Except when you say 'implies' it could also imply that it acts like it is curved ( mathematically ) without necessarily being so, as space-time is not a thing, but a mathematical construct. It ( curvature ) cannot be measured directly; other effects are needed to 'imply' curvature.
Remember that what we consider 'things' are three dimensional, and to join time to the three spatial dimensions into combined space-time, requires the time dimension to be orthogonal and therefore imaginary ( i ).

But as I said, your view of 'reality' might be different from mine, and while I can't falsify yours, you cannot falsify mine ✌ .

1 hour ago, KJW said:

If you shine a beam of light upward

BTW, if one considers how both the energy and frequency of a beam of light changes as it moves away from a gravitational source, one can derive the proportionality of energy and frequency. In other words, the proportionality of energy and frequency does not come from quantum mechanics. However, the proportionality constant h for a single photon does not emerge from this and must come from quantum mechanics.

40 minutes ago, MigL said:

Except when you say 'implies' it could also imply that it acts like it is curved ( mathematically ) without necessarily being so, as space-time is not a thing, but a mathematical construct. It ( curvature ) cannot be measured directly; other effects are needed to 'imply' curvature.
Remember that what we consider 'things' are three dimensional, and to join time to the three spatial dimensions into combined space-time, requires the time dimension to be orthogonal and therefore imaginary ( i ).

But as I said, your view of 'reality' might be different from mine, and while I can't falsify yours, you cannot falsify mine ✌ .

Well, it wasn't very long ago that I said I rejected the statement "a map is not the terrain", so I tend to accept mathematical descriptions of reality as truth about reality. I see no problem with imbuing physical reality with mathematical properties. And I see no problem with inferring truth about reality such as the reality of spacetime. Reality is clearly four-dimensional to me. Any description of a physical field will include time as well as the three dimensions of space as the domain. That the four-dimensional space has a metric also seems natural, particularly considering the principle of relativity. That the local metric is Minkowskian can be justified by the need for time and space to be distinct. That the global spacetime is generally curved seems obvious on the basis of how unlikely it is for it to be everywhere flat.

Edited September 7Sep 7 by KJW

2 hours ago, MigL said:

It ( curvature ) cannot be measured directly

I can't accept that spacetime curvature cannot be measured directly. However, I do accept that measuring it may be more problematic than the mathematics suggests. For example, I know that local spacetime is Minkowskian. Therefore, to measure spacetime curvature requires that one measure spacetime over an extended region, which seems to conflict with the mathematical notion that spacetime curvature exists at each point. The smallness of the magnitude of spacetime curvature certainly makes spacetime curvature difficult to measure with all but the most precise instrumentation, but I see this discussion as about measuring spacetime curvature in principle rather than the practical limitations of current technology.

2 hours ago, MigL said:

you cannot falsify mine ✌ .

That would depend on what you say. If you stick with empirical facts, you will be on solid ground. But I think it is important to be able to explain those facts. And that involves a certain amount of risk because one is no longer in the comfort zone of empirical facts. But I think having a good grounding in both physics and mathematics would help one to avoid the pitfalls that many people in the Speculations forum make.

3 hours ago, KJW said:

I tend to accept mathematical descriptions of reality as truth about reality. I see no problem with imbuing physical reality with mathematical properties

Does that view extend to Quantum Mechanics ?
Do you believe 'reality' is a probabilistic mathematical superposition of states until an observation/interaction collapses that 'reality' to a particle ? or do you believe in an underlying actual reality with non-local characteristics ?
While I have no problem with no local reality for quantum effects, I cannot male the jump to considering space-time as 'something' that can be curved, stretched, compressed, or even cut like a fabric.

31 minutes ago, MigL said:

Does that view extend to Quantum Mechanics ?
Do you believe 'reality' is a probabilistic mathematical superposition of states until an observation/interaction collapses that 'reality' to a particle ? or do you believe in an underlying actual reality with non-local characteristics ?
While I have no problem with no local reality for quantum effects, I cannot male the jump to considering space-time as 'something' that can be curved, stretched, compressed, or even cut like a fabric.

