I Questions concerning the geometry of spacetime

[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?

 

[PeterDonis]

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

Strictly speaking, you can't even compare the two. They're numbers from the metric of two different manifolds.

If there is some physical link between the Euclidean geometry you are considering and the Minkowski spacetime you are considering (for example, if the Euclidean geometry is the geometry of a spacelike slice of Minkowski spacetime--"space" at one instant of "time"), then you can use the same unit for both. Then 1 meter in the Euclidean geometry would be the same as 1 meter of the MInkowski spacetime that it's a spacelike slice of, because you chose the units that way.

 

[student34]

Strictly speaking, you can't even compare the two. They're numbers from the metric of two different manifolds.

If there is some physical link between the Euclidean geometry you are considering and the Minkowski spacetime you are considering (for example, if the Euclidean geometry is the geometry of a spacelike slice of Minkowski spacetime--"space" at one instant of "time"), then you can use the same unit for both. Then 1 meter in the Euclidean geometry would be the same as 1 meter of the MInkowski spacetime that it's a spacelike slice of, because you chose the units that way.

I think I went off focus when I brought up Euclidean geometry. A better question is what is the difference between a meter of spacelike distance and a meter of timelike distance?

 

[PeterDonis]

what is the difference between a meter of spacelike distance and a meter of timelike distance?

That the first is spacelike and the second is timelike. The first is measured with a ruler and the second with a clock.

 

[student34]

That the first is spacelike and the second is timelike. The first is measured with a ruler and the second with a clock.

I can also measure a meter of space with time, the distance light travels in 1/299792458s. Is there an intrinsic difference between a meter of each?

 

[PeterDonis]

I can also measure a meter of space with time

By using light and measuring its travel time, yes. In fact, this is how the meter is currently defined in SI units.

Is there an intrinsic difference between a meter of each?

Yes. The fact that you can use light travel time to measure distance in space does not make spacelike intervals the same as timelike intervals.

 

[student34]

Yes.

What is the difference?

The fact that you can use light travel time to measure distance in space does not make spacelike intervals the same as timelike intervals.

I agree.

 

[PeterDonis]

What is the difference?

That one is spacelike and one is timelike. There is no other answer. The physical fact in relativity is that these two kinds of things are fundamentally different. That's the way spacetime geometry works.

 

[jbriggs444]

What is the difference?

The question seems disingenuous. Physics, like all of the sciences, is experimental in nature.

Time is what clocks measure. We have well established and accurate agreement amount a large number of people and a large number of clocks. Spring clocks, pendulum clocks, crystal oscillators, water clocks, sun dials, sand clocks and atomic clocks. They all measure something. The something that they measure and agree upon is time.

Distance is what distance measuring devices measure. We have well established and accurate agreement among a large number of people and a large number of measuring devices. We use rulers, micrometers, feeler gauges, odometers, dead reckoning, radar, sonar and interferometry to measure distances. They all measure something. The something that they measure and agree upon is distance.

Some of the most precise and reproducible calibrations for distance measuring devices happen to be based on time measurements and an assumed (and well verified) consistency of the speed of light. So yes, there is a relationship between time and distance. But that does not make them the same thing.

I measure the "distance" between the world-line corresponding to Chicago and the world-line corresponding to Cleveland with distance measuring devices (probably an odometer), based in part on the assumption of a coordinate system in which the surface of the Earth is stationary.

I measure the "time" between my departure from Chicago and my arrival in Cleveland along a worldline corresponding to myself with a clock. Possibly a wristwatch, a dashboard clock or a cell phone.

 

[Sagittarius A-Star]

I can also measure a meter of space with time, the distance light travels in 1/299792458s. Is there an intrinsic difference between a meter of each?

Then you did this measurement of spatial distance not with a clock alone.

You can always use the combination of a clock and light as a ruler and the combination of a ruler and light as a clock.

Take as example the diagram with the rectangle in your OP. Assume, the vertical red line and blue line are worldlines of clocks, that are at rest in your reference frame, with a distance between each other of 1 meter.

How would you measure the time interval between two ticks of the red clock? I think it is obvious, that this cannot be measured with a ruler alone.

