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What is movement?


humility

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I was just wondering, what exactly is movement?  Like if we are all part of space, then what exactly is our movement within space? We can't move relative to space obviously, but what exactly is happening to space as we move?  Wow this is hard to explain. I have this weird incomplete picture in my head of space like a thick web of strings and when we move its like the mass of strings moves with us but pulls and sways on other masses of strings from other objects. Why strings instead of water I don't know, probably just my brain trying to picture einstein's sheet metaphor in 3d.

Point is I don't know how we move through space. I'm not even sure if space is an actual thing or just a word used as a placeholder.

Well maybe it'll be easier to explain if I explain why I started wondering. So I suddenly recalled that I read somewhere that time is literally finite and will eventually run out, but we won't notice as its running out will be a slowing and slowing down of time until the entire universe is frozen in a single moment.

 

So I thought huh, if time is a finite resource then that would make space finite as well.

 

And I thought if something is finite than it can probably be isolated. And I thought what would happen if you isolated a bit of space from all the other space. Would it vanish from perception, would say you isolated a person's space, would they have a period of time to live equal to the amount of space they take up or would they instantly freeze?

 

Then I thought well obviously two isolated bits of space can't move since space itself can't move otherwise space would be something you could move relative to. But then I thought but what actually happens to the space in a different bit when you move around, maybe you are moving relative to space when you move relative to an object. But then I thought wait but space and objects aren't the same thing. Objects are just in space. But then I remembered time  and space are the same thing and we are clearly a part of time. And then I realized I just didn't have a clue.

Edited by humility
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2 hours ago, humility said:

So I suddenly recalled that I read somewhere that time is literally finite and will eventually run out, but we won't notice as its running out will be a slowing and slowing down of time until the entire universe is frozen in a single moment.

No idea where you have read that time is finite. It seems nonsensical to me. So if you are worrying about it in your real life, just don't. The second part of the idea is interesting. If (huge 'if'!) time could come to a halt, would we notice? Well, I think special relativity gives the answer. If e.g. a spaceship is accelerating and nearing the speed of light from me, I would see that everything on board is slowing down more and more. For the inhabitants of the spaceship however nothing changes. For themselves, they keep their acceleration (which is nice, because they have artificial gravity, they can stand on the ground), and time on board goes on just as the are used to (but not if they look out of the window...). So I think, indeed, nobody would notice time slowing down, simply because time as we experience it is always local: everything moving along with us has the same time.

2 hours ago, humility said:

Like if we are all part of space, then what exactly is our movement within space?

I would say that space 'sec' does not exist, so you are right, we can't formally define movement as a change in position in space. But there are very practical ways to 'define' space. Take two objects in otherwise empty space: at least we can define a distance between these objects. And if this distance changes, we say the objects move in relation to each other. Now you can more and more objects, measure their distances and directions, and you get a kind of coordinate system that can be called 'space'. If most objects move slowly relative to each other the impression of the existence of space in itself becomes very strong. And then of course also the impression of (fast) movements relative to this 'space'.

2 hours ago, humility said:

I have this weird incomplete picture in my head of space like a thick web of strings and when we move its like the mass of strings moves with us but pulls and sways on other masses of strings from other objects.

Yes, you should get rid of this image. What you could do is just see them what they, in most pictures of 'space', really are: just coordinates: means of defining positions, nothing more, like meridians on the earth's surface. 

2 hours ago, humility said:

But then I remembered time  and space are the same thing and we are clearly a part of time

Time and space are definitely not the same. While the 3 space dimensions are more or less equivalent (you can just change the directions of your coordinate system and what was 'left' before is now 'in front', ie. just turn your body 90o ), you can't do this with time. If you know your Pythagoras:

The normal, space distance between two objects in a 3-dimensional coordinate system is:

distance = x2 + y2 + z2

Special relativity, when defining distances in spacetime, takes time into it (but multiplies it with c, the velocity of light, so it also becomes 'space-like') however, with a minus sign.

spacetime distance = x2 + y2 + z2 - (ct)2

Another (well I assume it is another) aspect of time that there is only one possible direction: we cannot go back in time. We always experience progression of time. Again, as said above, because everything moving along with us, has the same time. No way that the broken egg on the floor makes a travel back in time, and 'unbreaks' and nicely lands on your breakfast table.

