I showed you a few days ago i.e. related by the Euler angles with either circular or hyperbolic functions

You're right. But I'm a bit confused (or out of depth). Can you give an example of non-orthogonal axes? I realized some time ago that the reason I'm learning tensors and the related stuff slowly, is because of lack of good examples.

You're right. But I'm a bit confused (or out of depth). This is an excellent start! I mean it - the clue to understanding is a willingness to admit one's difficulties. We all have them and it's a life-long affliction.Can you give an example of non-orthogonal axes? I realized some while ago that the reason I'm learning tensors, and the related stuff slowly, is because of lack of good examples.First, any linear coordinates that are not mutually perpendicular are, by the definition, not orthogonal. As coordinates are entirely man-made and arbitrary, we could, if we chose, have non-perpendicular axes. Only a madman would do that, however.


Second, curvilinear coordinates don't even allow us to ask about orthogonality, although, oddly enough, a vector space defined on such coordinates may have the notion of an inner product. That's what I was trying to get at when we talked about tensor contraction.

On the fly, I can't decide whether a similar argument applies to spherical and cylindrical polar coordinates, but I suspect it does.

These things would be so much easier with a sketch.

(Maybe we should do vector spaces?)

Perhaps the following might be a more intuitive definition of orthogonality (restricted, as you'll see, to linear vector space):

If we take the "length" of a vector to be (x&middot;x)^1/2 = |x|, then x and y are orthogonal iff |x||y|cos &theta; = 0.

Remembering that cos &theta; = 0 when &theta; = 90 degrees, I think all should be clear.

Here is a nice animation i found in the following link.


http://www.astro.ucla.edu/~wright/Alfred-Brian-anim.gif

- http://www.astro.ucla.edu/~wright/Alfred-Brian-anim.html

It switches, gradually, between views of 2 different world lines, that is, how the space-time as viewed from each of the world-lines. The space co-ordinate looks skewed while trying to keep the time co-ordinates at the same angle.

everneo: Thanks for that, it's cute. But anyway, it's Saturday night, and I'm bored so...
Let's do this, I hope nobody finds it too patronising.

Take a point P in Euclidean 3-space, and ask how many vectors can pass through P. The answer is, of course, infinitely many. The set of all such vectors at P is referred to as a vector space.

Now vector spaces are said to be defined over a scalar field, which may be real or complex, and will inherit some, not all, of the operational properties of the underlying field. In this sense, the term "field" is simply shorthand for the rules of manipulation that apply to the vectors at P.
Now, we need a way of describing vectors at P, and we could, of course, throw down a set of fictitious coordinates, Cartesian, for example, but that's an ugly thing to do: nobody ever saw a coordinate! So let's rummage in our vector space and see what we can come up with.

First thing we notice is that the majority of our vectors can be described by reference to a limited number of other vectors at P. In 3-space, that number is 3 at a minimum. Specifically, any randomly chosen vector can be described as the sum of (scalar multiples) of at least 3 other vectors. This situation is referred to as linear dependence.

But, we can always find 3 vectors which are mutually linearly independent i.e. have zero extension along each other. All other vectors can be described by reference to these, and we will call these basis vectors Thus we can use these in place of ficitious Cartesian coordinates.

It is usually convenient, in Euclidean space, to insist on orthogonality of our basis, that being defined as in a previous post. Now here's the thing.

Our chosen basis isn't unique - there is an infinity of other equally valid choices, with the same properties, we could have made, and there is no "right" choice. But we can say this:

the number of elements in any basis of a vector space is the same as the number of any other basis of that space. Or, what amounts to almost the same thing:

the number of elements in any basis of a finite-dimensional vector space is the dimension of that space.

In fact, I geuss one would be hard-pressed to come up with an alternative definition of dimension.

More to come, if anybody wants it!

Everneo, that's exactly what I meant.

Aha! I think I've got it. By "skewed axes you mean B's are "skewed" relative to A's?

Yes, that's why I said was a bit confused. Your new post seems to say that only this is possible, anyway. That's what I wanted to know. Is orthogonality relative, all the time? I mean is there any sense in saying that two given axes can (say) never be orthogonal? If that isn't right, I don't understand what confused you earlier.

I mean is there any sense in saying that two given axes can (say) never be orthogonal? Hmm...I thought you had got it.

A condition for orthogonality is that it is, at least in principle, possible to establish a notion of length and angle (remember the definitions of these I gave you? A reminder .."orthogonal means x·y = 0, i.e. the projection of y on x is zero" and...."If we take the "length" of a vector to be (x·x)^1/2 = |x|, then x and y are orthogonal iff |x||y|cos θ = 0. Remembering that cos θ = 0 when θ = 90 degrees, I think all should be clear. "

Can you now see why this is only possible in a given vector space? Of course we can say, if we choose, that the minimum set of linearly independent vectors that constitute the basis of this space is orthogonal.

To describe a transformation as "linear orthogonal" simply means that it's a transformation in which lengths and angles are conserved. In general, these are rotations. So that vectors that are orthogonal in one space are orthogonal in another, provided the transformation itself is orthogonal.

But, and here is your problem - it makes no sense whatever to talk about the relative length and angle (i.e. orthogonality) between elements in one vector space and those in another.

Late as it is, I'm tempted to follow up my last "lecture". Maybe you should ask me some questions (if you have any) about my previous.

Hmm...I thought you had got it.

