Understanding Minkowski Space-time: An Explanation of Time as a Dimension

[calebhoilday]

If you have read my other threads, i am having trouble understanding special relativity. The issue seems to be my understanding of space-time.

Space-time infers to me that two events are not separated by only a length in three dimensions, but also time, with time being essentially indistinguishable from a length. This concept can be explained by considering a supernova occurring several thousand light years away and it being observed looking up at the sky. To the observer looking up in the sky, the event of looking up in the sky coincides with the supernova and are to the observer simultaneous. But in fact this occurred several thousand years ago and the event can only be recognised as occurring, once the information has traveled the distance. It might turn out in fact that the supernova wasn't a supernova, but a plane in the sky that has turned on a light. These events are not separated by a massive distance and so it can be considered that the fact that they are simultaneous according to the observer, means that the light only turned on a marginally before it was recognised.

This has been developed into a quantitative concept of four vector space (x,y,z,t) where time is indifferent to length. The resultant formula for space time is:
S^2 = X^2 + Y^2 + Z^2 -(CT)^2
S: the resultant distance between event A and event B
X: the difference between the x coordinates of event A and event B or (Xb-Xa)
Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
C: the speed of light in a vacuum (converts the time units into the length units used)
T: the difference in when event A occurs and when event B occurs, if they were in the same position (Ta-Tb)

Lets consider the supernova situation. Consider event A to be recognition of event B the supernova, the position of event A to be (0,0,0) and event B (100,0,0) units in light years. If in the same position then event A would occur 100 years after event B. As event A is the recognition of event B we know that the resultant distance in 4 vector space is zero and so a result for S must be zero.

S^2 = (100 light years)^2 + 0^2+ 0^2 -(C*100 years)^2
= 10,000-10,000
= 0
their for S = 0

This formula works perfectly in this situation, but in the next situation, something in my logic goes wrong. Event A is a bunch of 2012 believers looking to the sky expecting the end of the world and event B is a supernova, that emits massive amounts radiation, which when this radiation hits earth, will kill the majority of life. If event A and event B are in the same position then event B would occur 10,000 years later. event A has a position of (0,0,0) and event B a position of (10000,0,0). A prediction of what S will equal is based on my logic that (1) event A will happen (2) 10,000 years will pass (3) the supernova occurs and (4) 10,000 years pass before the radiation hits earth. My conclusion is that S=20,000 light years but the math gives a different result.
S^2 = (10,000 light years)^2 + 0^2+ 0^2 -(C*-10,000 years)^2
= 1,000,000 - 1,000,000
= 0
their for S = 0
According to the result found using the formula, the 2012 believers were right.

My problem with space-time seems to be a difficulty in recognising time behaves the same as a dimension such as x,y,z as my logic seems to consider time behave the same as S or the resultant. My logic would have the formula for S as:
S = (X^2+Y^2+Z^2)^0.5 -CT

This is causing me huge confusion and i can not put my finger on what i have done wrong. please help.

 

[yossell]

Hi Calebholiday,

There were a few things about your post that I found confusing - perhaps it's nothing, but perhaps if you cleared them up, it would be helpful in understanding the rest of your post.

Consider event A to be recognition of event B the supernova, the position of event A to be (0,0,0) and event B (100,0,0) units in light years.

Events should have 4-tuples as coordinates. Should this be:
event A to be (0,0,0,0) and event B (100,0,0,0) ?

If so, then you're saying that a supernova happens *here* in 100 *units* time - not light rays.

Or should it have been

event A to be (0,0,0,0) and event B (0,100,0,0)

So the event is NOW (in relative coordinate system), but 100 lightyears along the x axis?

If in the same position then event A would occur 100 years after event B.

What does this mean? A is an event - what does it mean to say if the event was in the same position? Do you intend to asking about a coordinate system where the spatial distance between events A and B is zero? This isn't possible if you had in mind the second case above, as A and B are space-like separated.

 

[Fredrik]

Please don't post the same thing in two different places. You can read my reply in the other thread: https://www.physicsforums.com/showthread.php?p=2801403.

