What is the true nature of time?

[Passionflower]

-I asked you several times what you mean by coordinates and yet you haven't answered that question to me.

Coordinates, as opposed to spacetime, is like a map of spacetime, a projection while spacetime is the reality we live in. Each observer in this spacetime can have a unique measure of time, this is unlike a Galilean spacetime where all time is identical for all observers.

-Yet, you keep using it as if spacetime has four dimensions and there are 3-spacial dimension and no time dimension. And yet, you never answered me what the fourth dimension would be if it is not time.

Observers observe slices of spacetime as space while time is orthogonal to this spactial slice. But different observers observe different slices. There is no single dimension in spacetime that is time.

For instance consider an accelerating observer in spacetime, this observer's worldline is curved, at each point we can make a foliation of spacetime that is space and time, sometimes called 3D+1, for this observer but it pseudo rotates in spacetime at each moment of the acceleration.

 

[cshum00]

Ok, thanks for trying to explaining things. I am still confused so i need more details in order to get the picture you have in your mind.

-First, how are you defining spacetime? You keep using spacetime while i keep telling you that the word spacetime uses time as a dimension. Use another word because spacetime can't be the word you are referring to (if you think that time is not a dimension).

Coordinates, as opposed to spacetime, is like a map of spacetime, a projection while spacetime is the reality we live in. Each observer in this spacetime can have a unique measure of time, this is unlike a Galilean spacetime where all time is identical for all observers.

-Ok, let's forget about spacetime at the moment. Let me try it with just physical space not spacetime. A map of physical space would be 3 coordinates; which is exactly the mathematical concept of 3 dimension. Then how come you say that coordinate doesn't mean dimension?
-Then this brings me to the next question of how are you defining dimension?

Observers observe slices of spacetime as space while time is orthogonal to this spactial slice. But different observers observe different slices. The is no single common dimension in spacetime that represents time.

-Again, having something orthogonal to something else is the idea of dimensions so that you can have an extra coordinate to navigate on while you are denying that time is not a dimension.

-And here is the last question, if a coordinate system is not the same as a dimensional system. Then what is the difference?

 

[Passionflower]

-Again, having something orthogonal to something else is the idea of dimensions so that you can have an extra coordinate to navigate on while you are denying that time is not a dimension.

Yes but this is a chart for a particular observer of spacetime, e.g. a particular foliation of space and time. Try to distinguish between a chart or a map of something and the real thing.

 

[cshum00]

Please answer the other questions as well.

Yes but this is a chart for a particular observer of spacetime, e.g. a particular foliation of space and time. Try to distinguish between a chart or a map of something and the real thing.

The problem is that you just tell me to compare it when i don't see the difference. Rather than just tell me to compare it, tell me what is the difference. That way get to the point faster.

 

[Passionflower]

Please answer the other questions as well.The problem is that you just tell me to compare it when i don't see the difference. Rather than just tell me to compare it, tell me what is the difference. That way get to the point faster.

Let's take an example, suppose we have an inertial observer who uses a chart of spacetime that maps his spatial dimensions the way he measures it and orthogonal to that he maps his time. He, for the sake of argument, defines that time in spacetime is orthogonal to his space. Now he accelerates, what will happen? Well the original chart no longer maps onto his space and time foliation. He could Fermi walker transport this chart so that at all times his foliation of space and time is as he measures it but then the chart pseudo rotates (pseudo because spacetime is a Minkowski spacetime and not an Euclidean spacetime) wrt spacetime. The one real dimension in spacetime he defined as time before acceleration is no longer time for him.

Spacetime as it exists in nature has four dimensions, it is however a mistake to claim that one of these dimensions is time, as I said before each observer can have a different view of what represents space and time and what for one observer is time may be a combination of space and time for another observer.

In general relativity the difference between a chart and spacetime itself becomes even more painful, especially in spacetimes that are non-stationary. Think in this context about the background independence of GR, something which distinguishes GR from QM.

