B Question about time and its relation to spacetime

[Nadeen]

As per my intro post, I don’t consider myself to have any science background beyond high school education.

In a philosophy thread about time on City-Data Forum, where most of the entries are just general unscientific musings, I saw the following post:

“Time is a distance in spacetime: v =x/t therefore t = x/v- ie- time is proportionally equivalent to 3D distance. Time is relative to speed also, hence Einstein's Special Theory about time & distance changing with speed.”

Where anyone makes categorical statements and accompanies them by a formula, I tend to check the validity of the post. In this case, the statement completely baffled me.

First, assuming that x refers to the position in space, is the formula correct? Should it not refer to differences in values and as such use the delta notation for each variable?

Second, does this formula apply to Special Relativity?

Third, Einstein is often quoted as having said that “time has no independent existence apart from the order of events by which we measure it.” If that is the case, is it correct to say that time is a property of spacetime? I learned from reading the Wiki article on Spacetime intervals, that depending on the value of spacetime, it can exhibit “timelike” property but that doesn’t mean that time is a distinct property of spacetime, just how we perceive it? Is that a correct assessment?

Fourth, if “time is the order of events by which we measure it”, is it appropriate to refer to distance? Should it not be referred to as spacetime interval between the events or is it the same thing? In which case could speed/velocity (I'm not sure which one is the correct term here) be independent of time and instead somehow be related to the difference between spacetime intervals of different events? (I’m not sure if how I word this makes sense.)

I am so sorry if none of the above makes any sense but I have thick skin so can handle criticism, and if you can direct me to other resources to possibly expand my understanding on the subject, I’d be much obliged.

 

[jbriggs444]

“Time is a distance in spacetime: v =x/t therefore t = x/v- ie- time is proportionally equivalent to 3D distance. Time is relative to speed also, hence Einstein's Special Theory about time & distance changing with speed.”

That statement looks to me like pure sophistry with no meaningful content.

Yes, it takes less time to drive from Washington, DC to New York city at 60 mph than at 30 mph. But so what?

If that is the case, is it correct to say that time is a property of spacetime? I learned from reading the Wiki article on Spacetime intervals, that depending on the value of spacetime, it can exhibit “timelike” property but that doesn’t mean that time is a distinct property of spacetime, just how we perceive it? Is that a correct assessment?

Given two events with a timelike separation, that separation is measurable and invariant. That is to say that it does not matter what frame of reference we use to make the measurement. The resulting measured time interval will be the same regardless. That quantity will be equal to the time that a properly running clock that drifts in free fall from the one event to the other will measure. [In special relativity this is unequivocally true. In general relativity there can sometimes be multiple free fall paths from one event to the other and, potentially, different elapsed times]

To put it briefly: Time is what a clock measures.

One can carve up space time into slices according to an agreed upon standard of simultaneity and judge "time" based on the slices: This slice is time 0, this slice is time 1, etc. This sort of time is not an invariant property. It depends on one's choice of simultaneity convention.

 

[Nadeen]

That statement looks to me like pure sophistry with no meaningful content.

Thank you, jbriggs444. I suspected as much but was afraid that in my own ignorance I may have missed something.

As to the main body of your response, may I ask... If the timelike separation between the two events is measurable and invariant, how does the time dilation fit into this picture? Or does it only apply to the relativity of simultaneity?

 

[jbriggs444]

Thank you, jbriggs444. I suspected as much but was afraid that in my own ignorance I may have missed something.

As to the main body of your response, may I ask... If the timelike separation between the two events is measurable and invariant, how does the time dilation fit into this picture? Or does it only apply to the relativity of simultaneity?

Just so. Time dilation is a coordinate effect. If I construct my planes of simultaneity so that clocks at "rest" tick off one second per coordinate second and if you construct yours similarly, our hyper-planes will be at an angle to one another. I will count your clocks (which I consider to be moving) to be slow. You will count my clocks (which you consider to be moving) also to be slow. There is no invariant fact of the matter. We are both right in our individual coordinate-relative sense.

But we will both agree on the interval between two identified events. We may ascribe different spatial separation and different temporal separation to the two events, but we will both get the same result if we take the difference between the two squared separations.

 

[Nadeen]

Thank you! This is the first time I've seen it described this way, really helpful visualisation. I don't pretend to having wrapped my head around it all but your examples made it so much more comprehensible.

 

[jbriggs444]

Thank you! This is the first time I've seen it described this way, really helpful visualisation. I don't pretend to having wrapped my head around it all but your examples made it so much more comprehensible.

This is the 'gist of Minkowski geometry. It is just like ordinary Euclidean geometry except that when you lay out Cartesian coordinates on it, instead of the "square root of sum of squares" metric (Pythagorean theorem), you get a "square root of difference of squares" metric -- sometimes known as a (+,+,+,-) signature.

 

[Nadeen]

This is the 'gist of Minkowski geometry. It is just like ordinary Euclidean geometry except that when you lay out Cartesian coordinates on it, instead of the "square root of sum of squares" metric (Pythagorean theorem), you get a "square root of difference of squares" metric -- sometimes known as a (+,+,+,-) signature.

I've actually come across Minkowski geometry for the first time today when I was reading up on the subject. Will definitely look more and see if I can find some lectures on YT later.

The example of the signature that you have given, is that where the time coordinate is placed last? The one I saw on Wiki was in the form (- + + +).

 

[jbriggs444]

I've actually come across Minkowski geometry for the first time today when I was reading up on the subject. Will definitely look more and see if I can find some lectures on YT later.

The example of the signature that you have given, is that where the time coordinate is placed last? The one I saw on Wiki was in the form (- + + +).

Yah. Six of one, half dozen of the other.

 

[Nadeen]

Yah. Six of one, half dozen of the other.

:) JB, I hope you don't mind me calling you JB, you, and all the others who welcomed me and for some astonishing reason liked my post, have made this old girl's day! Thank you.

 

[technosaurus]

Hello Nadeen,
This is an area that fascinates me, so your post caught my eye. To deal with the formula you quoted first: "v =x/t therefore t = x/v".
v represents velocity (i.e. speed in a given direction), x represents displacement, and t represents time.
This is more or less the same formula as for speed (speed = distance over time as opposed to velocity = displacement over time) except that we're talking about something that has direction (i.e. a vector quantity) as opposed to just magnitude (a scalar quantity).
The last bit "therefore t = x/v" is just stating the obvious - it's an algebraic re-arrangement of the original equation. I can only surmise that he/she was trying to make the point that time cannot exist independently of space, which is a conclusion you appear to have come to yourself. The statement taken as a whole, however, serves to confuse rather than clarify. A significantly more in-depth explanation is necessary.

So to answer your first question, yes the formula is correct, at least in non-relativistic terms. Which raises the first point - most of the formulae found in classical physics, of which this is one, are perfectly adequate to describe and explain most observable physical phenomena. Once we start encountering speeds that are a significant fraction of the speed of light, however, this is no longer the case. Remember, though, we are talking about very high speeds - light covers 300,000 kilometres in one second. The fastest speed at which human beings have traveled (relative to Earth) is just over 11 (eleven) km per second, achieved by the crew of the Apollo 10 space mission.

To answer your second question, no the formula as it stands does not apply to special relativity.

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