 

[student34]

That one is spacelike and one is timelike. There is no other answer. The physical fact in relativity is that these two kinds of things are fundamentally different. That's the way spacetime geometry works.

Well then I guess I hit the bottom here.

Does relativity have to go one step further and say that a postulate would be that the two must not be the same in terms of distance?

 

[jbriggs444]

Does relativity have to go one step further and say that a postulate would be that the two must not be the same in terms of distance?

What does it even mean to say that two different quantitative measures are "the same in terms of distance"? And is that not a bit self-referential when one of the measures is "distance"?

 

[student34]

Time is what clocks measure. We have well established and accurate agreement amount a large number of people and a large number of clocks. Spring clocks, pendulum clocks, crystal oscillators, water clocks, sun dials, sand clocks and atomic clocks. They all measure something. The something that they measure and agree upon is time.

So yes, there is a relationship between time and distance. But that does not make them the same thing.

In relativity, is there known to be an intrinsic relationship. For example, we know that a proton and an electron have an intrinsic relationship of mass.

 

[student34]

How would you measure the time interval between two ticks of the red clock? I think it is obvious, that this cannot be measured with a ruler alone.

I agree. But that is more of an extrinsic difference. There can be two different ways to measure the same thing.

 

[robphy]

Well then I guess I hit the bottom here.

Does relativity have to go one step further and say that a postulate would be that the two must not be the same in terms of distance?

Did you read...

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”

As I suggested earlier, in spite of your enthusiasm,
it's time to read a careful presentation of ideas (like in the above)
to guide your questioning.

In my opinion, it's difficult to field many of your questions because
you seem unaware of the big picture of how these ideas really fit together.
So, you reading a reference like that will likely help you understand the subject better.
...then ask more questions afterwards.

(An introduction to relativity in an introductory-physics textbook
is not sufficient preparation for the spacetime viewpoint. A reference like the above is.)

my $0.02

 

[student34]

What does it even mean to say that two different quantitative measures are "the same in terms of distance"? And is that not a bit self-referential when one of the measures is "distance"?

In other words, must relativity imply that 1m of space not be interchangable with 1m of time.

 

[jbriggs444]

In other words, must relativity imply that 1m of space not be interchangable with 1m of time.

If you have two events with a space-like separation, you cannot get a physical object to go from one to the other. If you have two events with a time-like separation, you can.

So the two types of separation are not interchangeable in reality. We like our models to match reality.

We usually use experiments that are less binary than "can we get there" and go for quantitative results like in the Bertozzi experiment.

 

[PeterDonis]

Does relativity have to go one step further and say that a postulate would be that the two must not be the same in terms of distance?

There is no "one step further". All of your questions are just rephrasing the same thing in different words: spacelike and timelike are different things.

 

[Nugatory]

In other words, must relativity imply that 1m of space not be interchangable with 1m of time.

It is pretty obvious that the two are not interchangeable because one is measured with a clock and the other with a meter stick. Therefore any correct theory will imply that; if you want to call this a statement about what the theory “must” do, you may.

 

[robphy]

There are more precise ways of describing what many of us are trying to tell you,
by using dot-products (which could be regarded as a way to geometrically formalize
measurements of physical quantities [modeled as geometrical objects]
with various measuring devices [modeled as certain unit vectors],
then making definitions).

The use of geometric units is done for consistency and convenience,
but one needs sufficient background understanding to see this.

In my opinion, to appreciate this viewpoint,
you need to understand the basics of spacetime geometry,
as presented in
Taylor and Wheeler's "Spacetime Physics (1st ed)" linked above.
However, I think you may benefit from
Bondi's "Relativity and Common Sense" first
because it emphasize the operational definitions of "time" and "space" coordinates
using light-rays and clocks,
and postpones the formulas and formalism (and use of geometric units) until later.

Until then, I think you are just getting caught up in the formalism
because you don't understand what the basics are (why relativity is formulated the way that it is).

Shameless plug? https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/

my $0.03

 

[PeroK]

In other words, must relativity imply that 1m of space not be interchangable with 1m of time.