 

[ghwellsjr]

Ever since your first post on this thread, you have not used the term "spacetime interval":

In Galilean spacetime time is surely a dimension, but is that the case in relativity?

I think in relativity time is the length of a path between two events in four dimensions. You think I am wrong?

Is that because you are talking about something entirely different?

 

[matheinste]

As I said before I think that time is a path in spacetime not a dimension of spacetime. Feel free to introduce mathematics to show how wrong I am.

Excuse the lack of rigor but Mathematically the definition of the dimension of a vector space is the least number of coordinate terms required to uniquely define the "position" of any (maybe abstract point) object within that vector space. However, when dealing with vector spaces modelling physical situations there are practical restrictions involved. We choose three spatial and one time axis, orthogonal to the spatial subspace, to suit our needs.
These axes are usually taken to be our physically defined dimensions althouggh they are really coordinate axes. So a dimension is really just a number describing a certain property of a vector space. We need to choose four coordinate to obtain four coordinates for the event we wish to define and we usually call these axes dimensions to suit our idea of what a dimension is physically.

A timelike interval, or path in spacetime can be regarded as a coordinate axis if the time axis is taken as part of the usual coordinate system asigned by an observer at rest with respect to that coordinate system. Of course in this somewhat special circumstance the other three coordinate numbers required as the spacetime coordinate of events along the time axis are all zero. But for non inertial observers this is not a practical proposition.


In the case of physical dimensions, for an observer at rest and so having the time dimension as one of his coordinate axes, each dimension for that observer is usually taken as being mutually orthogonal to the others, but this is really to suit physics and is not necessary mathematically where the only requirement is for the coordinate axes to be not linearly dependent.

Matheinste.

 

[Passionflower]

A timelike interval, or path in spacetime can be regarded as a coordinate axis if the time axis is taken as part of the usual coordinate system asigned by an observer at rest with respect to that coordinate system. Of course in this somewhat special circumstance the other three coordinate numbers required as the spacetime coordinate of events along the time axis are all zero. But for non inertial observers this is not a practical proposition.

A coordinate axis is not the same as a dimension of spacetime. Same story here, a chart used to map spacetime is mistaken for spacetime itself.

Again if we assume for the sake of argument that a particular dimension of spacetime is time then all observers would have to agree this dimension represents time but that is not the case as different observers are oriented differently in spacetime. What is time for one observer is a mixture of space and time for another observer.

 

[Dale]

There seems to be a lot of confusion on this thread with people "talking past" each other. In relativity treminology can often be confusing and the same word can be used to identify different concepts. So I hope the following helps:

1) Spacetime is a 4-dimensional pseudo-Riemannian manifold with a (-+++) signature. The dimensionality and the signature of the manifold are coordinate-independent properties of the manifold. One of these dimensions is singled out from the others (in a coordinate independent sense) by the signature and is called the "timelike" dimension. In this sense "time is a dimension of spacetime".

2) On the other hand the reason that spacetime is 4D is because at each point in the manifold you can construct an orthonormal basis (for the tangent space) with 4 basis vectors. One of these basis vectors will be timelike and the others will be spacelike. You can call the timelike basis vector "time", but it is not unique. There are an infinite number of possible sets of basis vectors at each point. So none of these individual time basis vectors can be said to be "the" time basis vector in a coordinate independent sense.

3) At each point along a worldline in spacetime it is possible to construct a tangent vector. This tangent vector can be classified as timelike, spacelike, or lightlike (null). If the tangent vector is timelike at every point along a worldline then the whole worldline is said to be timelike and it can represent the motion of a massive particle. The length of a timelike worldline is the called the proper time, and it is a coordinate independent scalar quantity.

Most of the confusion on this thread seems to be that everyone is using the same word for all three distinct concepts.

 

[Passionflower]

One of these dimensions is singled out from the others (in a coordinate independent sense) by the signature and is called the "timelike" dimension.