A condition for orthogonality is that it is, at least in principle, possible to establish a notion of length and angle (remember the definitions of these I gave you? A reminder .."orthogonal means x·y = 0, i.e. the projection of y on x is zero" and...."If we take the "length" of a vector to be (x·x)^1/2 = |x|, then x and y are orthogonal iff |x||y|cos θ = 0. Remembering that cos θ = 0 when θ = 90 degrees, I think all should be clear. "

Can you now see why this is only possible in a given vector space? Of course we can say, if we choose, that the minimum set of linearly independent vectors that constitute the basis of this space is orthogonal.

To describe a transformation as "linear orthogonal" simply means that it's a transformation in which lengths and angles are conserved. In general, these are rotations. So that vectors that are orthogonal in one space are orthogonal in another, provided the transformation itself is orthogonal.

But, and here is your problem - it makes no sense whatever to talk about the relative length and angle (i.e. orthogonality) between elements in one vector space and those in another.

Late as it is, I'm tempted to follow up my last "lecture". Maybe you should ask me some questions (if you have any) about my previous.

In SR Lorentz transformation, gamma factor applied directly to time and reciprocally to space to get the trasformed time and space. That is, the rotation on space coordinate is twice in the opposite direction of time's.

Orthogonality of co-ordinates is not maintained in LT since velocity of light is invariant (no rotation) in both the spaces as per SR. Hence necessarily the x-cordinate (space) need to be skewed in the other space else velocity of light will not be constant just as in galilean space where orgthogonality of the co-ordinates maintained in rotational transforms with varying c.

everneo: I don't how else to say it! A linear orthogonal transformation is one in which length and angle are preserved. Maybe you would prefer it if I used horrid SR jargon. A linear orthogonal transformation of reference frames (i.e. Lorentz transformation) is one in which the laws of physics remain the same in both frames. The two statements are equivalent, although the first is more precise.

If you take first postulate that velocity of light is constant in all frames, then rotational trasformation need to keep the world lines of the photon on both sides (45 degrees each) of time coordinate in both the spaces (say frames). This cannot be done if the transformation keeps orthogonality between space & time coordinate in the space that undergoes the transformation. LT is the projection of one space (rotated frame) on the other, right?.

everneo: I'll try one more time. Given a 2-dimensional space-time coordinate system, the worldline of A relatively at rest and B relatively in motion are easy to visualise. The operation on the coordinates which brings B to rest (in respect of which B's worldline now looks as A's did) is a rotation. Think of this as "transforming B to rest".

The whole point about light velocity is that it cannot be transformed to rest by this or any other operation.

What that means in visual terms is that the worldline of light rotates as the coordinates rotate. Try drawing it.

The whole point about light velocity is that it cannot be transformed to rest by this or any other operation.

Thats what i said :

..since velocity of light is invariant (no rotation) in both the spaces as per SR.

-----

What that means in visual terms is that the worldline of light rotates as the coordinates rotate. Try drawing it.

World line of photon does not rotate as the coordinates rotates.

Please note the animation on the previous page. As the animation switches between world lines of 2 events their y (time) co-ordinate rotates and the x (space) cordinate rotates twice in magnitude but in opposite direction(skewed), else world line of photon (which is not shown) undergoes rotation, not invariance.

could someone please explain me what space time is?

It's been seven pages and no one has answered that yet?

"Space-time" is exactly what it says. Physics deals with "events". Things that happen at a specific place (space) at a specific time. Special Relativity shows that they cannot be separated- whether or not events occur at different places but the same time (simultaneity) depends on the frame of reference from which they are observed. The combination of the two- what in relativity is called the "world path" is independent of the frame of reference. In classical mechanics, changing coordinates systems may mess up the separate components of a "position vector" but will not change the vector itself- that's why classical mechanics works with vectors rather than individual space coordinates. In relativity you lose invariance if you try to treat the three coordinates of space separately from the time coordinate. In order to maintain invariance you have to treat them as a single thing: "Space-time".

And the loss of invariance is because the speed of light is measured as 'c' (the same value) by observers in all frames. This leads to some bizarre physics, and some bizarre geometry, called the Minkowski space-time geometry. And this involves complex axes and rotaions through complex angles, as you can see a few posts earlier, where we were discussing this, before you interrupted.

Quarkhead,
Seems you're back. Don't forget this thread.

Back? No, not really. Got drunk and couldn't resist "showing off" about duals.

QuarkHead: So that was the reason you left. You got burned a few too many times trying to "show off" as you were only able to express logical fallacies and incorrect derivations. Did you find a dumbed down forum somewhere in which you might be able to appear intelligent? I hope so, since despite your incorrect derivations you often like to post, you do seem like an intelligent person to me.

sorry that didnt reply earlier
well. first of all, it is clear as a glass of iceberg water, that space and time are discrete and that they are mutualy bounded, the space can't exist without time and the time can't exist without space. Relativity always affects both the space and time, not only the time or the space.
the 2 big problems of SRT and GRT are:
1. They don't recognize and define clearly and correctly the absolute space
2. They can't work with discreet space-time

until it is solved SRT and GRT are doomed to be unapliable, the Minkowski space is blabla, the 4-dimensional tensors and the curved space, it can only bring some visual satisfaction, ofcourse it is permitted to plot everything in 3D or 4D graphics, anyway I dont understand for what is the big fuzz about ..., I also think that this is not the right way to understand spacetime, first of all, spacetime is deeply linked with energy and particles,it is not just some kind of emptiness, only particle physics can answer what is the spacetime