 

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3 hours ago, humility said:

I was just wondering, what exactly is movement?

A geometric relationship between reference frames in spacetime. In the simplest case of inertial motion, two reference frames being in relative motion quite simply means that they are rotated by a (hyperbolic) angle wrt each other in spacetime. Motion is purely a geometric phenomenon.

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3 hours ago, Scott Mayers said:

Your questions about space are a good start. Many scientists today don't like philosophy but is what is needed more now than ever if it is to progress beyond the limits of institutional authoritarianism. It is also needed if one is to internalize what you learn rather than restricting it to the 'faith' in those claiming authority wherever they exist.

!

Moderator Note

This is not the place to grind an axe, nor the place to post non-mainstream science responses. Discussion of an aether, except as an historical reference, is completely off-topic.

edit: upon further review, this was a massive hijack of the discussion, and the whole mess has been dropped in the trash can.

 
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Thank you swansont, +1

 

Now we are back on topic I can offer my thoughts on humility's question "What is movement?"

 

We say that if we can establish a comparative scale, possibly but not necessarily numeric, for some parameter or colllection of parameters that a change occurs when repetive observations of the system under observation return different scale values, after taking due allowance for observational error and variation.

Movement is the change of a restricted set of these parameters. That is it is a subset of change.

It is then necessary to examine more closely what we will allow as acceptable parameters.
This introduces modifying words or phrases to cotrol this.

Mechanical movement (which I think hunmiltity means) restricts the parameter to a position in space applied to material objects.

But we should consider other uses of movement for instance of immaterial objects eg the movement of heat in a bar or the movement of the stock exchange (ie a graph).

Edited by studiot
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4 hours ago, Eise said:

While the 3 space dimensions are more or less equivalent (you can just change the directions of your coordinate system and what was 'left' before is now 'in front', ie. just turn your body 90o ), you can't do this with time. If you know your Pythagoras:

The normal, space distance between two objects in a 3-dimensional coordinate system is:

distance = x2 + y2 + z2

Did you mean rather:

distance^2 =  x^2 + y^2 + z^2

(because Pythagoras equation is a^2+b^2=c^2)

therefor:

distance = sqrt( x^2 + y^2 + z^2 )

 

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3 minutes ago, Sensei said:

Did you mean rather:

distance^2 =  x^2 + y^2 + z^2

(because Pythagoras equation is a^2+b^2=c^2)

therefor:

distance = sqrt( x^2 + y^2 + z^2 )

 

Ups... Yes, of course. Slip of the mind...

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What does 'sec' mean? I assume you dont mean Secant and its not in the dictionary except as meaning very dry. 

On 10/15/2018 at 1:54 AM, Eise said:

.

I would say that space 'sec' does not exist, so you are right, we can't formally define movement as a change in position in space. But there are very practical ways to 'define' space. Take two objects in otherwise empty space: at least we can define a distance between these objects. And if this distance changes, we say the objects move in relation to each other. Now you can more and more objects, measure their distances and directions, and you get a kind of coordinate system that can be called 'space'. If most objects move slowly relative to each other the impression of the existence of space in itself becomes very strong. And then of course also the impression of (fast) movements relative to 

Time and space are definitely not the same. 

 

Wait... wait... so if the 'space' used to measure position and movement and time is not an actual thing, but just a placeholder so you can drop numbers into to measure where objects are in relation to each other.

 

Then that would mean that the space that gets curved by gravity, is completely unrelated to the space we move through?

 

Or that saying space curves was a misleading choice of words just to describe the change in relationship gravity had on objects moving.

 

In which case, if the sun's gravity wasnt curving space, how exactly did the light from stars behind our sun be visible even though they should have been blocked?

Should I create a different thread for that?

 

On 10/15/2018 at 2:11 AM, Markus Hanke said:

A geometric relationship between reference frames in spacetime. In the simplest case of inertial motion, two reference frames being in relative motion quite simply means that they are rotated by a (hyperbolic) angle wrt each other in spacetime. Motion is purely a geometric phenomenon.

Huh, that explanation creates a new and even wierder mental image in my head, where objects that are moving, arent actually in motion, but its more like space itself is moving them in those byperbola arcs in lazy circles around each other.

On 10/15/2018 at 6:21 AM, studiot said:

Thank you swansont, +1

 

Now we are back on topic I can offer my thoughts on humility's question "What is movement?"

