# Questions regarding traveling speed in time and gravity as a force

• I
I am an Engineer, so I have some knowledge about physics and calculus. I've seen relativity in university years ago but only briefly. The majority of my knowledge in SR and GR, which isn't much, are from reading, studying and searching online myself.

I have seen some definitions that I am not sure it's true, due to internet being a pace full of wrong facts. So, here I ask my doubts about relativity:

1- I've seen more than once people saying that we (as everything in the universe) travel at the speed of light. Just that inertial objetct travel with the speed of light in time and moveing objects travel with less speed in time and more in space. I understand that spacetime interval are invariant, but they aren't necessarily always equal to c, right? Also, the ct time-component of the metric are a convention thing, right? So that this coordinate get a unit of space? Anyway, I've seen no good reason or proof to believe that everything travel with c in spacetime.

2- I know that in GR gravity is not a force, but the curvatre of spacetime. However, while I can see why a moving object would be attracted to a massive object like the earth (follwing the geodesic path), I can't understand why a still object would start moving toward the ground, if there isn't a force pulling it to Earth. I've seen videos and explanations saying that this happens due to the curvature of spacetime, especially the time axis and that since we are always travelling in time, and the time axis is cuved due to the massive mass of a planet, then we travel toward the Earth
I think this is a terrible explanation, because they are treating time as a spatial dimension, and it isn't. As i understand, the "bending" or "curving" of time means time passing faster or slower to an object, and not affecting the object position in space. So, if someone could give me a light about this one would be very nice.

anuttarasammyak
Gold Member
Anyway, I've seen no good reason or proof to believe that everything travel with c in spacetime.
Neither do I. Except mass zero particles nothing moves with speed c.

• vanhees71
Ibix
2020 Award
I've seen more than once people saying that we (as everything in the universe) travel at the speed of light.
The "everything travels at ##c##" comes from the standard normalisation of the four velocity to have length ##c##. Interpreting that choice of normalisation as meaning "everything travels at ##c##" is circular, yes - all four velocities have the same magnitude because that's how they are defined. And it's not true for light anyway, where the tangent vector to its worldline is null so cannot be normalised to length ##c##.

However, the "##ct## thing" is not just a convention. Essentially, ##c## is a natural conversion factor between units of time and units of length. Not including it, or using a different speed, is analogous to taking a rectangle that's 1m on one side and 1000mm on the other and asserting that the diagonal must therefore be ##\sqrt{1^2+1000^2}## long.

I can't understand why a still object would start moving toward the ground, if there isn't a force pulling it to Earth.
Unfortunately the explanations you've seen are more or less correct. The point is that even when you aren't moving (by some definition) you are advancing in time. And spacetime near a mass is curved, and the unaccelerated path of an object in that spacetime gets closer to the object as a result.
I think this is a terrible explanation, because they are treating time as a spatial dimension, and it isn't.
Time is a dimension, but it is distinct from spatial ones. There is some crosstalk, though, in that the direction in spacetime that I call "time" you might call a mixture of time and some spatial direction.

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• Dale
anuttarasammyak
Gold Member
As i understand, the "bending" or "curving" of time means time passing faster or slower to an object, and not affecting the object position in space.
In weak limit of gravity, metric component ##g_{00}## and gravity potential have relation ##g_{00}=1+2V## where ##V=-\frac{GM}{Rc^2}## for our daily ground Earth case.

##g_{00}## decides not only time passing pace but also potential of attracting acceleration i.e.
In lower place time passing slower and gravity potential is lower. In higher place time passing faster and gravity potential is higher. Bodies even at still get acceleration of ##-\nabla V##.

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• Dale
Ibix
2020 Award
In weak limit of gravity, metric component ##g_{00}## and gravity potential have relation ##g_{00}=1+2V## where ##V=-\frac{GM}{Rc^2}## for our daily ground Earth case.
That much is always true (outside an event horizon, and assuming your zeroth basis vector is the timelike Killing vector). Where the weak field low speed stuff comes in is in showing that the Einstein field equations reduce to Poisson's equation.