 

[Naty1]

Try reading from Einstein:

http://www.einstein-online.info/

 

[calebhoilday]

Soz about the new thread... i did make some mistakes in the other; considered the thread had changed topic and decided i probably should have done a new thread.

Jargon and eloquence is once again is an issue of my posts. what is inferred by reading, is that event A and event B can have no distance separating them in space time, but have distance separating them in 3 dimensional space. If event B occurs simultaneously with event A, from event A's perspective, then there is no distance in space time separating them. As simultaneity is affected by distance in the 3 dimensions, it is necessary for one to consider what the duration between events is if they occupied the same position in x,y,z to obtain t. This is what was inferred at least from reading, but the fact that the formula doesn't correspond with the logic, suggests my inferences are wrong.

if X,Y,Z,T is eqivenlent to (Xb-Xa),(Yb-Ya),(Zb-Za),(Tb-Ta) then in the first situation, event A has the coordinates of (0,0,0,100) and event B (100,0,0,0). The reason i did not write it this way is because it is confusing. What is inferred is that event A and event B occupy the same postion in space time, but to use the 4 vector co-ordinates logically rules that out. They don't occupy the same position in space time according to the co-ordinates and hence my problem with 4 vector space.

Reading infers that space-time is the distance between two events, which is indistinguishable from the duration between two events. Is this inference wrong?

 

[Fredrik]

Jargon and eloquence is once again is an issue of my posts. what is inferred by reading, is that event A and event B can have no distance separating them in space time, but have distance separating them in 3 dimensional space.

That's right.

If event B occurs simultaneously with event A, from event A's perspective, then there is no distance in space time separating them.

That's wrong. First of all, there's no "event A's perspective". You can associate a coordinate system with a curve in spacetime, but not with a point. And "simultaneous in coordinate system S" only means that S assigns the same time coordinate to both events. The Minkowski square of the separation 4-vector has nothing to do with it.

As simultaneity is affected by distance in the 3 dimensions,

It isn't.

if X,Y,Z,T is eqivenlent to (Xb-Xa),(Yb-Ya),(Zb-Za),(Tb-Ta) then in the first situation, event A has the coordinates of (0,0,0,100) and event B (100,0,0,0).

If you meant (Xb-Xa,Yb-Ya,Zb-Za,Tb-Ta) and that t=0 at event B, then yes.

What is inferred is that event A and event B occupy the same postion in space time,

An event is a position in spacetime, so A and B "occupy the same position in spacetime" if and only if A=B.

They don't occupy the same position in space time according to the co-ordinates

They shouldn't.

Reading infers that space-time is the distance between two events, which is indistinguishable from the duration between two events. Is this inference wrong?

Minkowski spacetime is the set [tex]\mathbb R^4[/tex] with the standard vector space structure and a bilinear form g defined by [itex]g(x,y)=x_0y_0-x_1y_1-x_2y_2-x_3y_3[/itex]. The duration (in the coordinate system associated with the identity map) between x and y is [itex]|x_0-y_0|[/itex].

 

[calebhoilday]

I'm finding simultaneity difficult to describe. I considered that it would be best to describe it in terms of an observer at the position of each event and if the events occurred in the same position.

In the first situation the observer of the supernova considers the supernova to occur at upon observing the sky. If in the same position in space then event A (observation) occurs 100 years after event B (supernova). If an observer was in the position of the supernova, the supernova (event B) would occur 200 years before, someone on Earth observed the sky (event A). Simultaneity according to an observer is affected by resultant distance.

Does this make sense at all?

 

[Dale]

I think you misunderstand how times are assigned in SR. If I receive a signal now from an event 100 light years distant then I would say the event occurred in 1910. I.e. All observers in SR are intelligent enough to account for the finite speed of light in determining when something happened.

 

[calebhoilday]

This is the kind of thing that i needed to know, Thank you.

In what I would call a recognition event, where event A is the recognition of event B, that occurs a distance away from event A; does S in space-time =0 ?

 

[Fredrik]

I'm finding simultaneity difficult to describe. I considered that it would be best to describe it in terms of an observer at the position of each event and if the events occurred in the same position.