Is space time not just the set of intervals between events?

If the events are real aren't the intervals a property of those events?

Aren't properties as real as the objects themselves?(the events)

If the events follow quantum rules then would quantum spacetime reflect the behaviour of the quantum objects?

33 minutes ago, geordief said:

Aren't properties as real as the objects

What’s your definition of real? Do you mean real as in not an illusion, or real as in physically exists rather than being conceptual?

Can a frame-dependent quantity be considered real? Can it be real but lose that designation because someone is accelerating?

4 minutes ago, swansont said:

What’s your definition of real? Do you mean real as in not an illusion, or real as in physically exists rather than being conceptual?

Can a frame-dependent quantity be considered real? Can it be real but lose that designation because someone is accelerating?

I think I mean real as in that they exist (not necessarily physically -but possibly physically and at least as a property of something which is physical)

Not quite sure what you are asking in your second paragraph but does it boil down to asking whether what I see as real is equally as real as what you see as real even if we disagree as to what we are seeing?

If that is the nub of what are asking then that is a difficult question and perhaps I might say that we would have to look for the intersection of my perception of events and your perception of events and that this intersection would include everything that we can agree on (like perhaps the spacetime interval and perhaps other quantities I am unfamiliar with)

That sounds a very limited area of shared realities but perhaps it is a building block and once we know it is there we are free to gaily constuct cities in the sky of assumed realities that are probably true and most usefully so.

The alternative of imagining that things aren't real because we can't prove it for definite is too miserable an idea to hold in the head (I am not sure if anyone does-I don't except in a fleeting ,whimsical way from time to time)

Edited September 8Sep 8 by geordief

3 hours ago, MigL said:

Does that view extend to Quantum Mechanics ?
Do you believe 'reality' is a probabilistic mathematical superposition of states until an observation/interaction collapses that 'reality' to a particle ? or do you believe in an underlying actual reality with non-local characteristics ?

For me, it's the Many Worlds Interpretation. I'm not nearly as comfortable with quantum mechanics as I am with general relativity. I'm not sure that a "correct" quantum theory is even tractable. I like MWI because it provides a natural explanation of intrinic randomness. The mathematics of the Born rule points naturally to the MWI by removing wavefunction collapse and allowing all the eigenstates to prevail. Also, in the high-dimensional Hilbert space of classical-scale quantum physics, arbitrarily chosen vectors are almost certainly orthogonal, which not only provides an explanation of one aspect of the Born rule, but also why classical states don't exhibit interference, including why only a single eigenstate outcome is observed. A difficulty of MWI is explaining the nature of wavefunctions because all the "worlds" seem naturally to be weighted classically.

3 hours ago, MigL said:

I cannot make the jump to considering space-time as 'something' that can be curved, stretched, compressed,...

Could it be that you are trying to visualise this as if from outside of spacetime? No, spacetime curvature is intrinsic and doesn't need any higher-dimensional space in which it is embedded. I tend to visualise spacetime as a rectangular grid overlay in which the distances between the nodes do not necessarily obey Pythagoras' theorem. I became convinced of spacetime curvature when I saw the metrics of seemingly random coordinate systems over a flat space, and noted that these ostensibly didn't look any different from random metrics. But the random metrics will almost certainly not be flat. Inspecting the metrics themselves, unless they are especially simple or familiar, one can't tell if they are describing a flat spacetime, one has to obtain the Riemann tensor to determine this. In other words, given the infinitude of flat spacetime metrics, why not go all the way and consider the infinitude of all spacetime metrics. Anyway, I tend to regard spacetime curvature like the Mercator projection map of the world.

3 hours ago, MigL said:

... or even cut like a fabric.

I am somewhat ambivalent about spacetimes with non-trivial topologies. I have considered removing the entire interior of the Schwarzschild blackhole using a wormhole metric obtained from the Schwarzschild metric by a coordinate transformation. This raised an interesting question concerning the role of complex coordinates in physics.