• horacio torres and anuttarasammyak
Ibix
2020 Award
As i understand, the "bending" or "curving" of time means time passing faster or slower to an object, and not affecting the object position in space.
Forgot to say this earlier - you can't think of "the curving of time". It's always the curving of spacetime. Space and time are both aspects of spacetime.

• LBoy
I'm an engineer and I have knowledge of physics and calculus, even saw something about relativity in university, but not much. The majority of info I have about the topic I got from my own readings, studying and research online.

Here are a couple of concepts that I read more than once and that I find hard to understand:

1- I've seen people saying that everything is always travelling at the speed of light. It just happens that still objects are travelling at that speed in time, while moving objects are traveling at a combination of that speed in time and space. Now, I don't believe this is true, since I see no empyrical evidence of proof for such a claim. Also there makes no sense in saying that you travel in time with the speed of light (which unit is m/s). We travel in time by the rate of 1 second per second. I think people that say that might be getting a wrong concept from the time coordinate from the metric "ct". Isn't that c just a human convention, that is there to make the time coordinate unit spatial? Couldn't it be anyvalue?

2- Regarding gravity, I know that in GR gravity is not considered a force, but the curvature of spacetime. I can see why a moving object that goes in the way of a massive mass like a planet would have its trajectory changed by the curvature of spacetime to follow the geodesic path. However, I don't see why a still object in the presence of such massive mass would begin moving toward it, since there is not any force acting upon it.
The explanations I found were that everything is always traveling through time, and time is "curved" by the mass so it "bends" in the direction of the planet. This makes no sense to me, since time is not a spatial dimension, and while i understand it can be curved, since spacetime is one unified thing and mass curve spacetime, the "bending" of space, as I see, is just affecting the flow of time. Time curving means time flowing slower or faster for that object, and not making it travel in space.
I also saw explanation of how time would flow diferently inside the very object. So, if a ball is traveling parallel to Earth, the atoms of the side of the ball that is closer to Earth travel in time slower than the ones of the other side. This would cause the ball to turn and be "attracted" by Earth. Yet, I don't see this working with a still object and how it would start moving. Also, I find the explanation very hard to believe, since the difference in time flow in one side of the ball from the other would be ridiculously small. It don't seem suficient to give an aceleration of 9.8 m/s2. This is even worst the smaller the object you use.

I guess this is it for now.

Dale
Mentor
2020 Award
Please don't post messages on the same topic in different threads. I have merged the new thread with the existing one. Keep future messages on this same topic here in this thread, please.

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Dale
Mentor
2020 Award
Time curving means time flowing slower or faster for that object, and not making it travel in space.
Here is a good quick video that really shows the point effectively
The apple is initially at rest, staying at a constant position due to the force acting on it. Then, when that force is removed it travels in a straight line and the curvature of spacetime results in it starting to fall even though it was initially at rest.

• LBoy
The "everything travels at ##c##" comes from the standard normalisation of the four velocity to have length ##c##. Interpreting that choice of normalisation as meaning "everything travels at ##c##" is circular, yes - all four velocities have the same magnitude because that's how they are defined. And it's not true for light anyway, where the tangent vector to its worldline is null so cannot be normalised to length ##c##.

However, the "##ct## thing" is not just a convention. Essentially, ##c## is a natural conversion factor between units of time and units of length. Not including it, or using a different speed, is analogous to taking a rectangle that's 1m on one side and 1000mm on the other and asserting that the diagonal must therefore be ##\sqrt{1^2+1000^2}## long.

Unfortunately the explanations you've seen are more or less correct. The point is that even when you aren't moving (by some definition) you are advancing in time. And spacetime near a mass is curved, and the unaccelerated path of an object in that spacetime gets closer to the object as a result.

Time is a dimension, but it is distinct from spatial ones. There is some crosstalk, though, in that the direction in spacetime that I call "time" you might call a mixture of time and some spatial direction.
I disagree about the first part. Your rectangle example is treating two "different" units (meter and milimeters). We could just replace c by any chosen velocity (still in m/s) and still have the equations to work.

While we call spacetime and know that they are somehow realted, time is still the passing of time and shouldn't affect the direction an object travels in space. I can't grasp how affecting time can change the trajectory of an object. I know you will say that it's no affecting time, but *spacetime*.