Now you mean position in space, right? You can certainly consider two events on Earth, or two events at the star, but I don't see why you'd want to.

In the first situation the observer of the supernova considers the supernova to occur at upon observing the sky. If in the same position in space then event A (observation) occurs 100 years after event B (supernova). If an observer was in the position of the supernova, the supernova (event B) would occur 200 years before, someone on Earth observed the sky (event A). Simultaneity according to an observer is affected by resultant distance.

Does this make sense at all?

I'm afraid not. I can't make sense of it even when I take what you discussed with DaleSpam into account.

This is the kind of thing that i needed to know, Thank you.

You need to read about the standard synchronization convention. I linked to a post that has some of the details in my first reply. You should also be able to find an explanation in any SR book.

In what I would call a recognition event, where event A is the recognition of event B, that occurs a distance away from event A; does S in space-time =0 ?

Yes. The "S²" of the separation 4-vector is always =0 when a light signal goes from one of the events to the other (without being reflected somewhere along the way).

 

[calebhoilday]

Regarding the statement that doesn't make sense. What is stated is a reversal from 'situation one' in the first post. From the position of event A you are informed that event B has occurred at the same time event B occurs. From the position of event B, you are informed of event A's occurrence 200 years later. This is based on the distance in 3 dimensional space being 100 light years between and that event B occurs 100 years before event A

The "occurring according to" can be replaced by "informed of occurring" which can be differentiated from "occurring" due to the distance between events in 3 dimensional space.

When i was reading about Space-time i conceived 'situation two' quite early into the reading. Due to this I can not even understand the introduction, let alone any material that follows.

If you find my description of 'situation one' satisfactory, I would like to explain 'situation two' with a more satisfactory discourse. Hopefully then I can move past the obstacle and finally get the theory.

 

[Dale]

This is the kind of thing that i needed to know, Thank you.

In what I would call a recognition event, where event A is the recognition of event B, that occurs a distance away from event A; does S in space-time =0 ?

Yes. The Minkowski norm is degenerate in that sense. For the ordinary Euclidean norm if the distance s between two points A and B is 0 then A=B, but in the Minkowski norm if the spacetime interval s between two events A and B is zero then that does not imply that A=B.

If we have two events in units where c=1 with spacetime coordinates
[tex]A=(t_A,x_A,y_A,z_A)[/tex]
[tex]B=(t_B,x_B,y_B,z_B)[/tex]
[tex]s^2=(t_A-t_B)^2-(x_A-x_B)^2-(y_A-y_B)^2-(z_A-z_B)^2[/tex]

Then they are called "simultaneous" if [itex]t_A=t_B[/itex] and they are called "co-located" if [itex]x_A=x_B[/itex] and [itex]y_A=y_B[/itex] and [itex]z_A=z_B[/itex]. They are the same event if they are both simultaneous and co-located. Their separation is called "timelike" if [itex]s^2>0[/itex], or "spacelike" if [itex]s^2<0[/itex], and "lightlike" or "null" if [itex]s^2=0[/itex]. The separation is lightlike for any two events where a light signal is sent from one to the other, but since [itex]s^2=0[/itex] does not imply [itex]t_A=t_B[/itex] such events are not generally simultaneous.

 

[Fredrik]

From the position of event A you are informed that event B has occurred at the same time event B occurs. From the position of event B, you are informed of event A's occurrence 200 years later. This is based on the distance in 3 dimensional space being 100 light years between and that event B occurs 100 years before event A

I understand that you're talking about an event where the supernova is observed, and an event at the location of the supernova where the observation of the supernova is observed. But there's something very strange about how you're saying it. It sounds like you're saying that if you're at the location of the supernova 200 years after the star went supernova, this event is simultaneous with the observation event on Earth. It isn't. The second observation event occurs 100 years after the first, so they're certainly not "simultaneous", which means "occuring at the same time"...or to be more precise "assigned the same time coordinate".

When i was reading about Space-time i conceived 'situation two' quite early into the reading. Due to this I can not even understand the introduction, let alone any material that follows.