Well, torque and energy have the same units (Newton-metres). They are not the same thing. One $Nm$ of torque is not interchangeable with one $Nm$ of energy.

Also, if we continue with geometric units in relativity, we have mass measured in metres as well. The mass of the Sun, for example, is about $1.5 \ km$. That's something different again from a spacelike interval of $1.5 \ km$.

 

[jbriggs444]

In relativity, is there known to be an intrinsic relationship. For example, we know that a proton and an electron have an intrinsic relationship of mass.

That is not part of relativity.

 

[PeterDonis]

we know that a proton and an electron have an intrinsic relationship of mass.

What are you referring to here?

 

[vanhees71]

In relativity, is there known to be an intrinsic relationship. For example, we know that a proton and an electron have an intrinsic relationship of mass.

I do not know, what you mean by "intrinsic relationship". Elementary particles by definition are described by realizations of irreducible representations of the proper orthochronous Poincare group in terms of local fields. This implies that each particle is classified with the corresponding qualifiers of these representations, i.e., mass (squared), $m^2 \geq 0$, and Spin $s$ (leading to $2s+1$ polarization-degrees of freedom for massive and $2$ for $s \geq 1/2$ and $1$ for $s=0$ pliarization-degrees of freedom for massless particles).

Additionally there are the charges of the gauge symmetries of the Standard Model (color for the strong and weak isospin and hypercharge (or electric charge) for the weak and electromagnetic interactions).

 

[student34]

If you have two events with a space-like separation, you cannot get a physical object to go from one to the other. If you have two events with a time-like separation, you can.

Interesting way to think about this. But that just seems to be the geometry of the object's world line rather that an intrinsic difference of the "distance".

 

[student34]

It is pretty obvious that the two are not interchangeable because one is measured with a clock and the other with a meter stick. Therefore any correct theory will imply that; if you want to call this a statement about what the theory “must” do, you may.

Can't we measure the same thing two different ways?

 

[jbriggs444]

Interesting way to think about this. But that just seems to be the geometry of the object's world line rather that an intrinsic difference of the "distance".

If all world lines have this characteristic, it becomes useful to treat it as a global property rather than a one-off observation.

On the other hand, it seems pointless to discuss further. The theory works. Shut up and calculate already.

 

[student34]

There are more precise ways of describing what many of us are trying to tell you,
by using dot-products (which could be regarded as a way to geometrically formalize
measurements of physical quantities [modeled as geometrical objects]
with various measuring devices [modeled as certain unit vectors],
then making definitions).

The use of geometric units is done for consistency and convenience,
but one needs sufficient background understanding to see this.

In my opinion, to appreciate this viewpoint,
you need to understand the basics of spacetime geometry,
as presented in
Taylor and Wheeler's "Spacetime Physics (1st ed)" linked above.
However, I think you may benefit from
Bondi's "Relativity and Common Sense" first
because it emphasize the operational definitions of "time" and "space" coordinates
using light-rays and clocks,
and postpones the formulas and formalism (and use of geometric units) until later.

Until then, I think you are just getting caught up in the formalism
because you don't understand what the basics are (why relativity is formulated the way that it is).

Shameless plug? https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/

my $0.03

Thanks, I started reading the Bondi K-calculus link

 

[student34]

Well, torque and energy have the same units (Newton-metres). They are not the same thing. One $Nm$ of torque is not interchangeable with one $Nm$ of energy.

Just a meter with no other combination seems much more specific, but I am not saying that they necessarily have to be the same thing just because they are both using the same unit.

Also, if we continue with geometric units in relativity, we have mass measured in metres as well. The mass of the Sun, for example, is about $1.5 \ km$. That's something different again from a spacelike interval of $1.5 \ km$.

This is interesting. I did not know this.

 

[student34]

What are you referring to here?

I was just giving an example of what kind of relationship I am looking for when it comes to a meter of time and a meter of distance.

 

[student34]

I do not know, what you mean by "intrinsic relationship".

In what ways are time and space related? More specifically, in what ways is a separation between two points on a timelike interval the same as the separation between two points in space?