Do moving massive objects drag curved spacetime with them?

[pervect]

In the weak field, one can make an analogy between electromagnetism and gravity with a few other assumptions also necessary such as low velocities and others. This can be formalized as "gravitoelectromagnetism" or GEM for short. There's a wiki overview at https://en.wikipedia.org/w/index.php?title=Gravitoelectromagnetism&oldid=970487534

If we use this weak field analogy, we can regard gravity as a force, the coulomb force of electromagnetism between charges is, except for a necessary minus sign, is analogous to the gravitational force between masses. The sign issue comes into play because like charges repel, but "gravitational charge" is always positive, and two objects with positive masses, positive "gravitatioanl charge", attract each other, they do not repel each other.

If we have two moving charges, covariance demands that in the rest frame of said particles, there is only an electric force, while in a frame moving relative to the two charges, there is both an electric component and a magnetic component to the force.

Similar reasoning applies to gravity in the weak field using the GEM approximations.

So if we consider two stationary masses, we have only the "electric" GEM components betwen them. If we go to a frame in which the masses are moving, we find that there is both an "electric" force and a "magnetic" force. This is necessary for covariance, for our choice of frame not to matter to things we can observe.

In the strong field, where GEM doesn't work things are not so easy. There is an approach I like involving the Bel decomposition of the Riemann tensor, but I think what I'd write on that would not be helpful without a detailed knowledge of the Riemann tensor and tensor mathematics. I'm guessing that this is not a background you share, so it wouldn't be productive to go into it.

I would concentrate on understanding the origin of the magnetic force in special relativity first, and why it is needed for covariance. Then you can consider linear frame dragging to be a similar phenomenon - something that we need to make our theory covariant.

As far as experiment goes, though, we've directly confirmed frame-dragging from the rotating earth.

 

[hnaghieh]

if they see the object rotating, it's rotating; if they see it not rotating, it's not rotating.

Any observer who “see an object not rotating” could be at relative rest with respect to rotating object’s rest frame (rotating with them), therefore their “observations “ can not be considered as privileged and absolute. This was indeed one of the starting points of General relativity. The need for extension of the principle of relativity. I feel there is still a prevailing notion of spacetime as a “container” where “asymptotically flat space time”, a lorentzian manifold, “killing vector space”, “virtual particles”, “ world-lines”, “massive objects” etc. live, and move, which conflicts with everything STR and GR have said. That is what I am trying to clarify for myself.

 

[hnaghieh]

I'm guessing that this is not a background you share, so it wouldn't be productive to go into it.

I am not an expert but I am familiar with the mentioned topics. I would appreciate if you do include it in a step by step fashion so even I could follow it. Thank you.

 

[PeterDonis]

Any observer who “see an object not rotating” could be at relative rest with respect to rotating object’s rest frame (rotating with them), therefore their “observations “ can not be considered as privileged and absolute.

The "rest frame" I described is a physical property of the spacetime. In other words, while it is true that the laws of GR do not privilege any particular choice of frame, particular spacetimes that are solutions of those laws can have particular frames that are "privileged" in the sense that they match up with properties of the spacetime.

However, if you don't like my description in terms of this "rest frame", it is easy to restate it without using frames at all: An asymptotically flat spacetime that contains a non-rotating massive object has a property (hypersurface orthogonality of the timelike Killing vector field) that an asymptotically flat spacetime that contains a rotating massive object does not have. The presence or absence of this property is a geometric fact about the spacetime that is independent of any choice of reference frame.

I feel there is still a prevailing notion of spacetime as a “container” where “asymptotically flat space time”, a lorentzian manifold, “killing vector space”, “virtual particles”, “ world-lines”, “massive objects” etc. live, and move, which conflicts with everything STR and GR have said.

You are mistaken.

Spacetime is a dynamic geometric entity in GR--its geometry is determined by the distribution of stress-energy via the Einstein Field Equation. Properties like "asymptotically flat" and "timelike Killing vector field" are properties of particular solutions of the Einstein Field Equation--particular spacetime geometries that are determined by particular distributions of stress-energy (which is what the terms "non-rotating massive object" and "rotating massive object" that I used above describe--distributions of stress-energy that determine the spacetime geometry via the Einstein Field Equation).