Ok, so which one is the one singled out?

I am asking because I do not agree there is such a singled out dimension, I think 'rotating' our manifold gives us the same physical description, any direction can represent the timelike dimension.

You say in a coordinate independent sense, so let's say we have 5 observers going from event A to B with different path lengths. How, in a coordinate independent way, do they determine this singled out "timelike" dimension?

Do you agree or disagree that time for each of those observers is the length of the path on this manifold and that this is not represented by one single dimension of the manifold?

 

[Dale]

Ok, so which one is the one singled out?

The one with the negative signature.

I am asking because I do not agree there is such a singled out dimension, I think 'rotating' our manifold gives us the same physical description, any direction can represent the timelike dimension.

You can't rotate a manifold, that doesn't make sense.

You say in a coordinate independent sense, so let's say we have 5 observers going from event A to B with different path lengths. How, in a coordinate independent way, do they determine this singled out "timelike" dimension?

By looking at the signature. The observers have nothing to do with it. Remember, this refers to the dimensionality of the space, not some specific direction or vector within the space.

Do you agree or disagree that time for each of those observers is the length of the path on this manifold and that this is not represented by one single dimension of the manifold?

Yes, I agree that proper time (see 3 above) is the length of a timelike worldline. But there are other usages of the word "time", as I pointed out above.

 

[Passionflower]

Yes, I agree that proper time (see 3 above) is the length of a timelike worldline.

Ok, I am glad you agree.

But there are other usages of the word "time", as I pointed out above.

I must be slow. So in what way do you think the "timelike" dimension of the manifold is time?

 

[cshum00]

Let's take an example, suppose we have an inertial observer who uses a chart of spacetime that maps his spatial dimensions the way he measures it and orthogonal to that he maps his time. He, for the sake of argument, defines that time in spacetime is orthogonal to his space. Now he accelerates, what will happen? Well the original chart no longer maps onto his space and time foliation. He could Fermi walker transport this chart so that at all times his foliation of space and time is as he measures it but then the chart pseudo rotates (pseudo because spacetime is a Minkowski spacetime and not an Euclidean spacetime) wrt spacetime. The one real dimension in spacetime he defined as time before acceleration is no longer time for him.

The problem is that this not only happen to the time dimension. It also happens to the spatial dimensions. There is also length contraction when traveling at speed close to light, the observers won't agree on the distance seen to be traveled. In that case, not only time won't be a dimension but also space would be dimensionless according to your analogy.

Spacetime as it exists in nature has four dimensions, it is however a mistake to claim that one of these dimensions is time, as I said before each observer can have a different view of what represents space and time and what for one observer is time may be a combination of space and time for another observer.

If time is not the fourth dimension in the spacetime, then what is? Don't just tell me that "if time is the fourth dimension, what is its' length?" Tell me, in your analogy; what is the fourth dimension in spacetime if time isn't the one?

 

[Passionflower]

The problem is that this not only happen to the time dimension. It also happens to the spatial dimensions. There is also length contraction when traveling at speed close to light, the observers won't agree on the distance seen to be traveled. In that case, not only time won't be a dimension but also space would be dimensionless according to your analogy.

What is a spatial dimension for one observer can be a mixture of time and space for another observer.

If time is not the fourth dimension in the spacetime, then what is?

Not one single dimension of spacetime is space or time since this would imply an absolute space and time as in the case of Galilean spacetime.

As I wrote, by now like four times or more, what for one observer is time is a mixture of space and time for another observer or what for one observer is a spatial dimension is a mixture of space and time for another observer. While for an accelerating observer this constantly changes. Why does that seem so hard to understand?

 

[cshum00]

As I wrote, by now like four times or more, what for one observer is time is a mixture of space and time for another observer or what for one observer is a spatial dimension is a mixture of space and time for another observer. While for an accelerating observer this constantly changes. Why does that seem so hard to understand?

