Questions concerning the geometry of spacetime

[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.

 

[Mister T]

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)

 

[student34]

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?

I meant spatial distance.

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.

Yes, that makes sense.

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.

I agree. Yet we observe changes in the real world. The changes seem to be explained as time flowing by us/observation/consciousness or as we going through time.

 

[jbriggs444]

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 (-dtf^2)^(1/2)

In the context of special relativity, because the world line may not be a straight line (geodesic).

For example, one would not measure the length of a curving road in ordinary 2 dimensional Euclidean space by taking the square root of the sum of the squared difference between the endpoint coordinates. One would integrate along the path instead.

 

[robphy]

Spacetime Physics (1st edition)
Chapter One
Part One
https://www.eftaylor.com/download.html#special_relativity
It’s time to read “Parable of the Surveyors”

 

[Sagittarius A-Star]

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"

Only in case of the (-+++) convention ($ds^2=-c^2dt^2+dx^2+dy^2+dz^2$), not in case of the (+---) convention ($ds^2=c^2dt^2-dx^2-dy^2-dz^2$). See for this also the 2^nd part of posting #30 from @PeterDonis.

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)

See:
https://en.wikipedia.org/wiki/Proper_time#In_special_relativity

 

[robphy]

Spacetime Physics (1st edition)

https://www.eftaylor.com/download.html#special_relativity
It’s the proper time
to read about proper time
in Chapter One Part Two.

 

[PeterDonis]

if we use the spacetime interval formula then we will get a negative squared spacetime interval between "time-like separated events"

If you are using the appropriate signature convention, yes.

which I understand to be an imaginary number.

No. The sign of the squared interval is not taken into account when taking the square root. The sign of the squared interval just labels the interval as timelike or spacelike. It does not mean one of them corresponds to an interval that is an imaginary number.

(If you doubt this, consider that if we are using the timelike signature convention, then a spacelike squared interval is negative. Does that mean ordinary spatial distances are imaginary numbers? Of course not.)

 

[Dale]

I meant spatial distance.

A timelike interval is not a spatial distance.

Yet we observe changes in the real world.

Things change over space as well as time. So the fact that there is change in a particular direction in no way implies that anything is flowing.

The changes seem to be explained as time flowing by us/observation/consciousness or as we going through time

Certainly, that is a possible view which is compatible with the data. However, as I said before, it is not the geometrical view. In this thread you are asking about the geometrical view, so the non-geometrical view doesn’t belong in this thread. Not because it is inherently wrong, but it is just a topic for a different thread.

If you want to discuss time flowing instead of the geometrical view of time then let me know. I can easily close this thread and you can open a new one to discuss that instead.

 

[student34]

Thanks everyone, it was another helpful discussion.

To see if I learned anything, I will make a closing statement about what I learnt. Please correct me if I am wrong.

I knew there were differences between spatial properties and the temporal properties. However, I did not know that the temporal dimension does not share the same property of spatial distance as the spatial dimensions.

Having said that, I am still quite curious and perplexed about what this temporal dimension is, and how it can share so many spatial properties such as curvature, spatially measurable and intersect with spatial dimensions, yet does not have spatial distance.

 

[PeterDonis]

I knew there were differences between spatial properties and the temporal properties. However, I did not know that the temporal dimension does not share the same property of spatial distance as the spatial dimensions.

You still don't seem to have the right conceptual scheme. This might be because you don't even seem to have the right conceptual scheme for ordinary Euclidean geometry in ordinary Euclidean space, so let's start with that first.

What you are calling "spatial distance" is not a property of "spatial dimensions". It's a property of the metric, which, for our purposes here, you can think of as a function that takes two points and gives you a number, which in the case of Euclidean geometry we call the "distance" (or more precisely the "squared distance", since you have to take its square root to get what we normally call the distance). In Euclidean geometry, the metric is basically the usual Pythagorean theorem. A key property of this metric is that it is what is called "positive definite": the squared distance between any two distinct points is always a positive number. But this is not a property of any particular "dimension": it's a property of the geometry as a whole, because the metric is a property of the geometry as a whole.

Now consider the case of Minkowski spacetime, which is the geometry of spacetime in special relativity. The metric in this case is now not positive definite: the metric is still a function that takes two points and gives you a number, but now that number is not always positive. It can be positive, negative, or zero. (Purists would say that this means the thing we're calling the "metric" for Minkowski spacetime is really a "pseudometric", but we won't go into such fine points here.) But still, the metric is not a property of any particular "dimension", nor are the squared distances of different signs properties of different "dimensions". They're all just properties of the geometry as a whole.

Having said that, I am still quite curious and perplexed about what this temporal dimension is, and how it can share so many spatial properties such as curvature, spatially measurable and intersect with spatial dimensions, yet does not have spatial distance.

These questions also mostly come from having the wrong conceptual scheme. Hopefully the above helps.

However, you also mention other properties here: "curvature", "spatially measurable", and "intersect with spatial dimensions". Let's consider those briefly:

Curvature is not a property of a "dimension". It's a property of either a particular curve (curves can be "straight"--the technical term is "geodesic"--or not) or a geometric manifold as a whole (the Minkowski spacetime of SR is flat, but in General Relativity we also consider spacetimes that are curved). The latter kind of curvature is represented by a tensor, the Riemann curvature tensor.

Timelike intervals are not "spatially measurable", so I'm not sure what you're referring to with that.

Timelike curves can of course intersect spacelike curves, but this is not "dimensions" intersecting. "Dimensions", to the extent that concept even means anything in this context, aren't the kinds of things that can "intersect".

 

[student34]

What you are calling "spatial distance" is not a property of "spatial dimensions". It's a property of the metric, which, for our purposes here, you can think of as a function that takes two points and gives you a number, which in the case of Euclidean geometry we call the "distance" (or more precisely the "squared distance", since you have to take its square root to get what we normally call the distance). In Euclidean geometry, the metric is basically the usual Pythagorean theorem. A key property of this metric is that it is what is called "positive definite": the squared distance between any two distinct points is always a positive number. But this is not a property of any particular "dimension": it's a property of the geometry as a whole, because the metric is a property of the geometry as a whole.

Now consider the case of Minkowski spacetime, which is the geometry of spacetime in special relativity. The metric in this case is now not positive definite: the metric is still a function that takes two points and gives you a number, but now that number is not always positive. It can be positive, negative, or zero. (Purists would say that this means the thing we're calling the "metric" for Minkowski spacetime is really a "pseudometric", but we won't go into such fine points here.) But still, the metric is not a property of any particular "dimension", nor are the squared distances of different signs properties of different "dimensions". They're all just properties of the geometry as a whole.

So then is there a difference between 1 meter of Minkowski spacetime and 1 meter of Euclidean geometry?