What is the true nature of time?

[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?

 

[DaveC426913]

Passionflower, can what you are saying about time-as-a-path as opposed to time-as-a-dimension not just as easily be said about any of the spatial dimensions?

If I take a meandering route from Chicago to New York, I follow a path that moves through three spatial dimensions. Someone else might take a different, longer path. It seems analagous that you'd claim that x and y are not dimensions, since my personal x and y are different from someone else's.

 

[Passionflower]

If I take a meandering route from Chicago to New York, I follow a path that moves through three spatial dimensions. Someone else might take a different, longer path. It seems analagous that you'd claim that x and y are not dimensions, since my personal x and y are different from someone else's.

One difference is that in spacetime we have paths between events not paths between spatial locations.

If you and I leave Chicago at the same time and we arrive in New York at the same time and we take different paths our clocks may not agree on how long the trip took despite that we traversed through the same amount of, what DaleSpam calls, "timelike" dimension (See the above spacetime diagram)

For a good understanding I like everyone to contrast a Galilean spacetime and a Minkowski spacetime. Time is clearly a dimension in Galilean spacetime, I do not think anyone will disagree with that.

In Galilean spacetime our watches would always show the same time because in Galilean spacetime we actually could consider the time dimension as the time that a clock measures. We would simply look at how much we traversed in the time dimension to get the elapsed time of the trip. In Galilean spacetime the time and space dimensions are uniquely time and space for all observers.

But now contrast this with a Minkowski spacetime, here it is no longer that straightforward. In a Minkowski spacetime there is no longer a unique time and space dimension. Both you and I traversed the same amount of "timelike" dimension going from Chicago to New York, however our clocks still may not read the same time. We cannot simply take, as in the case of Galilean spacetime, the amount of time dimension traversed as the elapsed time, no instead we need to take the length of the path to get the elapsed time.

Minkowski recognized this very early when he made the famous statement:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
​

 

[cshum00]

Minkowski recognized this very early when he made the famous statement:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
​

You still haven't answered my three questions above. Please answer them.

Also, Minkowski's statement doesn't say anything about space and time not being dimensions. Rather, it is saying that space alone and time alone won't do the cut so we have to put them together.

And in my point of view, what the statement is saying is that since space was already 3-dimensions and time a dimension by itself; we combine them together so that we get a precise mathematical result of 4-dimensions which is called spacetime (which is exactly what i have been saying from the very beginning).

 

[Passionflower]

You still haven't answered my three questions above. Please answer them.

Also, Minkowski's statement doesn't say anything about space and time not being dimensions. Rather, it is saying that space alone and time alone won't do the cut so we have to put them together.

And in my point of view, what the statement is saying is that since space was already 3-dimensions and time a dimension by itself; we combine them together so that we get a precise mathematical result of 4-dimensions which is called spacetime (which is exactly what i have been saying from the very beginning).

Well please comment on the spacetime diagram below, the vertical axis is what DaleSpam calls the "timelike" dimension. We see five travelers between event A and B all taking different paths in spacetime.

So how do you think we calculate the time for the various travelers between these two events?

If time is a dimension, like in the case of Galilean spacetime, the elapsed time for all observers is the same, namely the height. But in case of a Minkowski spacetime it is the length of the paths (and in Minkowsi spacetime the paths that looks longest in the diagram are actually the shortest) and not the height. As you can see most of the paths are curved, which indicates those travelers underwent proper acceleration. To obtain the lengths we have to integrate. The obtained lengths are the elapsed times.
Do you agree with that?
[PLAIN]http://img713.imageshack.us/img713/9677/event.gif

 

[Dale]

So then explain yourself.

To locate any event requires three rods and one clock (minimally), thus there are three dimensions of space and one of time.

I claim that the time as measured by a clock between two events is the length of the path of this clock in spacetime. ... I claim that time for each clock is the path length calculated by using a Minkowski metric.

Sure, I clearly mentioned proper time as the third common usage of the word already.

You seem to think that I am saying you are wrong. I am not. I am simply pointing out that (as is common in relativity) the terminology is sloppy and there is more than one meaning in common usage.

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

Again, you are confusing the first and second usages. "Vertical" is a direction, not a dimension. In your spacetime diagram the vertical direction would be coordinate time for a reference frame where the starting and ending events are colocated.

 

[Passionflower]

You seem to think that I am saying you are wrong. I am not.

Well you can't have your cake and eat it too. If you agree with me I cannot understand how you can maintain that time is a dimension in a Minkowski (and Lorentzian) spacetime.

Please point out in the above referenced spacetime diagram where you think the time dimension is. If the diagram was a diagram of a Galilean spacetime I would agree that the vertical axis represents time but not in the case of a Minkowskian spacetime.

I am simply pointing out that (as is common in relativity) the terminology is sloppy and there is more than one meaning in common usage.

Well it looks you are fully supporting this sloppy usage. We now have people in this forum who think time is not calculated by taking the length of a path in spacetime because they are taught time is instead a dimension.You are of the opinion that a Minkowski spacetime has a time dimension am I right?

So then explain that for the 5 observers drawn (except the one having a straight line) in the spacetime diagram we cannot simply take the total amount progressed in this particular dimension to obtain the total time as we could have if the spacetime were Galilean?

To me that answer is simple: because we are not talking about a Galilean spacetime, instead we are talking about a Minkowski spacetime where the time is calculated by taking the length of the path not just the amount progressed against the, what you call, "timelike", dimension.

 

[Dale]

Well you can't have your cake and eat it too.

This is a rather absurd comment. I merely point out that a word has more than one definition (used by the community as a whole) and therefore I am trying to "have my cake and eat it too" according to you.

Well it looks you are fully supporting this sloppy usage.

On the contrary, I have been the only participant attempting to clarify usage. This thread is a prime example of how confusion can be fostered by participants not being specific and clear.

In particular I would recommend that you use the term "coordinate time" to indicate the 2nd meaning above and the term "proper time" to indicate the 3rd. Currently you are not clearly distinguishing between the two meanings in your posts.

 

[Passionflower]

In particular I would recommend that you use the term "coordinate time"

All observers in our universe observe proper time all the other times are really 'make believe times'. The term coordinate time hardly ever has a physical meaning in relativity especially in curved spacetime. Hardly an argument for calling such a thing time, let alone a dimension.

Comparing Galilean spacetime and Minkowski spacetime is a very good exercise. Trying to understand why in the case of Galilean spacetime, time can rightfully be called a dimension and why that is not correct for Minkowski spacetime.

I think there are two main problems in relativity education, first the "time is a dimension" argument and second the "acceleration does not matter" argument. With it, pages of confusion are created with people proclaiming paradoxes that are really not paradoxes at all. But the origin is bad education.

Perhaps you missed my question to you as you did not answer it:

Could you please point out in the above referenced spacetime diagram where you think the time dimension is?

 

[ghwellsjr]

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

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 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?

Passionflower, are you ever going to answer my question, are you talking about the "spacetime interval"?

 

[Passionflower]

Passionflower, are you ever going to answer my question, are you talking about the "spacetime interval"?

A spacetime interval is the distance between two events in spacetime this is not necessarily the same as the length of an observer's path between two events. In some cases however they could be identical namely in the case the observer takes the largest possible travel time between these events.