However, even though spacetime is like one unified thing, it still has two different concepts of a spatial dimension and a time dimension. I can see how bending spacetime would affect how moving objects travel in space (space is literally bent) and how time flows differently (time is bent). But not how travelling in time can affect your position in space.

@anuttarasammyak , sorry I don't understand your post.

@Dale Oh, I'm really sorry. I was trying to post this thread yesterday and got a login failure and I thought I lost it. Then I proceeded to re-write it now, without knowing it worked before. Thanks for mergin the threads, my questions stay basically the same.

About your video, why would the apple fall in the Einstein's example in the first place? I know it has to follow the geodesic path of the curved spacetime, but only if it is moving, right?

Dale
Mentor
2020 Award
I know it has to follow the geodesic path of the curved spacetime, but only if it is moving, right?
No. It has to follow the geodesic path in curved spacetime if it is not experiencing a real force (e.g. as measured by an accelerometer). Whether it is at rest or moving is irrelevant. A geodesic path is the path of a force-free object.

• vanhees71
No. It has to follow the geodesic path in curved spacetime if it is not experiencing a real force (e.g. as measured by an accelerometer). Whether it is at rest or moving is irrelevant. A geodesic path is the path of a force-free object.
I know this is the answer, but I can't understand why an object that has no forces acting upon it would move in space. I know it "moves" in time (as time passing), but moving in time won't (or shouldn't) change its position in space.

Dale
Mentor
2020 Award
I know this is the answer, but I can't understand why an object that has no forces acting upon it would move in space. I know it "moves" in time (as time passing), but moving in time won't (or shouldn't) change its position in space.
It isn’t good to think about “moving” in spacetime at all. A Newtonian point particle becomes a worldline in spacetime. You don’t think of the line as moving, it is just a line. That line has, for each value on the time axis, a corresponding value on the space axis.

So the slope of the line (wrt some specified axes) is what we think of as “moving” in Newtonian physics. If the worldline is parallel to the time axis in spacetime then it is at rest per Newton and if it is not parallel to the time axis then it is moving per Newton.

So now, if your time axis is curved then you cannot draw a straight line which remains parallel to it. Go ahead and try it. Print out a picture from the video and draw lots of straight lines. Notice when they become parallel to the time axis and when they do not. That is the Newtonian idea of rest and moving.

A straight worldline that begins parallel to the time axis will necessarily quickly begin to diverge from the axis as the axis curves.

• LBoy, sysprog and vanhees71
It isn’t good to think about “moving” in spacetime at all. A Newtonian point particle becomes a worldline in spacetime. You don’t think of the line as moving, it is just a line. That line has, for each value on the time axis, a corresponding value on the space axis.

So the slope of the line (wrt some specified axes) is what we think of as “moving” in Newtonian physics. If the worldline is parallel to the time axis in spacetime then it is at rest per Newton and if it is not parallel to the time axis then it is moving per Newton.

So now, if your time axis is curved then you cannot draw a straight line which remains parallel to it. Go ahead and try it. Print out a picture from the video and draw lots of straight lines. Notice when they become parallel to the time axis and when they do not. That is the Newtonian idea of rest and moving.

A straight worldline that begins parallel to the time axis will necessarily quickly begin to diverge from the axis as the axis curves.
I understand. My issue is that we are representing time (the x axis) as a spatial dimension the moment we bend it. What does it mean to "bend" time? As I see, it means to make the flow of time go faster or slower.
It isn't "wrong" to treat time with the same characteristics of space? Doesn't the concept of spacetime just means that space and time are related, yet not the same thing? For example, let's pretend that masses in our universe only bends time. What would that mean?
I believe my issue is in the concept of spacetime because I can't quite grasp the concept that space and time are the same thing. I can see how they are related, though, but not the same thing.

If, instead of representing spacetime as two axis we represent a 3 axis spatial place repeatedly as in frames of a movie, what would bend time mean?