 

We say that if we can establish a comparative scale, possibly but not necessarily numeric, for some parameter or colllection of parameters that a change occurs when repetive observations of the system under observation return different scale values, after taking due allowance for observational error and variation.

Movement is the change of a restricted set of these parameters. That is it is a subset of change.

It is then necessary to examine more closely what we will allow as acceptable parameters.
This introduces modifying words or phrases to cotrol this.

Mechanical movement (which I think hunmiltity means) restricts the parameter to a position in space applied to material objects.

But we should consider other uses of movement for instance of immaterial objects eg the movement of heat in a bar or the movement of the stock exchange (ie a graph).

Isnt that what Markus said? Am I missing something?

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17 minutes ago, humility said:

Isnt that what Markus said? Am I missing something?

No

 

Since you haven't given me a clue as to what you understand by I said, I don't have a clue and can't elaborate further.

 

So it is up to you?

Edited by studiot
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10 minutes ago, humility said:

Wait... wait... so if the 'space' used to measure position and movement and time is not an actual thing, but just a placeholder so you can drop numbers into to measure where objects are in relation to each other.

Then that would mean that the space that gets curved by gravity, is completely unrelated to the space we move through?

They are the same thing. When you walk 3 metres from your chair to the door, that distance is not an "actual thing" (depending, of course, on what you mean by "actual thing"). And that distance is exactly that is being measured and used in the description of motion etc.

12 minutes ago, humility said:

Or that saying space curves was a misleading choice of words just to describe the change in relationship gravity had on objects moving.

I don't think it is misleading. Although it might be more accurate to say that it is the geometry that relates our measurements that changes. But as those measurements are what define "space" and "time" (and their relationship) it is pretty much the same as saying that space-time is curved.

14 minutes ago, humility said:

In which case, if the sun's gravity wasnt curving space, how exactly did the light from stars behind our sun be visible even though they should have been blocked?

It is mass that curves space and time; one of the effects of that is the gravitational lensing of light around the Sun. Another effect is the apparent force that we call "gravity".

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6 hours ago, studiot said:

No

 

Since you haven't given me a clue as to what you understand by I said, I don't have a clue and can't elaborate further.

 

So it is up to you?

As I understood it, You described how movement is measured then went on towards linguistical definitions of the word.

 

6 hours ago, Strange said:

They are the same thing. When you walk 3 metres from your chair to the door, that distance is not an "actual thing" (depending, of course, on what you mean by "actual thing"). And that distance is exactly that is being measured and used in the description of motion etc.

I don't think it is misleading. Although it might be more accurate to say that it is the geometry that relates our measurements that changes. But as those measurements are what define "space" and "time" (and their relationship) it is pretty much the same as saying that space-time is curved.

It is mass that curves space and time; one of the effects of that is the gravitational lensing of light around the Sun. Another effect is the apparent force that we call "gravity".

By actual thing, I meant sometjing that can be influenced by external factors.  You know, if you have a big block its a thing because you can push and pull it and do stuff with it. But if you draw a picture of a big block, then that isnt a thing. The paper you drew on is a thing, the paint is a thing, but the apparent block, although you can measure its color and size, isnt a thing. No more than a cloud that looks like a marshmellow is actually anything more than a cloud.

 

Ive always thought of gravity as a thing since it could be warped and affected. And it seems like you are saying spacetime isnt real. Its just a conviently related set of equations that make nice pictures. 

 

At least this explains gravitons, I did wonder how gravity could be caused by space curving and gravitons. But if space doesnt curve then that means its all just gravitons curving the path of the objects. Since its the objects being affected, not the space the objects are in.

 

Ill have to drop spacetime as a word I use.  

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Just now, humility said:

By actual thing, I meant sometjing that can be influenced by external factors. 

Dreams can be influenced by external factors, so I suppose they are "actual things" by this definition.

In that case, our measurements of space and time (and the geometry relating them) are actual things because they are influenced by the presence of mass and energy, and by relative movement.

2 minutes ago, humility said:

Ive always thought of gravity as a thing since it could be warped and affected.

Gravity is a force we feel. That force is created by the geometry of space-time. 

3 minutes ago, humility said:

And it seems like you are saying spacetime isnt real. Its just a conviently related set of equations that make nice pictures. 