"Worldlines" do not "move"; they are curves in the 4-dimensional spacetime geometry.

"Virtual particles" are a quantum concept and are off topic in this forum, nor do they have anything to do with what is being discussed in this thread.

 

[hnaghieh]

particular spacetime geometries that are determined by particular distributions of stress-energy (which is what the terms "non-rotating massive object" and "rotating massive object" that I used above describe--distributions of stress-energy that determine the spacetime geometry via the Einstein Field Equation).

I agree. But may I ask you a question? Isn’ this distribution of stress energy that determines the spacetime geometry via Einstein’s Field Equation reference frame dependent? Aren’t The deviations from orthogonal timelike worldlines a geometric representation of the relative velocity of two respective frames under consideration?

 

[Nugatory]

Isn’ this distribution of stress energy that determines the spacetime geometry via Einstein’s Field Equation reference frame dependent?

No.
You should be thinking of a “frame” as a convention for assigning coordinate values to points in spacetime (not points in space!). Different frames will label points in spacetime differently, but changing the labels changes neither the spacetime nor the distribution of matter across that spacetime. An analogy: Using a different frame is like changing where we draw the zero meridian on the surface of the earth: every feature on the surface of the Earth will be at a different longitude, but the relationship between these features will be unchanged.

Aren’t the deviations from orthogonal timelike worldlines a geometric representation of the relative velocity of two respective frames under consideration?

I don’t understand what you’re trying to say here - I know what “relative velocity”, “orthogonal”, “timelike worldline”, “frames”, and “deviation” mean, but you have strung these words together in a way that makes no sense.

You might want to give Taylor and Wheeler’s book “Spacetime Physics” a try; it will help give you a solid grasp of some of the concepts that you’re misunderstanding.

 

[PeterDonis]

Isn’ this distribution of stress energy that determines the spacetime geometry via Einstein’s Field Equation reference frame dependent?

No. The whole point of using 4-vectors and tensors to describe things like the spacetime geometry (the metric tensor) and the distribution of matter and energy (the stress-energy tensor) is that such a description is frame-independent. That allows us to write physical laws in frame-independent form, as the principle of relativity requires.

Aren’t The deviations from orthogonal timelike worldlines a geometric representation of the relative velocity of two respective frames under consideration?

No. As I said, it is a frame-independent geometric property of the spacetime. The frame-independent geometric object that describes it is the metric tensor.

 

[PeterDonis]

The whole point of using 4-vectors and tensors to describe things like the spacetime geometry (the metric tensor) and the distribution of matter and energy (the stress-energy tensor) is that such a description is frame-independent. That allows us to write physical laws in frame-independent form, as the principle of relativity requires.

Btw, since electromagnetism has also been mentioned in this thread, it is worth noting that the physical laws of EM--Maxwell's Equations and the Lorentz force law--can also be written using 4-vectors and tensors in frame-independent form. The relevant objects are the charge-current 4-vector and the EM field tensor (an antisymmetric 2-index tensor).

 

[hnaghieh]

You should be thinking of a “frame” as a convention for assigning coordinate values to points in spacetime (not points in space!). Different frames will label points in spacetime differently, but changing the labels changes neither the spacetime nor the distribution of matter across that spacetime

So are you saying( I think you are not) that two observers in two frames K and k with relative motion will agree on simultaneity of two world events? Same as when they use the same frame(no relative motion between them)? My understanding Tensors is that Tesors are invariant object therefore they represent physical laws. They represent “relationships “ between the Compnents of tensor(vectors) while those components are frame dependent (the tensor itself represent invariant transformation relationships). An observer in the rest frame k of a moving object will not agree with observation of another observer at rest in frame K, not moving with frame k (at rest in a different frame)when they compare.

 

[Nugatory]

So are you saying( I think you are not) that two observers in two frames K and k with relative motion will agree on simultaneity of two world events?

