I Questions concerning the geometry of spacetime

[student34]

As long as saying that does not imply that it is a Euclidean rectangle, sure.As long as saying that does not imply that the plane is a Euclidean plane, sure.

What's the point of all this?

In general, I want to understand what can be said about the space of the rectangle. I also want to know about the properties of the temporal dimension and the relationship between the temporal dimension and the spatial dimensions.

I've already told you that. Please go back and read my previous posts again. If you honestly can't see how a clock and a ruler are physically different things measuring physically different quantities, this thread is pointless and we might as well close it.

You said, "measurements of physically different things", I thought this meant what the device is measuring, not the device itself.

This is verging on personal speculation, which is off limits here.

That is not what I have read. I have read that this is an implication to relativity. What else would compose a particles' world line?

Have you actually tried to study special relativity from a textbook?

Yes, I took it in a first-year physics course at university.

 

[PeterDonis]

In general, I want to understand what can be said about the space of the rectangle. I also want to know about the properties of the temporal dimension and the relationship between the temporal dimension and the spatial dimensions.

In other words, you want to understand special relativity. That means that, as I said before, you should go learn it from a textbook. We can't give you a complete course in SR here; that's way beyond the scope of a PF thread.

You said, "measurements of physically different things", I thought this meant what the device is measuring, not the device itself.

It means both. You have physically different devices (clock and ruler) measuring physically different things (proper time vs. proper length).

That is not what I have read.

Read where? Please give a specific reference.

In SR, particles are represented as worldlines, which are timelike curves (or null curves if the "particles" are photons--light rays) in spacetime. Neither of the things you said are saying that ("string through time" could sort of be interpreted that way, but could also be interpreted other ways, and "spacetime is granular" is completely off topic for relativity, that's a speculation in quantum gravity for which there is no evidence).

 

[PeterDonis]

I took it in a first-year physics course at university.

Then I am flabbergasted that you would be asking the questions you are asking and would not understand the basics of Minkowskian geometry already.

 

[PeterDonis]

the measurement of arc lengths along a spacelike dimension is not actually a "direct" measure.

This is true, but you don't even need to get to this point to see that arc length along a spacelike curve is physically different from arc length along a timelike curve. Just the fact that you have two different measuring devices is enough. The points you raise are additional valid points that further emphasize the difference, yes.

 

[student34]

No to both. A spacetime-diagram is a picture of physical reality. It is not exactly identical to it.

• The spatial (x,y)-coordinates on a paper belong to Euclidean geometry. The squared distance between two points on the paper is equal to $\Delta x^2 + \Delta y^2$.
• The spacetime (ct,x)-coordinates in reality have Minkowski-geometry (scenario without the other 2 spatial dimensions): The squared spacetime-interval between two events is either defined as $c^2\Delta t^2 - \Delta x^2$ or as $-c^2\Delta t^2 + \Delta x^2$. This is convention.

You must consider this difference between reality and a picture of it, when reading a spacetime diagram.

Then this is where I am confused. It is said that we travel through time at c. In one second I "travelled" 299,792,458 meters through time. But then if I use the spacetime interval formula, I get an imaginary number. Ok, I think I can learn something new here.

 

[PeroK]

Then this is where I am confused. It is said that we travel through time at c.

$c$ is a speed, in terms of distance per unit time. So, traveling "through time at $c$", requires a different definition of speed to start with.

 

[PeterDonis]

It is said that we travel through time at c.

Said where? Please give a reference.

 

[robphy]

Yes, I took it in a first-year physics course at university.

Then I am flabbergasted that you would be asking the questions you are asking and would not understand the basics of Minkowskian geometry already.

Sadly, the typical first-year course that introduces relativity
does not develop Minkowski spacetime geometry.
Those that do often hardly scratch the surface of the "geometry" part of it
... the emphasis is typically on the lorentz transformations.
(In more advanced treatments which deal with 4-vectors, many do not discuss the "geometry",
but merely the component-wise additivity and transformation properties.)

 

[Dale]

Then I am flabbergasted that you would be asking the questions you are asking and would not understand the basics of Minkowskian geometry already.

A first year physics course doesn’t have time for a good treatment of spacetime and relativity. They have to focus on forces and energy and torque and so forth. I wouldn’t expect that a first year physics course would cover relativity at any depth

 

[robphy]

A first year physics course doesn’t have time for a good treatment of spacetime and relativity. They have to focus on forces and energy and torque and so forth. I wouldn’t expect that a first year physics course would cover relativity at any depth

In short,
a first year physics course doesn't have space and time
for a good treatment of spacetime.

 

[Nugatory]

It is said that we travel through time at c. In one second I "travelled" 299,792,458 meters through time.

You’ll hear this in non-serious presentations, sometimes even from well-regarded physicists in a well-intentioned attempt to get the idea across without inflicting excessive math on the audience. It’s not exactly wrong, but it is not a sound basis for any deeper understanding. In particular…

But then if I use the spacetime interval formula, I get an imaginary number.