I'm still not sure what the source of the confusion is. If A, B and C are the three events "star goes nova", "light from nova reaches Earth" and "light reflected by a mirror on Earth comes back to the position of the nova", and we define t=0 and x=0 at event B, then the coordinates (written in the form (t,x)) of these events are:

A: (-100,100)
B: (0,0)
C: (100,100)

Nothing more needs to be said to describe this sequence of events.

 

[calebhoilday]

based on the intelligence assumption replace 'simultaneous' with 'informed of'. if the supernova happens in 1910, Earth is 100 light years away and someone observed the supernova on Earth in 2010, then if you assumed the position of the supernova, you would be informed of observation on Earth in 2110.

This make sense?

 

[calebhoilday]

Situation Number Two

Event A: 2012 theorists expecting impending doom (0,0,0,0)
Event B: Supernova in distant space (10000,0,0,10000)

S^2 = X^2 + Y^2 + Z^2 -(CT)^2
S: the resultant distance between event A and event B
X: the difference between the x coordinates of event A and event B or (Xb-Xa)
Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
C: the speed of light in a vacuum (converts the time units into the length units used)
T: the difference in when event A occurs and when event B occurs, if they were in the same position (Ta-Tb)

my logic would say in this instance before doing the calculations, that S should be 20,000 light years. As event A will occur 10,000 years will pass, the supernova will occur and then 10,000 years will pass before position A will be informed of event B.

The problem is if you do the calculation using the formula S=0 not 20,000

How is the logic wrong when it is correct in situation 1 ?

 

[yossell]

Am still having problems with your notion of 'informed of', and what the events are.

EITHER
Event A - theorists *expecting* impending doom? Do you mean, they are about to be doomed by the explosion, or they somehow come to form the belief that the Earth will be, at some point, wiped out by this explosion? These are two different events.

If the idea is that they are *about* to be wiped out by the effects of the supernova, and event A is their being wiped out, event B the supernova explosion, then I see why you expect s^2 to be relevant to the time they think passes between the two events. BUT IN THAT CASE the supernova explosion could not possibly have the coordinates you give it. Its coordinates place it in the future AND far away - it's t coordinate would have to be negative if its effects were about to be felt.

OR: Event A - the moment at which theorists, due to a keep theoretical calculation, realize that far off system will go supernova in (10000 0 0 10000), and realize the effects will reach the earth, destroying everything. In that case, the coordinates make sense, BUT IN THIS CASE there's no reason to expect s^2 to give information about THE TIME at which the impact will happen. s^2 only concerns the relations A and B - the event of realisation and the event of the supernova. You need to introduce a new event C: the moment of impact, and talk about the s^2 of this event from A.

Interestingly, the event of realisation and the event of supernova lie on the Earth's light cone. But that's not germane to your problem.

 

[Fredrik]

based on the intelligence assumption replace 'simultaneous' with 'informed of'. if the supernova happens in 1910, Earth is 100 light years away and someone observed the supernova on Earth in 2010, then if you assumed the position of the supernova, you would be informed of observation on Earth in 2110.

This make sense?

Yes.

Situation Number Two

Event A: 2012 theorists expecting impending doom (0,0,0,0)
Event B: Supernova in distant space (10000,0,0,10000)
...
As event A will occur 10,000 years will pass, the supernova will occur and then 10,000 years will pass before position A will be informed of event B.

So this time we're talking about a star 10000 light-years away that goes supernova 10000 years from now.

S^2 = X^2 + Y^2 + Z^2 -(CT)^2
S: the resultant distance between event A and event B
X: the difference between the x coordinates of event A and event B or (Xb-Xa)
Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
C: the speed of light in a vacuum (converts the time units into the length units used)
T: the difference in when event A occurs and when event B occurs, if they were in the same position (Ta-Tb)

As I said before, I wouldn't call S "distance". It's just the non-negative square root of the Minkowski square of the 4-vector (X,Y,Z,T).

Your definition of T doesn't make any sense. It's like talking about the difference between the numbers 7 and 5 if they were the same number. Why don't you define it as the difference between the time coordinates of event A and event B?

my logic would say in this instance before doing the calculations, that S should be 20,000 light years.

Then you haven't really looked at the definition of S². Did you read DaleSpam's comments about "separation"? You should.