You are right that I am not saying that. "Simultaneous" means "has the same time coordinate", so is frame dependent and tells us almost nothing about the relationship between two events in spacetime. The frame-independent concepts that describe this relationship are "spacelike separated", "lightlike separated", and "timelike separated".

(Why did I say "tells us almost nothing"? It turns out that if two events are spacelike separated in flat spacetime, then with a bit of algebra we can show that there exists an inertial frame in which two events have the same time coordinate. But it's the spacelike separation that is the physical fact in this situation, and the simultaneity is a just a happy but limited consequence).

 

[PeterDonis]

the Compnents of tensor(vectors)

Vectors are not components of tensors. Vectors are invariant objects just like tensors; they are best thought of as one-index tensors.

An observer in the rest frame k of a moving object will not agree with observation of another observer at rest in frame K, not moving with frame k (at rest in a different frame)when they compare.

You need to stop thinking in terms of frames; it is only confusing you.

What is happening physically is simply that observers in relative motion will make different observations--for example, they will measure light signals coming from the same source to have different frequencies. But all of these observations, including all the differences in the observations made by observers in relative motion, can be described entirely in terms of invariants. There is no need to bring frames into it at all.

Frames are a convenience for calculations, but conceptually they often cause more confusion than they solve.

 

[Nugatory]

An observer in the rest frame k of a moving object will not agree with observation of another observer at rest in frame K, not moving with frame k (at rest in a different frame)when they compare.

That is not correct, although it is a very common misunderstanding (perhaps the root of all relativity misunderstandings). In fact, they will agree about every observation either makes - otherwise we would have hopeless paradoxes in which (for example) whether a bug is squashed or not is frame-dependent. The difference is in how they explain these observations.

For example: I'm sure you've read in a million popular treatments of relativity that if you and I are moving relative to one another, you will "observe" that a meter stick at rest relative to me is length-contracted to the shorter length $\sqrt{1-v^2/c^2}$ where $v$ is our relative velocity. But that's not actually what you observe. What you observe is the time at which each end of the moving meter stick is at various points in space (you can either put a detector taking time-stamped photos at these points, or you can use your eyes and watch the meter stick as moves past you as long as you allow for the time it took the light to get to your eyes). Using these observations, you infer - not "observe"! - the length of the meter stick; it's just the distance between where the ends were at the same time, and it will be less than one meter.

You will explain the results reported by your detectors by saying that the clocks in the detectors are properly synchronized and the detectors are less than one meter apart; the detectors put the same timestamp on their detections because the moving meter stick is length contracted.

I will explain these results by saying that the clocks in your detectors are not synchronized so that despite the identical timestamps the two detections do not represent the positions of the ends of the stick at the same time.

Both explanations are equally valid. And crucially we agree about what is actually observed: the timestamps your detectors produced as the ends of the meter stick passes them.

 

[alantheastronomer]

What happens is that, instead of falling straight in towards the mass, the infalling particle acquires a gradually increasing angular velocity in the direction of rotation of the mass. So its trajectory in space looks like an ingoing spiral.

Similarly, irrespective of rotation, would that also be true for a pair of compact objects in orbit? Does their orbital motion also result in frame-dragging?

 

[hnaghieh]

Frames are a convenience for calculations, but conceptually they often cause more confusion than they solve.