It obscures the crucial distinction between timelike and spacelike intervals, which is captured in the sign of the square of the interval.

If you are serious about learning relativity, you will have to unlearn that bit about “traveling through time at c” and start learning from a real textbook. Taylor and Wheeler’s “Spacetime Physics” would be my first choice.

 

[vanhees71]

Here's my attempt to explain SR:

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[PeterDonis]

Sadly, the typical first-year course that introduces relativity
does not develop Minkowski spacetime geometry.

A first year physics course doesn’t have time for a good treatment of spacetime and relativity.

This is disappointing to me, but evidently I am not familiar with the current state of pedagogy in this area.

 

[Herman Trivilino]

This is disappointing to me, but evidently I am not familiar with the current state of pedagogy in this area.

It's not clear what the OP meant by "took [SR] in a first-year physics course at university". I assumed it was a typical first-year course in SR alone, not the first course in the introductory sequence. Sometimes the introductory sequence ends with a course in "modern physics" where the amount of time spent on SR will vary widely.

Regardless, the instructor may not have addressed the geometric approach to SR in much depth.

 

[robphy]

It's not clear what the OP meant by "took [SR] in a first-year physics course at university". I assumed it was a typical first-year course in SR alone, not the first course in the introductory sequence. Sometimes the introductory sequence ends with a course in "modern physics" where the amount of time spent on SR will vary widely.

Regardless, the instructor may not have addressed the geometric approach to SR in much depth.

Maybe you meant "first course in SR alone".
"First-year courses in SR alone" are very rare.
There is one first-year course I know of that deals with SR in some depth:
Tom Moore's SIx Ideas that Shaped Physics unit R
http://www.physics.pomona.edu/sixideas/sequences.html

possibly interesting:
http://www.physics.pomona.edu/sixideas/AAPTW20.pdf
http://www.physics.pomona.edu/sixideas/70Syllabus.pdf

 

[student34]

$c$ is a speed, in terms of distance per unit time. So, traveling "through time at $c$", requires a different definition of speed to start with.

It looks like I am going to walk into a semantical nightmare, but here goes anyways.

From what I understand, time has a direction, and it "flows" one way. So either we travel/flow through the time dimension, or time flows past us. I don't know if the difference matters for the purposes of my thread.

I am hoping to understand if the "distance" on the time axis is an actual distance, as in something we could measure with meters, or if it is the imaginary distance using the spacetime interval. I feel like I can make progress here by understanding the difference between the two "distances" through time.

 

[student34]

It's not clear what the OP meant by "took [SR] in a first-year physics course at university". I assumed it was a typical first-year course in SR alone, not the first course in the introductory sequence. Sometimes the introductory sequence ends with a course in "modern physics" where the amount of time spent on SR will vary widely.

Regardless, the instructor may not have addressed the geometric approach to SR in much depth.

I took a first-year physics course that had a chapter on relativity. We learned about Lorentz transformations and other basic principles, but not go very far in depth. And that was about 6 years ago.

 

[PeroK]

It looks like I am going to walk into a semantical nightmare, but here goes anyways.

Semantics only becomes a problem when you start to talk about physics, rather than doing physics. I'd say get on with solving problems (using spacetime diagrams or not) and let the conceptual side sink in as you do so.

That said, I think the transition to a fully geometric view of spacetime takes some time for your brain to assimilate. I didn't really understand the geometric view until I started to learn GR.

 

[student34]

You’ll hear this in non-serious presentations, sometimes even from well-regarded physicists in a well-intentioned attempt to get the idea across without inflicting excessive math on the audience. It’s not exactly wrong, but it is not a sound basis for any deeper understanding. In particular…
It obscures the crucial distinction between timelike and spacelike intervals, which is captured in the sign of the square of the interval.

If you are serious about learning relativity, you will have to unlearn that bit about “traveling through time at c” and start learning from a real textbook. Taylor and Wheeler’s “Spacetime Physics” would be my first choice.

I am serious. I have tried going on my own to study university courses using the textbooks needed for courses at the university, but it was an incredibly slow process for me.

I seem to learn much faster discussing the topics and getting feedback with others. However, I definitely read a lot from reputable sources as I dig further into the subject.

 

[student34]

Semantics only becomes a problem when you start to talk about physics, rather than doing physics.

Lol! Yeah, good point.

 

[PeterDonis]

From what I understand, time has a direction, and it "flows" one way.

The way this is modeled in SR is that, along every timelike worldline, the proper time, which is the "arc length" parameter along the worldline, increases in one direction. We call that direction the "future".

I am hoping to understand if the "distance" on the time axis is an actual distance, as in something we could measure with meters, or if it is the imaginary distance using the spacetime interval.