Space-time is a set of coordinates that we use for measurement. The equations relate space, time and the thongs that influence them.

Whether any of those things are "real" is not something I can comment on.

4 minutes ago, humility said:

At least this explains gravitons, I did wonder how gravity could be caused by space curving and gravitons. But if space doesnt curve then that means its all just gravitons curving the path of the objects. Since its the objects being affected, not the space the objects are in.

This conclusion seems to contradict everything that has been said.

We have a model of how curving spacetime causes gravity.

We have no model at all of how gravitons could cause gravity. 

5 minutes ago, humility said:

Ill have to drop spacetime as a word I use.  

Up to you. The best alternative then is to go back to Newtonian gravity where gravity is just a force proportional to mass. 

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1 minute ago, Strange said:

Dreams can be influenced by external factors, so I suppose they are "actual things" by this definition.

In that case, our measurements of space and time (and the geometry relating them) are actual things because they are influenced by the presence of mass and energy, and by relative movement.

Gravity is a force we feel. That force is created by the geometry of space-time. 

Space-time is a set of coordinates that we use for measurement. The equations relate space, time and the thongs that influence them.

Whether any of those things are "real" is not something I can comment on.

This conclusion seems to contradict everything that has been said.

We have a model of how curving spacetime causes gravity.

We have no model at all of how gravitons could cause gravity. 

Up to you. The best alternative then is to go back to Newtonian gravity where gravity is just a force proportional to mass. 

There are probably some vocabulary words Im missing that is causing this confusion. 

My initial question was making the assumption that space is basically its own force. Because its always described as if it was some kind of object that we are just sitting in. An object that gets curved and twisted by gravity, torn apart by black holes, and of course science fiction likes using warping of space to do all sorts of stupid things.

 

But thinking of space and time as just the numbers used to describe the relationship objects have with each other.  Really strongly changes my perspective on everything.

 

And it means however the force of gravity works, its affecting objects directly instead of the emptiness the objects are sitting in. 

 

 

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7 minutes ago, humility said:

My initial question was making the assumption that space is basically its own force. Because its always described as if it was some kind of object that we are just sitting in. An object that gets curved and twisted by gravity, torn apart by black holes, and of course science fiction likes using warping of space to do all sorts of stupid things.

Space is just distances (in three dimensions). Would you call the distance between you and your coffee cup an "object"? 

8 minutes ago, humility said:

But thinking of space and time as just the numbers used to describe the relationship objects have with each other.  Really strongly changes my perspective on everything.

It does require a new way of looking at things.

The curvature of spacetime caused by mass describes the way that the relative paths of particles moving forwards in time change in the spatial direction(s). 

As an analogy, imagine two people walking forwards, side by side, on a flat plane. Their paths will remain parallel over time. We can consider the direction they are walking as the "time" dimension (they are moving steadily into the future) and the distance between them as one of the space dimensions.

On the the flat plane, the distance between them doesn't change over time. Now put them on the surface of the Earth and have them both walk towards the North Pole (along lines of longitude). As they move forwards (in time) they get closer together. No force is acting on them, it is just a consequence of the curved geometry they are travelling in. You can consider them falling towards one another because of the gravity of the curved space-time they are in.

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1 hour ago, humility said:

As I understood it, You described how movement is measured then went on towards linguistical definitions of the word.

Thank you for letting me know this.

The above quote is enough to see that you completely missed the point of my post.

I'm sorry if it was unclear, did you have trouble with the English or what?

 

I distinguish between all types of change and movement which is a special type of change.

For instance as concrete or mud or porridge dries it hardens.

And yes the hardening can be measured.

So there is change.

But there is no movement.

 

For movement I further distinguish between the movement of a material object such as a football.

and the movement of a non material object such as a shadow.

 

Does this help to move our discussion forward?

 

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5 hours ago, Strange said:

Space is just distances (in three dimensions). Would you call the distance between you and your coffee cup an "object"? 

It does require a new way of looking at things.

The curvature of spacetime caused by mass describes the way that the relative paths of particles moving forwards in time change in the spatial direction(s). 

As an analogy, imagine two people walking forwards, side by side, on a flat plane. Their paths will remain parallel over time. We can consider the direction they are walking as the "time" dimension (they are moving steadily into the future) and the distance between them as one of the space dimensions.