Ibix
2020 Award
I disagree about the first part. Your rectangle example is treating two "different" units (meter and milimeters). We could just replace c by any chosen velocity (still in m/s) and still have the equations to work.
And if you do, you will find that your chosen velocity is now the one that is the same in all frames (which means that your revised model no longer matches reality), or else that your invariant interval isn't invariant while the one using ##c## remains invariant. This is exactly analogous to the situation where you measure ##x## distances in one unit and ##y## distances in another - "lengths" defined this way are not invariant under rotation, or if I wish to keep "length" invariant I need to introduce scaling along with rotation, which is not what happens in reality.
My issue is that we are representing time (the x axis) as a spatial dimension the moment we bend it.
No we are not. We are treating all four dimensions in the same way, except that the metric signature picks one out as different from the other three. That signature difference is what defines the timelike direction. Having any curvature at all entails curvature in a timelike plane - since there is no unique time direction you cannot have curvature that only affects planes perpendicular to time.

• LBoy, sysprog, vanhees71 and 1 other person
Dale
Mentor
2020 Award
My issue is that we are representing time (the x axis) as a spatial dimension the moment we bend it.
This is incorrect. Including time as part of spacetime does not mean that we are treating it as a spatial axis.

In any case, this "issue" isn't a scientific objection. The scientific question is: Does the resulting theory match the experimental data? It does. Even if we were treating time as a spatial dimension (which we are not), if the resulting theory gave correct results then it is correct to do so.

Doesn't the concept of spacetime just means that space and time are related, yet not the same thing? ... I can't quite grasp the concept that space and time are the same thing. I can see how they are related, though, but not the same thing.
Space and time are not the same thing. They are related, but not the same.

The relationship is expressed in the metric which (in an inertial frame) can be written ##ds^2= -c^2 dt^2 + dx^2 + dy^2 + dz^2##. So in the metric you can clearly see that there are two important distinctions that make time different from space. First, there is only one time dimension and there are three spatial dimensions, and second the time has a different sign in the metric than space.

This preserves the distinction between time and space. For example, because of the difference in signs intervals with ##ds^2 < 0## are measured with a clock while intervals with ##ds^2>0## are measured with a ruler. Because of the difference in the number of dimensions you can draw a closed spacelike loop but not a closed timelike loop.

Because clocks measure ##ds^2## (or more specifically they measure ##\sqrt{-ds^2}##) it is clear that this spacetime (which has the above relationship between space and time) has physical significance. You cannot treat time or space separately, but they are different parts of the same Lorentzian manifold.

What does it mean to "bend" time?
It means exactly what is shown in the video. It means that the time axis is not a straight line.

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• LBoy, Nugatory, vanhees71 and 2 others
@Ibix I understand. You are saying that if we replace the c in the time coordinate of the metric for another chosen number, than the speed limit would be that of the new number and not c, which is incorrect. I got it right? And does this means than that we are indeed...travelling throught time.. with the speed of light? What would that even mean? I would love to know the answer in a theory way, not mathematically, so I can understand that related to our universe.

@Dale Ok, let's jsut try to leave the equations aside for a moment and try to see what meaning this have in our physical universe. What does "bend time" means? Axis and worldlines are things to ilustrate and help understanding the concepts. But in our 3 spatial dimensional and 1 time dimensional universe what does it mean to bend time?

I already know the answer will be: it's not bending time, but bending spacetime. Let me rephrase it then:
Bending spacetime will have effects on how space works and on how time works. A bent space means that your trajectory will bend, because space is where you move spatially. So, what effect in time (and our perception of it) does bending spacetime has?

Ibix
2020 Award
I got it right?
Yes.
And does this means than that we are indeed...travelling throught time.. with the speed of light?
No. That is popsci nonsense, albeit often promulgated by Brian Greene.

We typically normalise the tangent vector to a worldline to have length ##c## and call this the four velocity, since it's the derivative of your four position with respect to your proper time. This is a velocity in the geometric sense of the word (a normalised vector tangent to a curve), but not really in the physics sense of speed.
What does "bend time" means?
Nothing. It's spacetime that's curved.
A bent space means that your trajectory will bend, because space is where you move spatially. So, what effect in time (and our perception of it) does bending spacetime has?
You can't separate curved spacetime into curved space and curved time. Curvature is measured in terms of what happens to a vector transported around a small loop. A loop has 2d extent, and necessarily includes both spatial and temporal extent. (Edit: it's possible to have a purely spatial loop, although a change of coordinates can make it not purely spatial, but there is only one timelike dimension.)

