Is time a true variable in the scheme of things?

[cyboman]

This is a fascinating discussion. And much of it is over my head.

Perhaps, considering the fact that C is constant could be useful? Such that, it isn't so much that time is faster or slower. It's that when measuring velocity C is always C, so time and distance must change in order to account for the different states of frames. This is only noticeable as you approach C.

I think what the OP may be confusing, and I think this is suggested, is that it is not the passage of time that changes. As said, it always ticks. Think the film, Back to the Future (A personal fav), for the dog, nothing changed in his frame, the clock ticked the same. It's the difference between the two frames and the fact the velocity C is constant. So the others parameters of distance and time must dilate / contract.

Don't know if that is at all correct. I'm channeling my understanding of special and general I read from many years ago.

 

[cyboman]

Whatever the process that controls the length of the line, the straight line is always the shortest.

Right, I hope I'm not complicating the analogy by stating this: Note that the shortest path in space time is a curved line not a straight one. Since mass bends space-time.

 

[PeroK]

Right, I hope I'm not complicating the analogy by stating this: Note that the shortest path in space time is a curved line not a straight one. Since mass bends space-time.

In fact, in flat spacetime the straight line is the longest path between two points.

 

[cyboman]

In fact, in flat spacetime the straight line is the longest path between two points.

From what I understand. Spacetime is not flat.

 

[PeroK]

From what I understand. Spacetime is not flat.

It's locally flat and for many applications, e.g. high energy particle physics, only SR is required. The curved spacetime of GR is relevant to the solar system and galaxies etc.

In any case, in both flat and curved spacetime particles travel on paths, whether straight or not, that maximise the spacetime distance travelled.

 

[cyboman]

It's locally flat and for many applications, e.g. high energy particle physics, only SR is required. The curved spacetime of GR is relevant to the solar system and galaxies etc.

In any case, in both flat and curved spacetime particles travel on paths, whether straight or not, that maximise the spacetime distance travelled.

Right, but from my understanding, if you're ever think space-time is flat, it's because that perception has mathematical or visual cognitive advantages for thinking of it that way. In truth, it is curved.

Perhaps, it's only usefully considered flat because you are "zoomed" so far in.

 

[cyboman]

It's locally flat and for many applications, e.g. high energy particle physics, only SR is required. The curved spacetime of GR is relevant to the solar system and galaxies etc.

In any case, in both flat and curved spacetime particles travel on paths, whether straight or not, that maximise the spacetime distance travelled.

My intuition is that they are never traveling on straight paths. That may seem so in a localized frame. But ultimately, any particle traveling will follow a non-euclidean arc along curved space-time.

 

[weirdoguy]

Right, but from my understanding, if you're ever think space-time is flat, it's because that perception has mathematical or visual cognitive advantages for thinking of it that way. In truth, it is curved

No. Specetime of special relativity is flat in the mathematical sense, i.e. its curvature tensor vanishes globally.

 

[cyboman]

No. Specetime of special relativity is flat in the mathematical sense, i.e. its curvature tensor vanishes globally.

This is perhaps, over my head mathematically. But I would contend, the mathematics are not euclidean. Or flat. Einstein had to produce his own mathematics to deal with this space.

From an astronomical viewpoint, mass effects space-time, and the bodies that interact with that gravity / forces follow arcs, not straight lines. Because the space itself, is curved.

 

[PeroK]

My intuition is that they are never traveling on straight paths. That may seem so in a localized frame. But ultimately, any particle traveling will follow a non-euclidean arc along curved space-time.

Nature is under no obligation to follow your intuition.

If by non-Euclidean arc you mean a geodesic of the curved spacetime then, ironically, many people use that as the generalized definition of a straight line!

 

[cyboman]

[NQUOTE="cyboman, post: 6171798, member: 470031"]
My intuition is that they are never traveling on straight paths. That may seem so in a localized frame. But ultimately, any particle traveling will follow a non-euclidean arc along curved space-time.