In order to make any calculation we need data or measurements. In order to have measurements we need units of measurements(yard sticks and clocks). Yard sticks and clocks are frame dependent in the sense that two reference coordinate systems in relative motion will have different units (due to relativistic effects on the yard sticks and clocks, etc) when they compare each other’s measurements of a world event they will not agree on their measurement or the laws of nature unless they know the transformation relationship connecting their coordinate frames units of measurements ( Lorentz transformations). This is solely because the speed of the signals used (light) to make “time”mearurments or “length” measurements, is finite and constant. Individually they won’t infer that anything (length contraction, time dilation) has taken place in their respective reference coordinate frames. But when they compare their determination their units of “time” and “length” won’t be the same. That’s why we have Lorentz transformation to make the LAWS invariant. That is why the laws are not coordinate dependent, but their components (based on a specific measurement by a specific yard stick etc ) is coordinate dependent. A simple test. Take Two solid discs. Put a pin through its center. This pin can only be rigidly connected to one of the two discs if there is a relative rotational motion between the two discs. If we are rigidly connected to the rotating disc and have no external reference point we will not notice a motion (let’s forget the accelerameter because of our rigidly connected assumption). Same is for linear motion. The motion of a frame can not be determined by any parameter from within the reference frame (from within a ship). Solutions to any invariant law must be a limited case of the invarent law, which is a specific frame dependent component of that law. How do we distinguish between invariant laws and their coordinate dependent components or solutions and do not attribute the same invariant properties to the solutions. My point is that in order to do that we must be clear about our chosen frame of reference. And more importantly how can we ensure that the view point of the observer is actually that of the rest frame observer of the chosen frame?

 

[Nugatory]

The motion of a frame can not be determined by any parameter from within the reference frame (from within a ship).

The “motion of a frame“ cannot be determined for about the same reason that I cannot determine the color of love or the weight of an idea - the notion is meaningless because frames aren’t things that can move. A frame is a mathematical convention for assigning coordinates to points in spacetime (these points are called “events” in the language of relativistic physics) and mathematical conventions aren’t things that move around in space.

Yes, I know you’ve heard people use the term “a moving reference frame” or “a frame moving relative to me” or similar... but that just shows that natural language isn’t always used precisely. It would be more accurate to say “a reference frame which assigns coordinates to events in such a way that the spatial coordinates of my position are a function of the time coordinate”.

Until you have resolved your confusion about what frames do and don’t do, you will find it very difficult to make sense of relativity. At this point all I can do is repeat my recommendation of the Taylor and Wheeler book.

 

[hnaghieh]

don’t understand what you’re trying to say here - I know what “relative velocity”, “orthogonal”, “timelike worldline”, “frames”, and “deviation” mean, but you have strung these words together in a way that makes no sense.

I apologize for lack of clarity in my statement. I assumed you were referring to minkowskie’s geographical representation of time like and space like worldlines as orthogonal axies on a spacetime diagram. And That he represented the respective coordinate axies of a reference frame with a relative motion with respective to the former as lines with acute angles with respect to the former orthogonal lines. Hence I called those acute angled lines as deviation from orthogonal lines representing a relative velocity of the two respective frames. I hope this clarifies a little.

 

[hnaghieh]

The “motion of a frame“ cannot be determined

This is Galilean principle of relativity. Precursor to Einstein’s principle of relativity. Not related to “color love”or “weight of an idea “ which are not physical parameters. It simply means motion is relative not absolute. We Need a frame of reference in order to give its magnitude or its direction. Once the frame of reference is chosen(which means the units of measurement is chosen) then we can physically measure it. Without that reference frame we will not be able to measure relative parameters.

 

[hnaghieh]

(you can either put a detector taking time-stamped photos at these points,

This is the crucial point. A detector making “time” measurement can not make “space” at the same time with unrestricted accuracy because of finite velocity of light. These two parameters (time and space)are orthogonal parameters. In the time it takes for the signal to reach the detector, the moving object has moved a definite amount which is the limit of accuracy of the measurements for the times of the two end points. Einstein called them “A”time and “B” time. There is no common and absolute AB time. (In quantum domain there is a further source of indeterminacy due to size of the signal used to determine spatial meaurmets of an entity).

 

[hnaghieh]

Vectors are not components of tensors. Vectors are invariant objects just like tensors; they are best thought of as one-index tensors.

I totally agree and apologize for sloppy statement. Instead of just “vector” i should have put The words “components of vector “ in the parentheses. What I meant was components of tensor, as well as the components of vectors, are coordinate frame dependent.