On the the flat plane, the distance between them doesn't change over time. Now put them on the surface of the Earth and have them both walk towards the North Pole (along lines of longitude). As they move forwards (in time) they get closer together. No force is acting on them, it is just a consequence of the curved geometry they are travelling in. You can consider them falling towards one another because of the gravity of the curved space-time they are in.

Your analogy doesnt work because earth is still an object. 

 

So one can say that magnets curve the space of other magnets then? Like those magnetic spinning toys that just keep spinning for years before running out of energy.

 

Also wouldnt it be better to just stop referring to space directly then? If you want to be super accurate with all terms, why say "Gravity curves space" instead of saying "Gravity pulls objects into hyperbolic arcs" Since its not actually curving space, its curving trajectories.

 

5 hours ago, studiot said:

Thank you for letting me know this.

The above quote is enough to see that you completely missed the point of my post.

I'm sorry if it was unclear, did you have trouble with the English or what?

 

I distinguish between all types of change and movement which is a special type of change.

For instance as concrete or mud or porridge dries it hardens.

And yes the hardening can be measured.

So there is change.

But there is no movement.

 

For movement I further distinguish between the movement of a material object such as a football.

and the movement of a non material object such as a shadow.

 

Does this help to move our discussion forward?

 

Nope, I still have no idea what you are trying to say. 

Edited by humility
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48 minutes ago, humility said:

 So one can say that magnets curve the space of other magnets then? Like those magnetic spinning toys that just keep spinning for years before running out of energy.

Mass/energy acts on spacetime by warping/curving it. We feel that warping/curving as gravity....a spinning massive object like the Earth, will twist spacetime within its vicinity which we call frame dragging or the Lense Thirring effect. 

Quote

 Also wouldnt it be better to just stop referring to space directly then? If you want to be super accurate with all terms, why say "Gravity curves space" instead of saying "Gravity pulls objects into hyperbolic arcs" Since its not actually curving space, its curving trajectories.

As I have just said, it is mass/energy that curves spacetime or mis-shapes it in someway [gravitational lensing, gravitational waves, Frame dragging] and we feel that as gravity. So gravity is simply geometry...the geometry of spacetime as dictated by mass/energy....or as a famous physicist named John Wheeler once put it, "Spacetime tells matter how to move; matter tells spacetime how to curve". 

Quote

Your analogy doesnt work because earth is still an object

Yes, it is a spherical object, or more correctly an oblate spheroid and the analogy by Strange is relevant. 

Also a point concerning gravitational lensing particularly,  light/photons will follow similar geodesics in spacetime.

Edited by beecee
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4 hours ago, beecee said:

Mass/energy acts on spacetime by warping/curving it. We feel that warping/curving as gravity....a spinning massive object like the Earth, will twist spacetime within its vicinity which we call frame dragging or the Lense Thirring effect. 

As I have just said, it is mass/energy that curves spacetime or mis-shapes it in someway [gravitational lensing, gravitational waves, Frame dragging] and we feel that as gravity. So gravity is simply geometry...the geometry of spacetime as dictated by mass/energy....or as a famous physicist named John Wheeler once put it, "Spacetime tells matter how to move; matter tells spacetime how to curve". 

Yes, it is a spherical object, or more correctly an oblate spheroid and the analogy by Strange is relevant. 

Also a point concerning gravitational lensing particularly,  light/photons will follow similar geodesics in spacetime.

So Space is like an object that is being warped by gravity?  And not just the measurement of distance?

I feel like you kerp going back and forth. Either space gets warped by gravity, or its a measurement of distance. I cant understand how it could be both.

 

Actually I see how it could be both, but it seems like you keep insisting space isnt something getting warped by gravity then saying it is.

Edited by humility
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5 hours ago, humility said:

Your analogy doesnt work because earth is still an object. 

The analogy is supposed to show how changing the geometry of the lines that the people are moving on can cause an apparent force between them.  Forget the "Earth" part of it and focus on the geometry of the lines of longitude.

5 hours ago, humility said:

So one can say that magnets curve the space of other magnets then?

You can certainly say that magnets curve the electromagnetic field. Which is a reasonable analogy for how mass affects the spacetime field.

5 hours ago, humility said:

Also wouldnt it be better to just stop referring to space directly then? If you want to be super accurate with all terms, why say "Gravity curves space"

Gravity doesn't curve space. Mass curves spacetime. One consequence of that is gravity.