Nature is under no obligation to follow your intuition.

If by non-Euclidean arc you mean a geodesic of the curved spacetime then, ironically, many people use that as the generalized definition of a straight line!
[/QUOTE]

OK, but that's not intrinsically, a straight line. It's actually curved. Mathematically, locally, straight perhaps, but to qualify it that way would not be accurate. It is perhaps, locally straight, but ultimately curved. Those of us that are not in the deep algebra of it all need to understand that difference.

 

[PeroK]

From an astronomical viewpoint, mass effects space-time, and the bodies that interact with that gravity / forces follow arcs, not straight lines. Because the space itself, is curved.

Before Einstein it was clear that the Earth traveled in a curved (almost circular) orbit through space around the sun.

Ironically, Einstein found that, in a way, the Earth follows a straight line through curved spacetime.

 

[PeroK]

From an astronomical viewpoint, mass effects space-time, and the bodies that interact with that gravity / forces follow arcs, not straight lines. Because the space itself, is curved.

Before Einstein it was clear that the Earth traveled in a curved (almost circular) orbit through space around the sun.

Ironically, Einstein found that, in a way, the Earth follows a straight line through curved spacetime.

 

[cyboman]

Before Einstein it was clear that the Earth traveled in a curved (almost circular) orbit through space around the sun.

Ironically, Einstein found that, in a way, the Earth follows a straight line through curved spacetime.

I disagree, it's semantic I think. But the path is not straight unless you are taking into account the space being curved. And for most analysis that would look like an arc not a straight line. But I admit, perhaps you understand it more than me and to you it looks straight. I simply don't see it that way. I see your perspective as a localized perspective.

I think it may be a mathematical visualization vs an intuitive one. We could argue forever which is more correct. I think perhaps they both are.

And it is a matter of relativity.

 

[cyboman]

Before Einstein it was clear that the Earth traveled in a curved (almost circular) orbit through space around the sun.

Ironically, Einstein found that, in a way, the Earth follows a straight line through curved spacetime.

A straight line through curved space time is an arc.

 

[PeroK]

From an astronomical viewpoint, mass effects space-time, and the bodies that interact with that gravity / forces follow arcs, not straight lines. Because the space itself, is curved.

Before Einstein it was clear that the Earth traveled in a curved (almost circular) orbit through space around the sun.

Ironically, Einstein found that, in a way, the Earth follows a straight line through curved spacetime.

 

[PeroK]

A straight line through curved space time is an arc.

You ought to research the term "geodesic".

In any case, the geodesics of spacetime, which are the paths that particles and planets naturally take, are paths of maximal spacetime distance.

 

[cyboman]

You ought to research the term "geodesic".

In any case, the geodesics of spacetime, which are the paths that particles and planets naturally take, are paths of maximal spacetime distance.

So from an objective frame, does the particle follow a straight line, or does it follow a geodesic arc along space-time? What is your frame of reference?

 

[PeroK]

So from an objective frame, does the particle follow a straight line, or does it follow a geodesic arc along space-time? What is your frame of reference?

Geodesics are independent of frame of reference.
As I said, many people consider a geodesic as the definition of a straight line.

Personally, I reserve straight line for Euclidean geometry and simply use geodesic.

But, there is no other possible definition of a straight line in curved spacetime. It's either a geodesic or left undefined. An arc is likewise an undefined term.

One problem with the question is that descriptions like straight line and arc depend on your coordinates. Unless you give them some coordinate free description, like shortest distance between two points, or maximal spacetime distance between two points.

You can define a straight line in these ways in classical mechanics and SR. And you can extend the latter definition to curved spacetime if you wish. But unless you do that straight line has no meaning in curved spacetime.

 

[cyboman]

Geodesics are independent of frame of reference.
As I said, many people consider a geodesic as the definition of a straight line.

Personally, I reserve straight line for Euclidean geometry and simply use geodesic.