S² is always =0 when there's light going from one of the events to the other. I have no idea how you got the number 20000.

As event A will occur 10,000 years will pass, the supernova will occur and then 10,000 years will pass before position A will be informed of event B.

The problem is if you do the calculation using the formula S=0 not 20,000

How is the logic wrong when it is correct in situation 1 ?

It's not correct in any situation.

 

[calebhoilday]

Imagine event A is like the end of a countdown, to the instant when a whole buch of people will think the world will end, based on some new age belief. If S=0 then the radiation traveling at C from event B should hit Earth during event A. Common sense says that's not the case in this situation and the 2012 theorists are out by 20,000 years as to when one could expect the cataclysmic event.

 

[Fredrik]

If S=0 then the radiation traveling at C from event B should hit Earth during event A.

That would only make sense if S² is the non-negative square of a time difference and it clearly isn't defined that way.

 

[yossell]

Imagine event A is like the end of a countdown, to the instant when a whole buch of people will think the world will end, based on some new age belief. If S=0 then the radiation traveling at C from event B should hit Earth during event A. Common sense says that's not the case in this situation and the 2012 theorists are out by 20,000 years as to when one could expect the cataclysmic event.

If this is what you have in mind then how could event A have coordinate (0,0,0,0) while event B has coordinate (10000,0,0,10000) as you say in post 15? If A is the origin of the coordinate system, and event B has coordinate (10000,0,0,10000), then, in this coordinate system, event B takes place 10000 years after event A and 10000 units away - i.e. after event A and far away.

In this situation, the only thing that could be inferred was the new age hippies were woefully wrong. But I take it that's not the intended scenario

 

[Dale]

If S=0 then the radiation traveling at C from event B should hit Earth during event A.

No, A is on the past light cone of B, so light can go from A to B. You have it backwards here.

 

[Dale]

Space - time is actually the idea for jumping from ordinary space [x, y, z] = [length,width,height] to imaginary time [it = imaginary time]

Hi ArabianKnight, welcome to PF!

The idea of imaginary time has been out of usage for decades. It can be fun in special relativity, but does not generalize to general relativity. So it is not used except as a sort of historical detour.

 

[calebhoilday]

The point of the problem is I don't think S=0, but when I enter the figures into the formula I get S=0. I think S=20,000 light years.

Logic also tells me that the result I obtain using the formula, would be the result of my logic if I swapped event A and event B.

The formula obtains the right result in situation one, but if I were Minkowski I would have derived
S= (X^2+Y^2+Z^2)^0.5 -CT
S: the resultant distance between event A and event B in space-time
X: the difference between the x coordinates of event A and event B or (Xb-Xa)
Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
C: the speed of light in a vacuum (converts the time units into the length units used)
T: the difference in when event A occurs and when event B occurs (Ta-Tb)

With this formula you get S=0 in situation one and S=20,000 in situation two

Can someone please set out situation two and use the actual formula to get a result.

 

[Fredrik]

The point of the problem is I don't think S=0, but when I enter the figures into the formula I get S=0. I think S=20,000 light years.

You're wrong. We have told you many times now that S²=0. Why are you ignoring that?

The formula obtains the right result in situation one

You keep suggesting that it can give you an incorrect result. That doesn't make any sense at all. It's like saying that something that we have defined to be A "really is" B (with B≠A). If you think it can be, you need to think hard about what the word "definition" means.

Can someone please set out situation two and use the actual formula to get a result.

What do you need that I didn't already tell you, e.g. at the end of post #13?

 

[Dale]

The point of the problem is I don't think S=0, but when I enter the figures into the formula I get S=0. I think S=20,000 light years.

Your calculation is correct, your intuition is wrong. That is pretty common in relativity, our brains are just not wired for instinctively understanding relativistic effects.

Logic also tells me that the result I obtain using the formula, would be the result of my logic if I swapped event A and event B.

Yes, S is symmetric under parity inversion.