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• LBoy, vanhees71 and ergospherical
Dale
Mentor
2020 Award
What does it mean to "bend" time?
It means exactly what the video shows. It means that the time axis is not straight.

Axis and worldlines are things to ilustrate and help understanding the concepts. But in our 3 spatial dimensional and 1 time dimensional universe what does it mean to bend time?
In this case the axis and worldline thing is more important than you are implying, and cannot be dismissed from the discussion. The reason is that your whole question is based on trying to understand why curved spacetime can make something that is initially at rest begin to move. But there is no such thing “in our 3 spatial dimensional and 1 time dimensional universe” itself that corresponds to the concept of “at rest” or “still”.

The concept of “at rest” means that you have a time axis and a worldline and they are parallel. There is nothing in the universe that corresponds to “at rest” without drawing that time axis. So you cannot divorce axes and worldlines from a question about objects at rest. They are part of the question itself.

So, once again, “curved time” means that the time axis is not straight. In other words, you have chosen a definition of “at rest” such that accelerometers at rest do not read 0.

I already know the answer will be: it's not bending time, but bending spacetime.
I differ slightly with my colleagues here. I have no objection to talking about bending time. It is, as I said, having a time axis which is not straight. This is a concept that does not exist without defining an axis, which is an arbitrary thing, not part of the universe. So it is a fine concept but it is “relative” or “frame variant” and thus not part of nature itself.

In contrast, bending spacetime means that there is tidal gravity. And this is an invariant concept that can be defined without reference to some reference frame.

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• PeroK and vanhees71
ergospherical
Gold Member
Hm as @Ibix already said, "curved time" doesn't mean anything; curvature is a property of a manifold and is characterised by the change ##\Delta X_{\nu} = \displaystyle{\oint} \Gamma^{\mu}_{\nu \rho} X_{\mu} dx^{\rho} = \dfrac{1}{2} R^{\mu}_{\nu \rho \sigma} X_{\mu} \Delta S^{\rho \sigma}## of a vector ##\mathbf{X}## after parallel transport around an infinitesimal closed contour. Non-vanishing of the tensor ##R^{\mu}_{\nu \rho \sigma}## implies the manifold is curved!

[##R^{\mu}_{\nu \rho \sigma} := \partial_{\rho} \Gamma^{\mu}_{\nu \sigma} - \partial_{\sigma} \Gamma^{\mu}_{\nu \rho} + \Gamma^{\mu}_{\alpha \rho} \Gamma^{\alpha}_{\nu \sigma} - \Gamma^{\mu}_{\alpha \sigma} \Gamma^{\alpha}_{\nu \sigma}## is called the Riemann tensor.]

• LBoy and vanhees71
It means exactly what the video shows. It means that the time axis is not straight.

I wanted to know what it means physically, in our real world. math is a tool that we use to help us comprehend , discover and work with theories. Including things that are abstract.
I don't know you guys, but for me, to really, trully understand a theory or concept in physics I must be able to understand what that concept means in our real world. I can't be satisfied with just the abstract math, even though I know it is really important for better understanding and even developing those concepts.

So, when I asked what does it means to bend time, I'm not asking to bend the axis representation of time. There is no axis in our universe. Axis are just a tool to measure some variable, like a rule.

In this case the axis and worldline thing is more important than you are implying, and cannot be dismissed from the discussion. The reason is that your whole question is based on trying to understand why curved spacetime can make something that is initially at rest begin to move. But there is no such thing “in our 3 spatial dimensional and 1 time dimensional universe” itself that corresponds to the concept of “at rest” or “still”.
I meant still in the spatial dimensions, since no regular object on Earth can be "still" in time.

The concept of “at rest” means that you have a time axis and a worldline and they are parallel. There is nothing in the universe that corresponds to “at rest” without drawing that time axis. So you cannot divorce axes and worldlines from a question about objects at rest. They are part of the question itself.