But, there is no other possible definition of a straight line in curved spacetime. It's either a geodesic or left undefined. An arc is likewise an undefined term.

One problem with the question is that descriptions like straight line and arc depend on your coordinates. Unless you give them some coordinate free description, like shortest distance between two points, or maximal spacetime distance between two points.

You can define a straight line in these ways in classical mechanics and SR. And you can extend the latter definition to curved spacetime if you wish. But unless you do that straight line has no meaning in curved spacetime.

Fascinating.

I guess I was thinking that the conventional Newtonian idea of a straight line, in a curved space-time is actually an arc. This is what I come to understand of SR. The shortest distant turns out to be a curve, not a line, because the space itself is bent and non-euclidean due to gravity.

 

[PeroK]

Fascinating.

I guess I was thinking that the conventional Newtonian idea of a straight line, in a curved space-time is actually an arc. This is what I come to understand of SR. The shortest distant turns out to be a curve, not a line, because the space itself is bent and non-euclidean due to gravity.

If you take an example from Newtonian physics. A ball falls straight down under gravity. That is spatially a straight line in the Earth's reference frame. But if you plot height against time, then as the ball is accelerating it's path in Newtonian spacetime is curved.

Whereas a ball moving at constant velocity would follow a straight line through Newtonian spacetime.

This is also true in the flat spacetime of SR.

But, in GR the ball falling under gravity is not accelerating. In the sense that it feels no force and has no intrinsic or "proper" acceleration.

Whereas the ball rolling along a table does feel an upward force from the table, so is accelerating.

The situation in GR is somewhat reversed. And, it is not just semantics to say that the ball in free fall - or a planet in orbit - is following a natural, geodesic "straight" path through spacetime. And the ball rolling at constant velocity is not following a geodesic path but being accelerated in a "curved" path.

 

[Mister T]

But I would contend, the mathematics are not euclidean. Or flat.

The spacetime of special relativity is flat, but it is not Euclidean.

 

[Mister T]

Note that the shortest path in space time is a curved line not a straight one.

There is no shortest path through spacetime for objects with mass. However short a path you choose, a shorter one can always be found.

Just as, on that flat sheet of paper I was talking about, there is no longest path between the two dots. However long a path you choose, a longer one can always be found.

 

[cyboman]

Thanks for the examples!

That is spatially a straight line in the Earth's reference frame.

But isn't it true that that's because it locally only appears that way. Because you are in effect zoomed in so far. It looks like it's straight, but if you go far enough out it's actually happening in curved space-time. So like everything else, the ball exists in curved space-time due to the mass of the Earth.

This is also true in the flat spacetime of SR.

But when you say flat spacetime, are you not assuming there is no gravity. Empty spacetime is flat where no masses are curving it.

And the ball rolling at constant velocity is not following a geodesic path but being accelerated in a "curved" path.

Is a geodesic path not curved? Zoomed in locally it appears straight. It's as close to straight as you can get, but it's still a curve. So, there are no straight lines in curved space-time. Isn't that correct? If there is no Earth then the space-time is flat and you could say there are straight lines right.

 

[cyboman]

The spacetime of special relativity is flat, but it is not Euclidean.

It's only flat if there is no masses curving it though right? In the context here the Earth is curving space-time.

I'm wondering, could one argue that everywhere in the universe space-time has some curvature?

 

[cyboman]

There is no shortest path through spacetime for objects with mass. However short a path you choose, a shorter one can always be found.

Just as, on that flat sheet of paper I was talking about, there is no longest path between the two dots. However long a path you choose, a longer one can always be found.

I'm lost on this one. I thought that an object moves along the shortest path between two points in space-time.

 

[weirdoguy]

No, as been already pointed out:

In fact, in flat spacetime the straight line is the longest path between two points.

In fact, from the mathematical point of view, the path has to extremize certain functional, which means it's not always minimal, but can be maximal.

 

[jbriggs444]

I'm lost on this one. I thought that an object moves along the shortest path between two points in space-time.