 

[hnaghieh]

The “motion of a frame“ cannot be determined for about the same reason that I cannot determine the color of love or the weight of an idea - the notion is meaningless because frames aren’t things that can move. A frame is a mathematical convention for assigning coordinates to points in spacetime (these points are called “events” in the language of relativistic physics) and mathematical conventions aren’t things that move around in space.

Yes, I know you’ve heard people use the term “a moving reference frame” or “a frame moving relative to me” or similar... but that just shows that natural language isn’t always used precisely. It would be more accurate to say “a reference frame which assigns coordinates to events in such a way that the spatial coordinates of my position are a function of the time coordinate”.

Until you have resolved your confusion about what frames do and don’t do, you will find it very difficult to make sense of relativity. At this point all I can do is repeat my recommendation of the Taylor and Wheeler book.

One of The people I have heard using these terms: “a moving reference frame” or “ a frame moving relative to me” is the great inventor of all this. Mr Albert Einstein in his celebrated paper “on the electrodymamics of moving bodies”

 

[pervect]

In order to make any calculation we need data or measurements. In order to have measurements we need units of measurements(yard sticks and clocks). Yard sticks and clocks are frame dependent in the sense that two reference coordinate systems in relative motion will have different units (due to relativistic effects on the yard sticks and clocks, etc) when they compare each other’s measurements of a world event they will not agree on their measurement or the laws of nature unless they know the transformation relationship connecting their coordinate frames units of measurements ( Lorentz transformations).

I agree with much , though not all, of this. Let me try and give you the outline of a modern, relativistic perspective.

Distances as measured by yardsticks are frame dependent quantities, agreed. The time interval measured by a single, specific clock is, however, a frame independent quantity.

Time intervals measured by combining the results of more than one clock via some synchronization scheme are frame dependent, because the synchronization scheme is frame dependent.

Neither the frame dependent distances, nor the frame dependent non-proper time intervals measured by multiple clocks are invariant quantities.

When you specify both of these frame dependent quantities, though, and combine them into a 4-vector, the 4-vector itself is regarded as frame independent, even though the components of the 4-vector are frame dependent.

The reason the 4-vector is regarded as frame independent is that if you know all the components in one frame you can, as you mentioned, use the Lorentz transform to find the components in any other frame. So, the 4-vector itself is regarded as a "geometric entity" that exists regardless of a specific choice of frame. The actual measurements that some observer made in some particular frame are not the 4-vector, but the components of the 4-vector in the observer's frame. Knowing the components of the 4-vector, and the specifics of the observer, one knows the four-vector.

Working with frame independent objects is a very powerful technique that makes errors much less likely. All the information you need to find the physical measurements are collected together into one place.

To give a specific example relative to the previous discussion. In special relativity, the electric field and the magnetic field are not tensors. They are parts of a larger object that is a tensor, the Farday tensor, which is a rank 2 tensor.

Knowing the electric fields at a point according to one observer doesn't let you know what the electric fields at that point are for another observer. Knowing both the electric and magnetic fields at that poitn according to an observer gives you the components of the Farday tensor, and you can use the approrpriate generalization of the Lorentz transform to find the electric and magnetic fields at that point for any observer you choose.

 

[PeterDonis]

would that also be true for a pair of compact objects in orbit?

Do you mean in orbit about each other? That's a different spacetime geometry, and "frame dragging" is not really a useful concept for that case.

 

[PeterDonis]

Yard sticks and clocks are frame dependent

No, they are not. Objects like this can be represented by 4-vectors and tensors, i.e., by frame invariant objects.

You really, really need to spend some time learning about the formulation of relativity using vectors and tensors. The classic reference for this is the GR textbook by Misner, Thorne, and Wheeler, which is a heavy lift but which makes it abundantly clear how it all works.

 

[PeterDonis]

the laws are not coordinate dependent, but their components (based on a specific measurement by a specific yard stick etc ) is coordinate dependent

No. The components of vectors and tensors are indeed frame-dependent, but they are not the same as actual results of measurements by actual physical things like yard sticks. Confusing these two very distinct and different things is unfortunately a common error, but it is nevertheless an error. You will continue to be confused unless you correct this error in your thinking and learn how results of actual measurements by actual physical things like yard sticks are actually represented in physics: by invariants.