Edited September 8Sep 8 by KJW

16 hours ago, KJW said:

  18 hours ago, Eise said:

Yes, it is. So what? I see Kip Thorne as an authority. Don't you agree? (Standard text book on Gravity, one of of scientific minds behind LIGO.)

Perhaps. But I will look at the mathematics before I look at who wrote it. And yes, I've read stuff by Kip Thorne.

But obviously not everything... From Wikipedia:

In Poincaré's conventionalist views, the essential criteria according to which one should select a Euclidean versus non-Euclidean geometry would be economy and simplicity. A realist would say that Einstein discovered spacetime to be non-Euclidean. A conventionalist would say that Einstein merely found it more convenient to use non-Euclidean geometry. The conventionalist would maintain that Einstein's analysis said nothing about what the geometry of spacetime really is.

Such being said,

1. Is it possible to represent general relativity in terms of flat spacetime?

2. Are there any situations where a flat spacetime interpretation of general relativity may be more convenient than the usual curved spacetime interpretation?

In response to the first question, a number of authors including Deser, Grishchuk, Rosen, Weinberg, etc. have provided various formulations of gravitation as a field in a flat manifold. Those theories are variously called "bimetric gravity", the "field-theoretical approach to general relativity", and so forth. Kip Thorne has provided a popular review of these theories.

The flat spacetime paradigm posits that matter creates a gravitational field that causes rulers to shrink when they are turned from circumferential orientation to radial, and that causes the ticking rates of clocks to dilate. The flat spacetime paradigm is fully equivalent to the curved spacetime paradigm in that they both represent the same physical phenomena. However, their mathematical formulations are entirely different. Working physicists routinely switch between using curved and flat spacetime techniques depending on the requirements of the problem. The flat spacetime paradigm is convenient when performing approximate calculations in weak fields. Hence, flat spacetime techniques tend be used when solving gravitational wave problems, while curved spacetime techniques tend be used in the analysis of black holes.

Maybe look it up in MTW? One of the advantages that is also mentioned is that the concepts of flat spacetime paradigm is closer to QED, and so might help to unify the standard model with gravity. But that is still speculative.

17 hours ago, KJW said:

But don't take that to mean that I don't know what I'm talking about. What I'm saying is that I have a different mindset. Are you a physicist?

Of course you know what you are talking about! But I assume Kip Thorne is too. And no, I only studied physics as subsidiary subject.

13 hours ago, KJW said:

I can't accept that spacetime curvature cannot be measured directly.

Now that is a huge methodological error you are making here. Just because GR is so successful, you think it is measured directly?

8 hours ago, KJW said:

For me, it's the Many Worlds Interpretation.

Well, that is consistent with your idea that spacetime really is curved: because predictions are correct, it is the real description of reality with the respective calculation methods. One can use the Schrödinger equation for (probability) predictions in QM, so the wave function is real, which automatically leads to MWI. But as you of course know, the empirical content of MWI is not different from other interpretations. That's why it is called an interpretation. And the same might be valid for GR's curved spacetime.

Edited September 8Sep 8 by Eise

The anti-time,
i wonder how will it act when somehow it comes in contact with time,but its too theoretical and imaginary.

Time flows, so can this anti time be opposite.
if there's anti time then there must be anti-space too as Spacetime=Space+Time.
So this can be a whole another dimension.

Our space is expanding,so can this anti-space contracting.

13 hours ago, geordief said:

Not quite sure what you are asking in your second paragraph but does it boil down to asking whether what I see as real is equally as real as what you see as real even if we disagree as to what we are seeing?

It’s not mere disagreement. Physics tells us both of our observations are correct. The abstract existence of the property, in general, is what we can agree on. KE, for example, is a property that is real and not an illusion. The value that it has, though, is not real in this same sense.

So can a particular spacetime interval be considered real, even if we agree the property exists?

24 minutes ago, swansont said:

It’s not mere disagreement. Physics tells us both of our observations are correct. The abstract existence of the property, in general, is what we can agree on. KE, for example, is a property that is real and not an illusion. The value that it has, though, is not real in this same sense.

