What are the possible formations of just the dimension of time in a Minkowski space?

[Mates]

Did I make a mistake in my diagram? The diagram on the right has 2 blue objects and the red object coming towards them very quickly (let's say at a speed that would cause a 50% length contraction). Wouldn't the red object have to be twice as close to both blue objects? If this is correct, how can the red object be twice as close to each blue object?

 

[topsquark]

Did I make a mistake in my diagram? The diagram on the right has 2 blue objects and the red object coming towards them very quickly (let's say at a speed that would cause a 50% length contraction). Wouldn't the red object have to be twice as close to both blue objects? If this is correct, how can the red object be twice as close to each blue object?

The diagram on the left has a length contraction between the blue lines so I see no problems.

-Dan

 

[Mates]

Yes, they would be parallel lines. (However, not to muddy things, parallel and perpendicular are both defined differently for intervals. Using the 3D concept, ie. the Pythagorean theorem, then yes they are parallel.)

-Dan

It seems very interesting to me what you say here in brackets. Looking at the diagram on the right side, are you saying that the worldlines are not parallel along the time dimension?

Since we cannot look down on the crossing of the x and t dimension like we see in the graph, is it possible to look down on just the time dimension the way that we are looking down on the x dimension in the graph?

 

[topsquark]

It seems very interesting to me what you say here in brackets. Looking at the diagram on the right side, are you saying that the worldlines are not parallel along the time dimension?

Since we cannot look down on the crossing of the x and t dimension like we see in the graph, is it possible to look down on just the time dimension the way that we are looking down on the x dimension in the graph?

Yes, you can take a projection along the time axis like you would any other axis. Really there is nothing that special about the time axis, it's just that it's a different geometry than what we are used to seeing in everyday life.

Parallel lines never meet in any geometry. That's how they are defined. However parallel lines in Minkowski space can appear curved in regular 3D space so they don't always appear to be lines. That's what I meant.

-Dan

 

[Mates]

Yes, you can take a projection along the time axis like you would any other axis. Really there is nothing that special about the time axis, it's just that it's a different geometry than what we are used to seeing in everyday life.

Parallel lines never meet in any geometry. That's how they are defined. However parallel lines in Minkowski space can appear curved in regular 3D space so they don't always appear to be lines. That's what I meant.

-Dan

I am still a little confused. Hopefully help with this next question will clear things up for me.

Suppose this block universe in the diagram is all that exists and without the red world line. Assume this structure of 2 world lines is finite, say, the total time is only 100 meters long.

On the diagram, the worldlines would seem to have to be parralel as per the definition of parallel that you mentioned.

And from what I understand, the lines are straight and are always exactly the same distance apart. And I think that I can even say that the two lines are the same length - correct me if I am wrong. And assume we can only work with 2 dimensions because I want to use special relativity as a guide.

Given all of these constraints and properties, doesn't this structure have to exist as a Euclidean rectangle (without a top and bottom)? Or is there a way that this can be something else because of it being in a Minkowski space?

 

[topsquark]

I am still a little confused. Hopefully help with this next question will clear things up for me.

Suppose this block universe in the diagram is all that exists and without the red world line. Assume this structure of 2 world lines is finite, say, the total time is only 100 meters long.

On the diagram, the worldlines would seem to have to be parralel as per the definition of parallel that you mentioned.

And from what I understand, the lines are straight and are always exactly the same distance apart. And I think that I can even say that the two lines are the same length - correct me if I am wrong. And assume we can only work with 2 dimensions because I want to use special relativity as a guide.

Given all of these constraints and properties, doesn't this structure have to exist as a Euclidean rectangle (without a top and bottom)? Or is there a way that this can be something else because of it being in a Minkowski space?

I am racking my brains (and the internet) to show you a picture of how this appears. If you are looking for what I think you are looking for a Euclidean diagram will be transformed by the Lorentz transformation to an image that appears to have a focal point in the distance. (I'm not an artist... I think that's the word.) You would never be able to see it in a real diagram but you can see evidence of it in Astronomy. Locally it will look like a rectangle but the sides curve a bit.

-Dan

 

[Mates]

I am racking my brains (and the internet) to show you a picture of how this appears. If you are looking for what I think you are looking for a Euclidean diagram will be transformed by the Lorentz transformation to an image that appears to have a focal point in the distance. (I'm not an artist... I think that's the word.) You would never be able to see it in a real diagram but you can see evidence of it in Astronomy. Locally it will look like a rectangle but the sides curve a bit.

-Dan

So does this mean that even a single vertical world line is not straight in a Euclidean space?

 

[topsquark]

So does this mean that even a single vertical world line is not straight in a Euclidean space?

If you are moving in the direction of the line the line will appear to be straight in both spaces. It's only when there is motion perpendicular to the line that the line appears to curve.

-Dan

 

[Mates]

If you are moving in the direction of the line the line will appear to be straight in both spaces. It's only when there is motion perpendicular to the line that the line appears to curve.

-Dan

Then in the diagram, would the blue line be curved because the red line is approaching it? If so, can we visualize this curve in Euclidean space?

 

[topsquark]

Then in the diagram, would the blue line be curved because the red line is approaching it? If so, can we visualize this curve in Euclidean space?

Yes, the blue lines would appear to be curved according to the moving object.

-Dan

 

[Mates]

Yes, the blue lines would appear to be curved according to the moving object.

-Dan

Is this only appearance, from photons or some kind of illusion, or are the blue lines actually curved for the "red-line observer"?