It's neither. It's a time. Elapsed time is a physically different thing from spatial distance. It's not an "actual distance". It's not an "imaginary distance". It's an elapsed time.

I think you're making this much harder than it needs to be. What I've just said above should be obvious. In this case, the obvious is actually true. You shouldn't be looking for reasons to doubt it. You should be looking for how to fit this obvious truth into your understanding of relativity.

 

[Nugatory]

was an incredibly slow process for me

What is your timeframe for “incredibly slow”?
Someone can put six serious good faith months into that Taylor and Wheeler book with us here to help them over the hard spots, and they’ll have covered stuff that took some of the smartest people who ever lived most of a century to figure out. I’d call that “astoundingly fast”.

 

[student34]

The way this is modeled in SR is that, along every timelike worldline, the proper time, which is the "arc length" parameter along the worldline, increases in one direction. We call that direction the "future".It's neither. It's a time. Elapsed time is a physically different thing from spatial distance. It's not an "actual distance". It's not an "imaginary distance". It's an elapsed time.

I think you're making this much harder than it needs to be. What I've just said above should be obvious. In this case, the obvious is actually true. You shouldn't be looking for reasons to doubt it. You should be looking for how to fit this obvious truth into your understanding of relativity.

So then why hasn't anyone taken issue with my diagram? I gave the rectangle a length of 2 meters. Nobody commented on that.

 

[PeterDonis]

So then why hasn't anyone taken issue with my diagram? I gave the rectangle a length of 2 meters.

Meters is a perfectly acceptable unit of time in relativity. One meter of time is the time it takes light to travel 1 meter, or about 3.3 nanoseconds.

 

[student34]

What is your timeframe for “incredibly slow”?
Someone can put six serious good faith months into that Taylor and Wheeler book with us here to help them over the hard spots, and they’ll have covered stuff that took some of the smartest people who ever lived most of a century to figure out. I’d call that “astoundingly fast”.

Yeah, that time frame seems reasonable.

 

[Sagittarius A-Star]

In one second I "travelled" 299,792,458 meters through time. But then if I use the spacetime interval formula, I get an imaginary number. Ok, I think I can learn something new here.

It is correct, that the squared spacetime-interval can be negative.

If you use for example the (-+++) convention $ds^2=-c^2dt^2+dx^2+dy^2+dz^2$ , then the squared spacetime-interval between time-like separated events is negative, but the elapsed proper time along a world-line connecting both events, which is in this case the integral over ${d\tau} = \frac{1}{c}\sqrt{-ds^2}$, is real. See also a related posting of @vanhees71 .

It is always ensured, that directly mensurable physical quantities, like time-intervals (measured with clocks) and spatial distances (measured with rulers), are real.

 

[student34]

Meters is a perfectly acceptable unit of time in relativity. One meter of time is the time it takes light to travel 1 meter, or about 3.3 nanoseconds.

But you are also saying that it is not an actual distance. I am confused.

 

[PeterDonis]

you are also saying that it is not an actual distance. I am confused.

Why? The fact that we are using "meters" as a unit of time does not make time the same thing as spatial distance. It just means we have chosen units in which the speed of light is $1$. Such units are often used in relativity.

 

[Dale]

I am hoping to understand if the "distance" on the time axis is an actual distance, as in something we could measure with meters

I am not sure what you mean by “actual distance”. It is a spacetime interval. Why should it be anything other than what it is?

Spacetime intervals can be either timelike or spacelike. Timelike intervals are measured with clocks and spacelike intervals are measured with rulers. Either way the result of such a measurement can be reported in meters, regardless of the device used.

From what I understand, time has a direction, and it "flows" one way. So either we travel/flow through the time dimension, or time flows past us.

This isn’t the geometrical view. In the geometrical view time is just another dimension. There is a future time direction and a past time direction but there is no time flow. If you draw a line on a page there is no sense that the page flows along the line. The line simply has some extent on the page without requiring the page to move under the line or the line to move on the page.

 

[student34]

It is correct, that the squared spacetime-interval can be negative.

If you use for example the (-+++) convention $ds^2=-dt^2+dx^2+dy^2+dz^2$ , then the squared spacetime-interval between time-like separated events is negative, but the elapsed proper time along a world-line connecting both events, which is in this case the integral over ${d\tau} = \sqrt{-ds^2}$, is real. See also a related posting of @vanhees71 .

It is always ensured, that directly mensurable physical quantities, like time-intervals (measured with clocks) and spatial distances (measured with rulers), are real.

Thank you very much for this because this is exactly where I am confused. If I am understanding you correctly, you say that if we use the spacetime interval formula then we will get a negative squared spacetime interval between "time-like separated events", which I understand to be an imaginary number. However, I do not understand what is different about an "elapsed proper time along a world line" which you say can be real. Why are we taking its integral instead of just (-dt_f^2)^(1/2)