It is almost impossible to understand you for me because you keep using words that have solid definitions in a totally different meaning.

What is a spatial dimension for one observer can be a mixture of time and space for another observer.

Not one single dimension of spacetime is space or time since this would imply an absolute space and time as in the case of Galilean spacetime.

Ok, so you are saying that neither space or time are dimensions? I agree with you that in Galilean spacetime thinks of absolute time and space but even if you are in Minkowski spacetime you still uses both space and time as dimensions.

 

[Passionflower]

It is almost impossible to understand you for me because you keep using words that have solid definitions in a totally different meaning.

All right let's turn things around.

Suppose we have an observer that determined the 'timelike' dimension of spacetime as DaleSpam described (I think this does not make any sense, but for the sake of argument I assume we found it) is the time dimension. Now this observer starts to accelerate for 5 seconds? What do you think will happen? Where is the time dimension after 2 seconds and where is it after 4 seconds. Please answer that question.

 

[cshum00]

All right let's turn things around.

Suppose we have an observer that determined the 'timelike' dimension of spacetime as DaleSpam described (I think this does not make any sense, but for the sake of argument I assume we found it) is the time dimension. Now this observer starts to accelerate for 5 seconds? What do you think will happen? Where is the time dimension after 2 seconds and where is it after 4 seconds. Please answer that question.

The spacetime view of the observer changes meaning each dimensions gets transformed around. Just as simple as that.

Edit: Let me as you this. What is so exceptional about "dimension" that time and space can't be a dimension?

 

[Passionflower]

The spacetime view of the observer changes meaning each dimensions gets transformed around. Just as simple as that.

But there is only one spacetime right and one dimension is time (at least that is what you claim) right? Or are you saying that there are many spacetimes?

 

[cshum00]

But there is only one spacetime right and one dimension is time (at least that is what you claim) right? Or are you saying that there are many spacetimes?

Are you playing philosophy? Yes, there is "one" time dimension. Just because it gets transformed it doesn't mean that it is not itself. Just because you grew older doesn't mean that you are not yourself. Yes, in philosophy you can play with the words and way that the you one second ago is not the you at the moment. But in your logic you are creating an infinite amount of yourself everytime the present becomes the past; and each one of them is a different you.

I knew you were getting there. You have been using random words or trying to find a specific word so that you can mock around with it. I love philosophy myself but there are reasons things have a solid definitions in science so that they don't get tossed around with multiple meanings.

There is nothing wrong with transforming the dimension since the current reference frame has been changed due to acceleration; it is the same spacetime dimension on a different shape.

 

[cshum00]

But there is only one spacetime right and one dimension is time (at least that is what you claim) right? Or are you saying that there are many spacetimes?

Are you playing philosophy? Yes, there is "one" time dimension. Just because it gets transformed it doesn't mean that it is not itself. Just because you grew older doesn't mean that you are not yourself. Yes, in philosophy you can play with the words and say that the you one second ago is not the you at the moment. But in your logic you are creating an infinite amount of yourself everytime the present becomes the past; and each one of them is a different you. And each of those different you(s) are independent at freewill form each other.

I knew you were getting there. You have been using random words or trying to find a specific word so that you can mock around with it. I love philosophy myself but there are reasons things have a solid definitions in science so that they don't get tossed around with multiple meanings.

There is nothing wrong with transforming the dimension since the current reference frame has been changed due to acceleration; it is the same spacetime dimension on a different shape.

 

[Passionflower]

There is nothing wrong with transforming the dimension since the current reference frame has been changed due to acceleration; it is the same spacetime dimension on a different shape.

So the spacetime dimension gets transformed, only for the accelerating observer or all observers? Hopefully you will now see that you cannot maintain that one single dimension of spacetime is time. Or do you still think that the spacetime we live in gets transformed by an accelerating observer? If so, how does this transform impact other observers?

