Questions concerning the geometry of spacetime

[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?

 

[student34]

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

I meant that that would not seem to necessarily mean that the two "distances" were different .

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

If GR were the theory of everything and had no paradoxes (like the grandfather paradox) I would agree.

 

[PeterDonis]

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.

That doesn't help, because I don't know what kind of relationship you are even talking about when you say:

a proton and an electron have an intrinsic relationship of mass.

What kind of relationship are you talking about here? There is no such relationship in physics that I am aware of.

 

[PeterDonis]

If GR were the theory of everything and had no paradoxes (like the grandfather paradox)

What are you talking about here? There is no "grandfather paradox" in GR. All solutions in GR are self-consistent.

I think you have read way too much pop science and not enough actual science.

 

[PeterDonis]

in what ways is a separation between two points on a timelike interval the same as the separation between two points in space?

There aren't any. Timelike intervals and spacelike intervals are fundamentally different. No matter how many times you try to ask about this in different words, the answer is not going to change. Why is this such a problem?

 

[student34]

That doesn't help, because I don't know what kind of relationship you are even talking about when you say:What kind of relationship are you talking about here? There is no such relationship in physics that I am aware of.

They both have mass. That is their intrinsic relationship. Or more specifically, they both have that as an intrinsic property.