So can a particular spacetime interval be considered real, even if we agree the property exists?

I thought the particular spacetime interval was definitely real whether or not we agree upon it in different accelerated frames or not (I was under the impression that they did agree but perhaps I was mistaken and the agreement only applies between unaccelerated frames

I also thought it was possible for an observer X in a frame accelerated wrt observer Y to calculate the spacetime interval as observed by Y if the observations were indeed different.

(Perhaps I am mistaken in this also :-(

In any event I have been under the (mistaken?) impression these last 10/15 years that the spacetime interval was the one physical quantity that all observers could agree upon.

Edited September 8Sep 8 by geordief

15 hours ago, MigL said:

While I have no problem with no local reality for quantum effects, I cannot male the jump to considering space-time as 'something' that can be curved, stretched, compressed, or even cut like a fabric.

That's what I was clumsily trying to get earlier with an inept analogy. Is maybe one way to express that is that something like a metric tensor will describe what happens with moving particles AS IF there is a curvature, AS IF there is an inherent geometry that causes photons and fermions to move along a certain path? What the "as if" means to me is that measurement can lead to accurate predictions of future action without asserting anything metaphysical. I'm not sure quite where @KJW lands on this.

1 hour ago, geordief said:

I thought the particular spacetime interval was definitely real whether or not we agree upon it in different accelerated frames or not (I was under the impression that they did agree but perhaps I was mistaken and the agreement only applies between unaccelerated frames

It’s invariant in inertial frames (everyone agrees on the value), but not if the observer is accelerated.

1 hour ago, geordief said:

I also thought it was possible for an observer X in a frame accelerated wrt observer Y to calculate the spacetime interval as observed by Y if the observations were indeed different.

(Perhaps I am mistaken in this also :-(

You could account for it, I suppose, much like adding in pseudoforces lets you apply Newton’s laws

1 hour ago, geordief said:

In any event I have been under the (mistaken?) impression these last 10/15 years that the spacetime interval was the one physical quantity that all observers could agree upon.

Agreeing if you’re in an inertial frame. Rest mass and charge, too. Any invariant quantity. But the interval is not a property of an object, as such. It’s a value for two events.

1 hour ago, swansont said:

It’s invariant in inertial frames (everyone agrees on the value), but not if the observer is accelerated.

You could account for it, I suppose, much like adding in pseudoforces lets you apply Newton’s laws

Agreeing if you’re in an inertial frame. Rest mass and charge, too. Any invariant quantity. But the interval is not a property of an object, as such. It’s a value for two events.

A property of the system?

Is the system as (physically) real as the objects (events?) that populate it?

1 hour ago, geordief said:

A property of the system?

Is the system as (physically) real as the objects (events?) that populate it?

In the spacetime interval example, it could be describing two events. It’s a system only because we describe it as such. Events don’t have to be related

1 hour ago, swansont said:

In the spacetime interval example, it could be describing two events. It’s a system only because we describe it as such. Events don’t have to be related

Are you saying that events have to be within the "causal window" to be related?**

Might my post stand up if i limited the events to that subset?

Or would you stilll answer that there was no intrinsic system of events ,but just that we put that construction on them?

**perhaps it was a more general observation?

1 hour ago, geordief said:

Are you saying that events have to be within the "causal window" to be related?**

Might my post stand up if i limited the events to that subset?

Or would you stilll answer that there was no intrinsic system of events ,but just that we put that construction on them?

**perhaps it was a more general observation?

I don’t see how that makes it a property.

11 hours ago, swansont said:

I don’t see how that makes it a property.

Is the question here whether a particular system can be considered as having a measurable property in that it contains(spacetime) intervals?

(trying to keep up with my own assertions)

Well ,those intervals could be evenly spaces (flat space?) or unevenly spaced in regions of the system that contain varying energy densities .

Could that system have a measurable property based on the intervals between its events?