Again, I don't even know if it is possible to explain (certainly it is not easy) what happens in our physical universe without the axis representation (or math), but it should be. Or else the theory is not what really happens in the universe and is just a tool to calculate relativistics movements. We know that this is not the case for Relativity, since it went through experimental tests.

A way of trying to do that is to work with thougths experiments imagining our real 3D space and a clock. Or just our real 3D space frame by frame as a movie to show the flow of time.

So, once again, “curved time” means that the time axis is not straight. In other words, you have chosen a definition of “at rest” such that accelerometers at rest do not read 0.

I differ slightly with my colleagues here. I have no objection to talking about bending time. It is, as I said, having a time axis which is not straight. This is a concept that does not exist without defining an axis, which is an arbitrary thing, not part of the universe. So it is a fine concept but it is “relative” or “frame variant” and thus not part of nature itself.

In contrast, bending spacetime means that there is tidal gravity. And this is an invariant concept that can be defined without reference to some reference frame.

Are you saying that the "bending" is not a real world thing? Is it just an abstract concept, for the math? @Ibix gave a similar answer about how bending applies to vector and loops. However, one of the major experiments that proved relativity right was the one with the solar eclipse, where the light of stars behind the sun could be seen, since space curved the rays into Earth.

Let's take a step back and see the effects that bending spacetime has on space. What happens to the space around us when a planet like Earth bends spacetime? What's the difference between the space around you if you are on Earth and if you are in deep intergalactic space? What is the physically definition of bending space, since space is continuous and all around us?

I might be asking very silly questions, but I'm just trying to understand the concept applied to our real world.

• LBoy and vanhees71
anuttarasammyak
Gold Member
@anuttarasammyak , sorry I don't understand your post.
It is just to give you an idea behind, not to annoy you with mathematics. Length in space is given by Pythagoras theorem say
$$dl^2=dx^2+dy^2+dz^2$$
Similarly world interval, "length" of spacetime, is given
$$ds^2=c^2dt^2-dl^2=c^2dt^2- dx^2-dy^2-dz^2=dx_0^2-dx_1^2-dx_2^2-dx_3^2$$
with ct,x,y,z renamed as ##x_0,x_1,x_2,x_3##. Though strange (+,-,-,-) signs appear, anyway, it is an integrated "length" of spacetime. c is constant of light speed. It is in SR, no curving of spacetime takes place. In GR spacetime is curved so it becomes mingled.
$$ds^2 = g_{00}dx^0 dx^0+ g_{01} dx^0 dx^1+...+g_{33}dx^3dx^3$$
where ##g_{ij}## is metric tensor with 10 different components which denotes how spacetime is curved. For SR, metric tensor is simple, i.e.,
$$g_{00}=-g_{11}=-g_{22}=-g_{33}=1,\ \ \ g_{ij}=0\ \ otherwise$$

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what really happens

Could you define what "really happens" means, and how to distinguish it from "not really"?

Ibix
2020 Award
I don't know you guys, but for me, to really, trully understand a theory or concept in physics I must be able to understand what that concept means in our real world.
If you only want to talk about "time curvature" then @Dale's answer (the time axis is not a geodesic) is the only one possible. I don't think it's particularly meaningful because it depends on your choice of time axis, but there isn't really any other interpretation of "time curvature" available.
I meant still in the spatial dimensions, since no regular object on Earth can be "still" in time.
"Not moving in space" is a frame dependent concept (a person moving with respect to you says that you are moving), so to be stationary you've picked a definition of space and hence a definition of time. If you accept "still in space" as a real concept you are thereby defining a time axis.

You can do physics without coordinates, but you have to let go of notions like "stationary in space" because defining "space" is picking coordinates.
Let's take a step back and see the effects that bending spacetime has on space. What happens to the space around us when a planet like Earth bends spacetime?
If you dig a tunnel through the center of the Earth and measure the length, ##d##, of the tunnel and the circumference, ##C##, of the planet you will find ##C\neq \pi d##. You will also find that clocks held at different altitudes will tick at different rates.

Note that both of these statements involve picking coordinates.