Most usually, the longest path. That is, the longest elapsed proper time from starting event to ending event.

 

[cyboman]

Most usually, the longest path. That is, the longest elapsed proper time from starting event to ending event.

I think this notion is beyond my capacity to understand. Reminds me of the time cone stuff I remember reading in A Brief History of Time. But I think I need to reread that and some of the explanations you all gave a few hundred times more to really get a handle. Fascinating stuff though!

Thanks for your explanations. Don't want to hijack the thread (hope I haven't).

 

[Eric Boudreau]

Let me give an example. Say we had a means of generating some "field" that caused something within that field to experience a different rate of elapsed time. To be able to do in the lab what currently requires a trip in space at a high speed or in a stronger gravitational field. We can not do this but the concept is valid as we know time can be manipulated. This is accepted.

Now, my question is there any way to speed up time for an object? What would that do for us? Well if you could control this locally it would make for faster computers. If a computer had to run for a week now to get a result if the computer was in a room where time was accellerated it might do a weeks calculation in a second rather than a week. The concept of a slower clock for our near light speed trip is accepted.

Is there any physics rule that would prohibit us from finding a way to speed up time? If time can be slowed, why not made faster?

I think the issue is obervational. Frame of reference is what gives rise to the differentiation between two time measuring devices. A discreet measurement of a system gives the properties of the system at the moment of observation. Any difference in dimensional properties would require a second observation. Time is always 1 or 0. The passage of time relative to the observer is the difference between the first observation and second observation of the system.

 

[cyboman]

If you take an example from Newtonian physics. A ball falls straight down under gravity. That is spatially a straight line in the Earth's reference frame. But if you plot height against time, then as the ball is accelerating it's path in Newtonian spacetime is curved.

Whereas a ball moving at constant velocity would follow a straight line through Newtonian spacetime.

This is also true in the flat spacetime of SR.

But, in GR the ball falling under gravity is not accelerating. In the sense that it feels no force and has no intrinsic or "proper" acceleration.

Whereas the ball rolling along a table does feel an upward force from the table, so is accelerating.

The situation in GR is somewhat reversed. And, it is not just semantics to say that the ball in free fall - or a planet in orbit - is following a natural, geodesic "straight" path through spacetime. And the ball rolling at constant velocity is not following a geodesic path but being accelerated in a "curved" path.

I was rereading this thread and it's really fascinating. I wanted to apologize for the self righteous way I came off in my discussion with you PeroK. Not sure where my head was at. I mean I'm disagreeing with your perspective on something you know way way more about than me. That's a bit arrogant. I'm reading this thread and I'm sounding like a mouse arguing with an owl about the techniques for detecting small prey from far above and ideal flight approaches for capturing said prey.

Anyway, just wanted to say thanks for entertaining my ramblings and having such patience with a laymen. Your explanations were succinct and thought provoking.

 

[phinds]

I think that those of you who are insisting that things in freefall in space follow curved lines are making the mistake of applying Euclidean Geometry in a domain where it is not relevant. Yes, in Euclidean Geometry those paths ARE curved, but so what? That is utterly irrelevant because the geometry of spacetime is not Euclidean, it is pseudo-Riemannian and in that geometry the paths are geodesics, which as has already been pointed out are considered by many to be a logical generalization of "straight line".

 

[jbriggs444]

I think that those of you who are insisting that things in freefall in space follow curved lines are making the mistake of applying Euclidean Geometry in a domain where it is not relevant. Yes, in Euclidean Geometry those paths ARE curved, but so what? That is utterly irrelevant because the geometry of spacetime is not Euclidean, it is pseudo-Riemannian and in that geometry the paths are geodesics, which as has already been pointed out are considered by many to be a logical generalization of "straight line".

Technically, I do not think this is quite the right distinction to make.