 

[cshum00]

So the spacetime dimension gets transformed, only for the accelerating observer or all observers? Hopefully you will now see that you cannot maintain that one single dimension of spacetime is time. Or do you still think that the spacetime we live in gets transformed by an accelerating observer? If so, how does this transform impact other observers?

Yep, no doubt about it. You are just turning this into something philosophical which can just drag on forever as long as we twist things up into our own advantage like all philosophical arguments do.

Let's make it this way, you say that the second observer have a different spacetime because the current view his/her spacetime is different from the first observer who is accelerating. It could just twist this around and say that it is the same spacetime that is just transformed to his current view. As a proof, the second observer only has to accelerate to the same syncrohization so that the spacetime view of his is just like the first observer. It is the same spacetime in a different transformed view due to the fact that he is not accelerating.

Let's end this pointless argument because you are just trying to use words to make a invalid argument when in the first place you had to change the original meaning of spacetime just to create an argument when it actually clearly states that it is 3-spacial dimensions and one time dimension.

 

[Passionflower]

Let's make it this way, you say that the second observer have a different spacetime because the current view his/her spacetime is different from the first observer who is accelerating.

No, I am not saying that at all, there is only one spacetime we are living in. I am saying that spacetime is 4 dimensional but, unlike in Galilean spacetime, no single dimension is time or space. Different classes of observers will measure space and time differently because their measure of space and time are pseudo rotated wrt each other in spacetime.

However we can clearly determine what is time in spacetime, time is an observer's path in spacetime. Clearly a path in spacetime and a dimension of spacetime are two different things.

By the way an observer observes only 3 dimensions, clocks record time.

it actually clearly states that it is 3-spacial dimensions and one time dimension.

What clearly states?

 

[cshum00]

No, I am not saying that at all, there is only one spacetime we are living in. I am saying that spacetime is 4 dimensional but, unlike in Galilean spacetime, no single dimension is time or space. Different classes of observers will measure space and time differently because their measure of space and time are pseudo rotated wrt each other in spacetime.

Here is the thing, if time was not a dimension we can't transform time meaning we can only transform the 3 spatial dimensions which is the problem what the Galilean transforms used to cause.

However we can clearly determine what is time in spacetime, time is an observer's path in spacetime. Clearly a path in spacetime and a dimension of spacetime are two different things.

Ok? Neither you or I ever said anything against that statement: a path in spacetime and a dimension of spacetime are two different things. So, how does this relate to anything we are talking about? You said that accelerating observer spacetime dimension looks different from a non-accelerating observer, then their spacetime dimensions must be different. I never said anything about path, i only said that they are the same spacetime dimensions and to proof it you only have to accelerate the non-accelerating observer to so that his spacetime view looks the same as the accelerating one. It is the same spacetime just that each of them are seeing different things due to their conditions. It is like having 2 observers one in front of a light distorting glass while another one standing front of a car. Both will see the same car but differently due to the different conditions. And yet, it is the same car that is standing in front of them.

What clearly states?

Spacetime clearly states that it is composed of 3-spacial dimensions and one time dimension.

 

[Passionflower]

Here is the thing, if time was not a dimension we can't transform time meaning we can only transform the 3 spatial dimensions which is the problem what the Galilean transforms used to cause.

How do you suppose to transform time, you can't transform time, time is what a clock measures.

Ok? Neither you or I ever said anything against that statement: a path in spacetime and a dimension of spacetime are two different things. So, how does this relate to anything we are talking about?

Well if we agree that the length of an observer's path, for instance between to events, is time you cannot also say that it is a dimension of spacetime.

You said that accelerating observer spacetime dimension looks different from a non-accelerating observer, then their spacetime dimensions must be different.

I never said that, please try to read clearly. An accelerating observer, for as long as he is accelerating, keeps (pseudo) rotating his spatial foliation wrt spacetime dimensions.