The formula obtains the right result in situation one, but if I were Minkowski I would have derived
S= (X^2+Y^2+Z^2)^0.5 -CT
S: the resultant distance between event A and event B in space-time
X: the difference between the x coordinates of event A and event B or (Xb-Xa)
Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
C: the speed of light in a vacuum (converts the time units into the length units used)
T: the difference in when event A occurs and when event B occurs (Ta-Tb)

That equation is not invariant under Lorentz transforms.

 

[jtbell]

The idea of imaginary time has been out of usage for decades. It can be fun in special relativity, but does not generalize to general relativity. So it is not used except as a sort of historical detour.

See for example "Farewell to ict", from a textbook written 37 years ago.

 

[calebhoilday]

Obviously my definition of S is wrong.

Here is situation three. An observer wishing to see a supernova looks at a night sky and at the same time 100 light years away a supernova happens. My logic says that S=100 light years, based on the distance the light has to cover before reaching Earth and that there exists no duration between events.

Co-ordinates of Event A (0,0,0,0)
Co-ordinates of Event B (100,0,0,0)

S^2 = X^2 + Y^2 + Z^2 -(CT)^2
S: the resultant distance between event A and event B
X: the difference between the x coordinates of event A and event B or (Xb-Xa)
Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
C: the speed of light in a vacuum (converts the time units into the length units used)
T: The duration between event A and event B (Ta-Tb)

S^2 = 100^2 - (C*0)^2
=100^2
their for S=100

My definition is consistant with the formula in situation one and situation three, just not situation two.

My definition of S: the resultant in length units (reason I consider it a length), between event A and event B in space-time. S is converted into time units by dividing S by C. S/C is considered by myself to be the duration between event A and being informed that event B has occurred from the position in 3 dimensional space that event A took place.

If the above was the definition for S, then in situation one, S/C would =0. The supernova occurs, the information concerning its occurrence travels through space and reaches event A's three dimensional position, at the same instant event A occurs.

If the above was the definition for S, then in situation three, S/C would =100 years. The supernova (event B) occurs at the same time as an observer looks at the sky (event A), 100 years pass as the light from the supernova moves through space, reaching the position of event A 100 years after event A.

If the above was the definition for S, then in situation two, S/C would =20,000 years.
event A occurs, 10,000 years pass, the supernova occurs, radiation from supernova travels 10,000 light years through space and reaches event A's position 20,000 years after event A.

So what is S? how does it differ from what i considered S to be?

 

[DrGreg]

calebhoilday, you seem to be getting tied in knots trying to work out a physical interpretation of what S is, where [itex]S^2 = X^2 + Y^2 + Z^2 -c^2T^2[/itex] in your notation.

The most important property of S is that it is invariant: if two different observers, traveling at different speeds, each calculate S in their own coordinate systems (i.e. each in coordinates in which the observer is at rest), between the same pair of events, then they both get exactly the same answer. The reason the formula for S is what it is, is because that's the simplest formula that has this property of invariance.

Now we can consider what S is in a few special cases.

• if T=0, i.e. if both events are simultaneous relative to the observer, then S is just ordinary Euclidean space-distance between the events
• if X=Y=Z=0, i.e. if both events occur at the same place relative to the observer, then [itex] cT = \sqrt{|S^2|}[/itex], so S is proportional to the time between the events
• if it is possible to send a light signal from one event to the other, then S=0.

Bearing in mind that S takes the same value in all coordinate systems, it plays a very similar role in 4-dimensional spacetime to Euclidean distance in ordinary 3D non-relativistic space, so some people may refer to S as "distance" (in quotation marks) in 4D space. This has the potential to be confused with ordinary distance in 3D space, so many people call S the "spacetime interval" instead.

Finally, the physical interpretation of S is as follows

• if S² > 0, there is an observer for whom the two events are simultaneous and for that observer S is the distance between the events. All other observers measure a longer distance, so S is the shortest space-distance that any observer can measure between the events.
• if S² < 0, there is an observer for whom the two events occur at the same place and for that observer S is proportional to the time between the events. All other observers measure a longer time, so S is proportional to the shortest time that any observer can measure between the events.
• if S² = 0, then it is possible to send a light signal from one event to the other.
  