As an aside is the spacetime interval also called or closely related to the metric?(have heard "metric" used very often with understanding it too well)

Metric just means a math tool which allows one a way to calculate a separation or distance between points. AFAICT, that's all it is. Instead of using something simple and algebraic, as one would for euclidean space, for a Riemannian manifold one might use a tensor. Or, as in the case of spacetime with an energy density, a tensor for a pseudo-Riemannian manifold. This is where my college math stopped, and my impression is that if we don't marinate in this stuff pre age 25, it's a really rocky road.

22 hours ago, geordief said:

As an aside is the spacetime interval also called or closely related to the metric?(have heard "metric" used very often with understanding it too well)

This is actually a very good question +1

and @TheVat has given a good shortform answer +1

If you like to explore this further the mathematics is extremely simple, yet reaches into most corners of the subject from classical mechanics to relativity to geometry to cryptography to error correction codes to statistics to topology to....

This all hinges on what is meant by 'distance' that is what we want from our 'tool'.

I would think that the concept of a 'metric space' is more fundamental.
"A set with the notion of distance between its elements." from Wiki.

The corresponding metric, is then, the mathematical tool to measure those distances.
And the example most people are familiar with is 3D Euclidian space with its usual distance between points.
But another example would be on a sphere with angular distance.
And not necessarily on a manifold, it could be the 'relation' between members of a set that aids transforms between them.

My apologies if the terms I use aren't up to Studiot's standards, but this isn't my area of expertise.

1 hour ago, studiot said:

This is actually a very good question +1

and @TheVat has given a good shortform answer +1

If you like to explore this further the mathematics is extremely simple, yet reaches into most corners of the subject from classical mechanics to relativity to geometry to cryptography to error correction codes to statistics to topology to....

This all hinges on what is meant by 'distance' that is what we want from our 'tool'.

When it comes to predicting physical outcomes in our(classical) universe is the spacetime interval the only tool that can be usefully so used?

I also wonder whether there are other higher dimensional spaces (called Hilbert spaces?) that I suspect may be used to model other phenomena.

If that is the case are there comparable tools for measuring distances in those spaces and would you be able to give me an example (or do they all just use the simple pythagorean system with extra dimensions?)

Btw do any of those higher dimensional spaces include a time axis that works in the same (imaginary?) way as in Minkowski space?

@MigL didn't see your post.

Yes,angular distance seems like another way of doing things(does that work in spacetime?).

Would that just be the same as polar coordinates,really?

And ,out of curiosity are the transforms you mention comparable or relevant to the dual space (which struggled with some years ago)

Edited September 10Sep 10 by geordief

3 hours ago, geordief said:

Yes,angular distance seems like another way of doing things(does that work in spacetime?).

Would that just be the same as polar coordinates,really?

Another excellent question. In physics, where you have some kind of centralized phenomenon, like a field around a concentration of mass or charge, or object moving around a central point, it can be useful to use a spherical coordinate system (which is the 3D version of the polar coordinate system). Since this is, again, where my rusty college math hits a wall, one of the physics grownups here would have to address how such a system would work for spacetime stuff. I would think that whenever you have rotations, there would be an advantage of polar/spherical over a Cartesian coordinate system. Dimly I recall a professor saying it doesn't matter which coordinate system you use, they can all work, but it's more a matter of which takes the least time and fuss.

I like @studiot comment that it all hinges on what is meant by distance, as to what is wanted from a math tool. For some reason, I am drawn more to the term "separation," which seems to me more neutral, maybe. When people say distance, it's too easy to get stuck thinking of it as linear.

21 minutes ago, TheVat said:

they can all work, but it's more a matter of which takes the least time and fuss.

Yeah.
As TheVat mentions, in spherically symmetric systems, spherical polar co-ordinates can vastly simplify solutions.
One example would be the solution of the Schrodinger equation for the Hydrogen atom.
It is a partial differential equation in (radius, theta, phi) where we can use separation of variables, as the spherical symmetry allows the 'removal' of the angular co-ordinates (theta, phi) leaving a differential equation in one variable (radius), that even I can solve.
I'm sure it could be done by someone ( else ) in cartesian co-ordinates, but it would be extremely messy.