It is not the difference between Euclidean and Pseudo-Riemannian that matters. It is the difference between Euclidean and non-Euclidean. Consider special relativity (SR) for a moment. In the flat Minkowski space of SR, geodesics are straight lines. Yet this is a pseudo-Riemannian geometry.

As I know you understand already, it is not that geodesics are curved. It is the space within which they exist that is curved.

It is only when we apply a non-default metric to the space in question that the notion of curvature becomes a meaningful concept. This is perhaps more easily seen if we go back to a two dimensional analogy in a space that is not pseudo-Riemannian -- paper maps of the surface of the Earth.

If we have a flat map of the surface of the Earth this will necessarily be some sort of projection. For instance a Mercator projection. If we look at lines on the map corresponding to straight roads on the Earth, some of those lines will be curved. They will be [sections of] great circle arcs. This is reflected in the distance metric.

This "metric" amounts to a big table of distances. For any pair of points on the Earth's surface metric would tell you how far it is [along the surface] from point A to point B. For instance, 800 miles from New York to Chicago.

If you translate this metric and present it in terms of map coordinates (for instance in terms of latitude and longitude if you are using a Mercator projection) then you will find that it does not match the standard Euclidean metric. The Euclidean metric would be, for instance:$$D = 60 * \sqrt{\Delta\text{ lat}^2+\Delta\text{long}^2}$$where D is in nautical miles and lat and long are in degrees. As should be obvious, this metric matches distances measured with ruler on a flat paper map but does not match distances measured with an odometer on a real earth.

Locally on this flat paper map we will almost always be able to find a metric which fits the Euclidean form ($D^2 = \Delta x^2 + \Delta y^2$) and locally approximates the true metric. We may have to put the local x and y axes at an angle. And we may have to scale them by a constant factor, but we can still obtain something Euclidean-looking. In the case of the Mercator projection we won't have to mess with the angles of the local x and y axes. With some other projections we might need to.

[It is "almost always" because you can get coordinate singularities. In the case of the Mercator projection you get a coordinate singularity at the North and South poles. Other projections tear or irreparably stretch the map in other places]

It is a similar situation when comparing the flat space of special relativity with the curved space of general relativity. Locally you can almost always fix things up so that the Minkowski metric ($D^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - \Delta t^2$) is approximated well. One may have to scale the axes and put them at odd angles, but you can still recover something Minkowski-looking.

[Again, you can get coordinate singularities -- for instance at the event horizon of a black hole when using Schwarzschild coordinates. This is in addition to true singularities such as at the "center" of a Schwarzschild black hole]

It is possible to apply the metric (as presented in terms of map coordinates) to figure out how to extend a line on the map so that it corresponds to a straight great circle arc on the surface of the Earth. Of course, this line will not be straight on the map. But it will be straight on the surface of the Earth.

Perceived geodesic curvature is all about the projection, not about the geometry.

Note: I have never taken a course covering differential geometry. Pretty much everything think I know has been absorbed over the years here.

 

[cyboman]

As I know you understand already, it is not that geodesics are curved. It is the space within which they exist that is curved.

If we have a flat map of the surface of the Earth this will necessarily be some sort of projection. For instance a Mercator projection. If we look at lines on the map corresponding to straight roads on the Earth, some of those lines will be curved. They will be [sections of] great circle arcs. This is reflected in the distance metric.

Much of this math is over my head but very fascinating nonetheless. I do like the analogy of the Earth and a map as a projection within that curved space. To us on the ground the road appears straight. But if we keep walking in that straight line, eventually we end up where we started. This reminds me of a mathematical concept I read in a fascinating book many years back. It talked about the notion of a finite but unbounded universe.

 

[jbriggs444]

This reminds me of a mathematical concept I read in a fascinating book many years back.

The one where I was first introduced to the notion of curved space was "Sphereland", I think. Possibly the same one that you are alluding to.

It is difficult to properly capture the idea of intrinsic curvature. As I recall, that book did a credible job of doing so in an entertaining and understandable manner.