• Dale
Dale
Mentor
2020 Award
I wanted to know what it means physically, in our real world.
If you want to know about the real world then you will need to ask questions about the real world. For instance, if you ask “what is a moral justification” then you can hardly complain when the answer involves some human-made system of morals. So instead of dismissing a correct answer to your question, either engage with the answer or ask a better question. Instead of all the effort you put into complaining about good answers, if you had put half that effort into improving your question or a quarter of that effort into understanding the answers, then you would have made some progress in understanding.

math is a tool that we use to help us comprehend , discover and work with theories. Including things that are abstract
Why would you avoid using a tool for comprehending if you are trying to comprehend?

for me, to really, trully understand a theory or concept in physics I must be able to understand what that concept means in our real world
Again, that is fine for concepts that are actually "in our real world", but your questions are full of concepts that don't exist in our real world but are only in our theories and math. You asked about time and you asked about space, but the real world only has spacetime. The separation of spacetime into time and space is a human convention (our choice of axes), not part of the real world. You asked about objects at rest, but the state of being at rest is also a human convention and not part of the real world.

The problem here is that you are asking questions about things that only exist in our theories and math and then complaining that the answers are not only about the real world. If you want answers that are only about the real world then you will need to ask questions that are only about the real world. If you ask questions that are a mix of real world and human convention then the answers will necessarily be a mix of real world and human convention.

I can't be satisfied with just the abstract math, even though I know it is really important for better understanding and even developing those concepts.
Frankly, this line of complaint is starting to irritate me. You asked a question and it was answered. Furthermore, I explained explicitly how the mathematical concepts in your question could be directly tied to real experimental result. As I already told you, curved time means that accelerometers which you have arbitrarily designated as being "at rest" read non-zero. If accelerometers at rest measure 0 then time is not curved, if accelerometers at rest measure non-zero then time is curved. I don't know how you can complain about "just the abstract math" when I already tied it in to real physical measurements.

I'm not asking to bend the axis representation of time. There is no axis in our universe.
Then according to you there is no time in our universe.

Stop blaming us if you are asking a question that you don't want to know the answer to. I am a physicist, not a psychic. I can't tell what question you actually want answered if you don't actually ask the question.

Again, I don't even know if it is possible to explain (certainly it is not easy) what happens in our physical universe without the axis representation (or math), but it should be.
Certainly, it is possible to explain what happens in our physical universe without the axis representation. The axis representation is about "time", so if you want answers that avoid the axes then you should ask about spacetime (the manifold with its metric) instead of space or time (the axes we arbitrarily draw on that manifold).

I don't think it could be done without math entirely. I think that you should revisit your anti-math stance. Of course, math is not sufficient. You need to also include the link between the math and actual physical experiments. But physics is more than just experimental data, it is also theory, and the theory is built on math. So to get a full understanding you must have both math and experiment.

A way of trying to do that is to work with thougths experiments imagining our real 3D space and a clock. Or just our real 3D space frame by frame as a movie to show the flow of time.
There is no such thing as our real 3D space. What is real is 4D spacetime.

Are you saying that the "bending" is not a real world thing?
I am saying that "bending of time" requires the specification of an arbitrary human convention of "time". This is done by declaring certain objects to be at rest. But there is no universal truth to what objects are at rest. Once you have made that arbitrary human specification of which objects are declared to be "at rest" (thereby fixing the time axis), then the real-world meaning of "time bending" is that accelerometers at rest do not read 0.

Let's take a step back and see the effects that bending spacetime has on space. What happens to the space around us when a planet like Earth bends spacetime? What's the difference between the space around you if you are on Earth and if you are in deep intergalactic space? What is the physically definition of bending space, since space is continuous and all around us?
Space is as much an arbitrary human invention as time. The physical thing is spacetime. I am not going to answer this question because you will just make the same complaints for the same reasons.

If you want a purely physical answer then you should ask a purely physical question like "what are the effects that bending spacetime has on physical objects". Stop asking about space or about time and then complaining that the answers aren't physical, when you have already been told that the question isn't physical. What exists is spacetime, the separation into space and time as separate things is simply choosing your coordinate axes, and no amount of complaining will change that. If you don't want to hear about axes then why do you keep asking about them?

Time is not the same as space, as discussed above, but the division of spacetime into separate concepts of time and space is artificial.

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