It is the same spacetime just that each of them are seeing different things due to their conditions. It is like having 2 observers one in front of a light distorting glass while another one standing front of a car. Both will see the same car but differently due to the different conditions. And yet, it is the same car that is standing in front of them.

That is simply incorrect.

Here is an analogy: think of spacetime as a fixed box, different classes of observers will be rotated wrt each other inside this box, the effect will be that lengths and durations are not perceived equally. The analogy is not perfect of course, fist of all we need a 4 dimensional box and second spacetime is not a Euclidean but a Minkowskian (or when curved a Lorentzian) manifold, and observers are pseudo rotated.

Spacetime clearly states that it is composed of 3-spacial dimensions and one time dimension.

Can we agree that spacetime is a real thing? Einstein's EFE represent a particular spacetime, obviously we cannot express our universe analytically in this equation because it is far to complicated but our universe is a spacetime nevertheless. Then what do you mean by 'spacetime states', you describe it as some kind of definition only.

 

[cshum00]

How do you suppose to transform time, you can't transform time, time is what a clock measures.

Lorentz transformation and or Minkowskian geometry takes time as a dimension and then transform time dimension in order to calculate time dilation.

Well if we agree that the length of an observer's path, for instance between to events, is time you cannot also say that it is a dimension of spacetime.

I never said anything about agreeing on the length of the path. I only said about the shape of the dimensions would transform.

I never said that, please try to read clearly. An accelerating observer, for as long as he is accelerating, keeps (pseudo) rotating his spatial foliation wrt spacetime dimensions.

Here is an analogy: think of spacetime as a fixed box, different classes of observers will be rotated wrt each other inside this box, the effect will be that lengths and durations are not perceived equally. The analogy is not perfect of course, fist of all we need a 4 dimensional box and second spacetime is not a Euclidean but a Minkowskian (or when curved a Lorentzian) manifold, and observers are pseudo rotated.

That is exactly i have been saying and that it requires to take time as a dimension. My analogy for the car and glass was a oversimplified analogy but the same.

Can we agree that spacetime is a real thing? Einstein's EFE represent a particular spacetime, obviously we cannot express our universe analytically in this equation because it is far to complicated but our universe is a spacetime nevertheless. Then what do you mean by 'spacetime states', you describe it as some kind of definition only.

I have no problem agreeing that spacetime is a real thing. The problem is that you say that spacetime are not made of dimensions.

Let me ask you this then. How in the world are you going to work on Minkowskian spacetime if time is not a dimension? Show me the mathematics.

 

[Passionflower]

The problem is that you say that spacetime are not made of dimensions.

I never said that. You really need to read more accurately.

How in the world are you going to work on Minkowskian spacetime if time is not a dimension? Show me the mathematics.

I do that all the time. To calculate an observer's time between two events one needs to take the length of the path, one generally does this by integration.

 

[cshum00]

I never said that. You really need to read more accurately.

Ok, let's try to match each other thoughts for a second and lay out our commonalities and differences.

First, let's start with what we have in common.
-We both agree that there is actually such thing as spacetime.
-We both agree that Galilean spacetime is somewhat faulty but Minkowski is correct.

What we don't agree on is:
-You say that one of spacetime dimensions is not time. I say time is one of spacetime dimensions.
-You say that Time is a path in spacetime not a dimension of spacetime. while i say that the path in spacetime is just a path in spacetime but not time.

So, we can conclude that our main conflict is with time and spacetime. I am saying that time is a dimension and it is also a dimension in spacetime.

So let's start with your analogy. You say that there are four dimensions in spacetime. Three of the four dimensions are spatial dimensions and there last one is something but not time. Time cannot be a dimension of spacetime because when an observer accelerates his old time dimension gets transformed and it is no longer the old time dimension. Therefore it can't be a dimension and neither the one for spacetime.