5 hours ago, humility said:

"Gravity pulls objects into hyperbolic arcs" Since its not actually curving space, its curving trajectories.

If you want to to stick to a totally Newtonian approximation, that's fine.

I get the impression you don't really want to understand, so I won't put any more effort into this.

31 minutes ago, humility said:

So Space is like an object that is being warped by gravity?  And not just the measurement of distance?

Space IS the measurement(s) of distance. That is all it is. And those measurements are changed by the presence of mass-energy.

So, not an object.

And not "warped by gravity".

 

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10 hours ago, Strange said:

Space is just distances (in three dimensions). Would you call the distance between you and your coffee cup an "object"? 

 

I would yes, if that distance could be twisted so my cup and me touch.

So mass waes

4 minutes ago, Strange said:

The analogy is supposed to show how changing the geometry of the lines that the people are moving on can cause an apparent force between them.  Forget the "Earth" part of it and focus on the geometry of the lines of longitude.

You can certainly say that magnets curve the electromagnetic field. Which is a reasonable analogy for how mass affects the spacetime field.

Gravity doesn't curve space. Mass curves spacetime. One consequence of that is gravity.

If you want to to stick to a totally Newtonian approximation, that's fine.

I get the impression you don't really want to understand, so I won't put any more effort into this.

Space IS the measurement(s) of distance. That is all it is. And those measurements are changed by the presence of mass-energy.

So, not an object.

And not "warped by gravity".

 

Maybe it is a problem with words. So mass warps space, and spacetime is a thing that gets warped but space is just measurement?

 

Wait you calked it a spacetime field, so there a field isnt a mathamatical word so there IS an actual object like thing and its called the spacetime field.

 

And I do want to understand, You are just terrible at explaining and seem to keep contradicting yourself.

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6 minutes ago, humility said:

I would yes, if that distance could be twisted so my cup and me touch.

That could, in principle happen. So call the distance between things an "object" if you like. It may cause confusion with other people's concept of "object" but I don't really care.

8 minutes ago, humility said:

Maybe it is a problem with words. So mass warps space, and spacetime is a thing that gets warped but space is just measurement?

Mass warps spacetime. Spacetime is the set of measurements we make of space and time. (You keep focussing on space, but time is affected as well.)

Quote

Wait you calked it a spacetime field, so there a field isnt a mathamatical word so there IS an actual object like thing and its called the spacetime field.

Field is a mathematical term (https://en.wikipedia.org/wiki/Field_(mathematics)).

But that isn't the same meaning as in physics (https://en.wikipedia.org/wiki/Field_(physics)). 

A field is the set of values at different locations in space and time. So the electromagnetic field describes how electric charges and magnetism interact. The spacetime field describes how mass and energy interact.

The electromagnetic field is described by Maxwell's equations. The spacetime field is described by the Einstein Field Equations.

(Again, whether these fields are "objects" or even "real" is not something I have an opinion on. It isn't relevant.)

15 minutes ago, humility said:

And I do want to understand, You are just terrible at explaining and seem to keep contradicting yourself.

OK. I will try and improve! :)

 

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54 minutes ago, humility said:

So Space is like an object that is being warped by gravity?  And not just the measurement of distance?

Umm, no...spacetime is warped/curved by mass energy...when that happens we get the sensation of gravity...so gravity is curved/warped spacetime.

Space as Strange said, is the measurement of distance, while spacetime is the  4D framework within which it is possible to locate events. The concept of spacetime follows from the observation that the speed of light is invariant, that is it does not vary with the motion of the emitter or the observer. Spacetime allows a description of reality that is common for all observers in the universe, regardless of their relative motion. Intervals of space and time considered separately are not the same for all observers.

Quote

So, not an object.

Space, spacetime are not physical entities, but in my opnion still real.....We feel the effects of warped spacetime as gravity, just as we see the effects of a magnetic field which also is not a actual physical entity.

Quote

And not "warped by gravity"

No, mass energy does the warping.

Edited by beecee
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Just now, beecee said:

Space as Strange said, is the measurement of distance, while spacetime is the  4D framework within which it is possible to locate events.

Just to expand on that, to avoid any confusion, "space" is a subset of spacetime: the 3 spatial dimensions. So spacetime is the measurements of distances and times. 

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