 

[Fredrik]

Here is situation three. An observer wishing to see a supernova looks at a night sky and at the same time 100 light years away a supernova happens. My logic says that S=100 light years, based on the distance the light has to cover before reaching Earth and that there exists no duration between events.

Co-ordinates of Event A (0,0,0,0)
Co-ordinates of Event B (100,0,0,0)

S^2 = X^2 + Y^2 + Z^2 -(CT)^2
S: the resultant distance between event A and event B
X: the difference between the x coordinates of event A and event B or (Xb-Xa)
Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
C: the speed of light in a vacuum (converts the time units into the length units used)
T: The duration between event A and event B (Ta-Tb)

S^2 = 100^2 - (C*0)^2
=100^2
their for S=100

Yes, when the two events are simultaneous, S² is just the square of the usual Euclidean distance in "space, at time 0". This is obvious from the definitions of S² and "simultaneous".

("Space, at time 0" is defined as a set of points that are all assigned time coordinate 0 by the coordinate system).

My definition of S: the resultant in length units (reason I consider it a length), between event A and event B in space-time. S is converted into time units by dividing S by C. S/C is considered by myself to be the duration between event A and being informed that event B has occurred from the position in 3 dimensional space that event A took place.

This doesn't define anything, because it's not clear what you mean by a "resultant in space-time". (I can't even guess what you have in mind). Also, when you try to define something, you need to give it a unique name. You defined S² earlier. If you want to define something different, you should at least pick another letter. What you're trying to do is the equivalent of trying to redefine "wednesday" to mean "dog" by saying that a "wednesday" is "can be black or some other color, and has legs", and adding "I consider them cute".

 

[calebhoilday]

A resultant is the equivalent of a set of vectors. But from now on I will consider S to be the space time interval.

When considering how time can be equivalent to length. The only thing that came to mind, was when viewing distant objects, you are viewing the past. As this interval separates events, I considered the recognition situation (situation one), to try get an understanding of what the interval actually is. S = 0 in this situation, i considered that the interval refers to the equivalent length, between events if simultaneous, that would result in the same duration between event A and event B e.g. If event A occurs and then 3 seconds pass before one would be informed of event B, then this is equivalent to event A and event B being 3 light seconds away in 3 dimensions if they are simultaneous.

Consider another situation just for clarity concerning the invariance aspect of S. If a spacecraft leaves a space station and has a velocity that causes time on board to dilate so time passes at half the rate compared to the space station . The spacecraft decelerates instantaneously after one year, according to the spacecraft , so that it has the same velocity as the space station.
Considering event A to be the take off and event B to be the deceleration, does the difference in duration between event A and event B, for the spacecraft (1 year) and the space station (2 years), hold any significance?

 

[Fredrik]

As this interval separates events

As evidenced by the fact that S² can be positive, zero or negative, and the fact that it can be zero without the two events being equal, it's not a measure of how "big" the separation is.

Considering event A to be the take off and event B to be the deceleration, does the difference in duration between event A and event B, for the spacecraft (1 year) and the space station (2 years), hold any significance?

Yes. To see this clearly, have the spaceship do the same thing again, but in the opposite direction. When it gets back to the space station, everyone on the station have aged 4 years, and everyone on the ship has aged 2 years. This is because human bodies can be thought of as really crappy clocks, and a clock measures the proper time of the curve in spacetime that represents its motion. (Note that there are lots of threads about this scenario already. This is a recent one).

 

[calebhoilday]

I get the twins paradox. I was wondering if Lorentz transformations affect the co-ordinates. Event A say happens in 2010 to both the the spaceship and the space station, when does event B happen? Would the spacecraft consider it to be 2011 and the space station consider it to be 2012?

 

[Fredrik]

I get the twins paradox. I was wondering if Lorentz transformations affect the co-ordinates. Event A say happens in 2010 to both the the spaceship and the space station, when does event B happen? Would the spacecraft consider it to be 2011 and the space station consider it to be 2012?

The space station would. To the people on the spacecraft , 1 year has passed since 2010. If they would say that the year is 2011 is another matter. It wouldn't make much sense to say that, because then they should also say that it's 2012 when they get back home and meet all the people who live in 2014.