Now, using the same analogy of time and because of time dilation the time dimension; it transforms time dimension so it can't be a dimension. There is also the other three spatial dimensions in spacetime. The three other spatial dimensions can have length contraction according to specail relativity; which is a transformation of three spatial dimensions. But according to your analogy, spatial dimensions can't be dimensions neither because it transforms just like time! Then if that is true, then what are the dimensions in spacetime?!

I do that all the time. To calculate an observer's time between two events one needs to take the length of the path, one generally does this by integration.

Ok, show them to me. Both the formulas and the derivations. I bet you that right from the beginning of the derivations they use time as a dimension. Show them to me otherwise i can't see the whole picture.

 

[Passionflower]

Ok, let's try to match each other thoughts for a second and lay out our commonalities and differences.

First, let's start with what we have in common.
-We both agree that there is actually such thing as spacetime.
-We both agree that Galilean spacetime is somewhat faulty but Minkowski is correct.

Yes a Galilean spacetime has a notion of absolute time, in fact this is nothing but the fourth dimension of Galilean spacetime. Adjusted for special relativity we have a Minkowski spacetime, adjusted for mass and energy we have a Lorentzian spacetime.

You say that one of spacetime dimensions is not time. I say time is one of spacetime dimensions.

Correct, none of the four dimensions of spacetime can be called time or space because that would imply absolute space and time. Each class of observers has their own notion of what orientation in spacetime consists of space (and orthogonal to that time). In other words what is space and time is observer dependent in relativity not a property of the spacetime dimensions.

-You say that Time is a path in spacetime not a dimension of spacetime. while i say that the path in spacetime is just a path in spacetime but not time.

Correct.

So let's start with your analogy. You say that there are four dimensions in spacetime. Three of the four dimensions are spatial dimensions and there last one is something but not time.

No that is not what I am saying, what I am saying is that each class of observers will observe an identical dimension of spacetime differently, some will say it is space while others will say it is some mixture of space and time. One observer is not more right in relativity than another observer so one must conclude that no single dimension of spacetime can rightfully be identified as time or space.

Time cannot be a dimension of spacetime because when an observer accelerates his old time dimension gets transformed and it is no longer the old time dimension. Therefore it can't be a dimension and neither the one for spacetime.

I agree that time is not a dimension of spacetime but I do not agree with the argumentation you describe above. However when an observer accelerates he constantly adjusts his notion of space wrt spacetime because he pseudo rotates in spacetime. Interestingly in this respect is Fermi Walker transport which illustrates this.

 

[cshum00]

Looks like we made some progress. Now that we have some stuff on the same page, let's try to do it similarly with the stuff we disagree on.

So our problem are still on
-Time
-Spacetime
-Spacail dimensions (new)

One observer is not more right in relativity than another observer so one must conclude that no single dimension of spacetime can rightfully be identified as time or space.

1) How does that one observer not being right in relativity than another observer conclude to no single dimension of spacetime can rightfully identified as time or space? You did a huge jump there. It almost seemed to me that you are trying to relate two completely unrelated subjects.

2) So, now you conclude that neither time or space are dimensions of spacetime? Then what are the dimensions of spacetime? You were saying that there were 4-dimensions from the beginning right? What are they?

3) You still haven't shown me the formulas and its derivation we talked about. Show them to me.

So, there are 3 questions above. Don't just pick and choose to answer the ones that favor you. Answer them completely.

 

[Dale]

So in what way do you think the "timelike" dimension of the manifold is time?

In the sense that it must be measured with clocks.

 

[Passionflower]

In the sense that it must be measured with clocks.

So are you saying that a clock does not measure the path length between two events in spacetime but instead measures an interval of a single dimension of spacetime?

So for two different observers traveling between these two events with different path lengths, would you claim that they each measure an interval of the same dimension of spacetime?

 

[ghwellsjr]

Passionflower, back in post #111, I asked you:

Ever since your first post on this thread, you have not used the term "spacetime interval":

In Galilean spacetime time is surely a dimension, but is that the case in relativity?

I think in relativity time is the length of a path between two events in four dimensions. You think I am wrong?

Is that because you are talking about something entirely different?

And now for the first time you are using the word "interval" in the same sentence with "spacetime" but not the term "spacetime interval":

So are you saying that a clock does not measure the path length between two events in spacetime but instead measures an interval of a single dimension of spacetime?

So for two different observers traveling between these two events with different path lengths, would you claim that they each measure an interval of the same dimension of spacetime?

So I still can't tell if you are talking about the "spacetime interval" or something else.

But, just in case you are talking about "spacetime interval", let me explain what it is and then you can tell me if it helps.

First you have to understand what an event is. It is nothing more than a specified location (in three dimensions) at a specified time as defined by a specified coordinate system. It does not necessarily have anything to do with observers or paths or any actual event, although it may. You can then transform the event (location plus time) to any other coordinate system and the numbers you get to describe the four components of the event could be totally different.

In Galilean spacetime, if you have two events, the spatial distance between any two events can be calculated by taking the square root of the sum of the squares of the differences in the three dimensions and the time difference is merely the difference in the two times. Then if you transform the two events into a different coordinate system, even though all the numbers are different to describe the locations and times of the two events, if you perform the same computation, you will get the same answers for the spatial distance and time difference between the same two events defined by the second coordinate system, even if this second coordinate system is in motion with respect to the first one.

By Galilean spacetime, we mean that the relative speed between the two coordinate sytems, otherwise known as frames of reference, is much less than the speed of light.

But if the two coordinate systems (frames of reference) have a high speed between them, then the calculations that we did under the Galilean spacetime do not give the same spatial distance and time difference in the two frames of reference. However, we can define a new "distance" or "difference" between the two events which is called the "spacetime interval" that will be the same no matter what frame of reference we do the computation in, but instead of getting two numbers, a spatial distance and time difference, we get just one, the spacetime interval, based on a calculation of the two previous values.

The computation is very similar to the spatial distance, in fact we start with that prior to taking the square root but instead we subtract the square of the time difference multiplied by the square of the speed of light.

It should be no surprise that this computaton yields a frame invariant quantity, since we use the Lorentz Transform to produce the numbers for the second frame of reference, and the transform guarantees that the spacetime interval is frame invariant.

Does that help or are you talking about something completely different?

 

[Dale]

So are you saying that a clock does not measure the path length between two events in spacetime but instead measures an interval of a single dimension of spacetime?

So for two different observers traveling between these two events with different path lengths, would you claim that they each measure an interval of the same dimension of spacetime?

What is an "interval of a dimension"?

Btw, it is hard to be sure, but I think you are confusing usages 1) and 2) above. I.e. You seem to always think in terms of a direction or a basis vector (2) instead of a dimension (1). The dimensionality of a space is a geometric property which exists independent of any coordinate system. Can you formulate your question without respect to any coordinate system? (e.g. Without any observers' perspective)

 

[Passionflower]

So in what way do you think the "timelike" dimension of the manifold is time?

In the sense that it must be measured with clocks.

So then explain yourself.

I claim that the time as measured by a clock between two events is the length of the path of this clock in spacetime. You claim that that the clocks measure the "timelike" dimension of spacetime to read time.

So consider a few clocks going from event A to event B, all have a different path length in spacetime. You agree that the manifold has four dimensions, are you perhaps claiming that their paths are all perpendicular to what you call the "timelike" dimension of the manifold so that they can measure the "timelike" dimension of spacetime?

I included a spacetime diagram showing the spacetime paths of those clocks going from event A to event B:
[PLAIN]http://img713.imageshack.us/img713/9677/event.gif
I claim that time for each clock is the path length calculated by using a Minkowski metric.

Now why do you think the vertical dimension, which is what you call the "timelike dimension of spacetime is time?