 

[PeterDonis]

This is Galilean principle of relativity. Precursor to Einstein’s principle of relativity.

The principle of relativity does not mean what you think it does. You are trying to apply the words without a proper understanding of the underlying concepts.

 

[PeterDonis]

Distances as measured by yardsticks are frame dependent quantities, agreed. The time interval measured by a single, specific clock is, however, a frame independent quantity.

Unfortunately, I think these two statements taken together, while I see what you mean by them, are highly likely to increase the OP's confusion rather than decrease it.

The spacelike interval between two particular spacelike separated events is invariant, just as the timelike interval between two particular timelike separated events is invariant. (Here I am assuming the interval is being measured along a unique geodesic between the two events, to avoid a lot of technicalities that are out of scope for the OP at this point in the discussion.) A particular timelike interval has an obvious physical interpretation as the time measured by a clock that travels between the two events (along the geodesic between them). So it's easy to see how the invariance of a timelike interval is physically realized.

It's harder to see how the invariance of a spacelike interval is realized, because nothing moves on spacelike worldlines, so a measurement of any spacelike interval requires at least two timelike worldlines to be involved (one passing through each endpoint of the spacelike interval). This seems to introduce frame dependence, but it actually doesn't. It just means that we have to specify two worldlines instead of one. But worldlines are invariants, and if we specify two worldlines, say of the two endpoints of a ruler, and pick a pair of spacelike separated events, one on each worldline, that invariant spacelike interval will have a physical interpretation that can be described as "distance measured by the ruler". The interpretation won't be as intuitively obvious as the one for a timelike interval, but it will still be enough to support the invariance of the interval.

When the OP talks about "distance measured by the ruler" in two different frames, he is talking about two different spacelike intervals. Yes, these two intervals will have different lengths. But that doesn't mean the intervals themselves are not invariant. The length of each interval is invariant. They're just different.

I know this all is obvious to you, but I don't think it's obvious to the OP, and given how this thread has gone up to now, I think it's worth belaboring points like this to try to make clear what is going on.

 

[hnaghieh]

Unfortunately, I think these two statements taken together, while I see what you mean by them, are highly likely to increase the OP's confusion rather than decrease it.

The spacelike interval between two particular spacelike separated events is invariant, just as the timelike interval between two particular timelike separated events is invariant. (Here I am assuming the interval is being measured along a unique geodesic between the two events, to avoid a lot of technicalities that are out of scope for the OP at this point in the discussion.) A particular timelike interval has an obvious physical interpretation as the time measured by a clock that travels between the two events (along the geodesic between them). So it's easy to see how the invariance of a timelike interval is physically realized.

It's harder to see how the invariance of a spacelike interval is realized, because nothing moves on spacelike worldlines, so a measurement of any spacelike interval requires at least two timelike worldlines to be involved (one passing through each endpoint of the spacelike interval). This seems to introduce frame dependence, but it actually doesn't. It just means that we have to specify two worldlines instead of one. But worldlines are invariants, and if we specify two worldlines, say of the two endpoints of a ruler, and pick a pair of spacelike separated events, one on each worldline, that invariant spacelike interval will have a physical interpretation that can be described as "distance measured by the ruler". The interpretation won't be as intuitively obvious as the one for a timelike interval, but it will still be enough to support the invariance of the interval.

When the OP talks about "distance measured by the ruler" in two different frames, he is talking about two different spacelike intervals. Yes, these two intervals will have different lengths. But that doesn't mean the intervals themselves are not invariant. The length of each interval is invariant. They're just different.

I know this all is obvious to you, but I don't think it's obvious to the OP, and given how this thread has gone up to now, I think it's worth belaboring points like this to try to make clear what is going on.

When the OP talks about "distance measured by the ruler" in two different frames, he is talking about two different spacelike intervals. Yes, these two intervals will have different lengths. But that doesn't mean the intervals themselves are not invariant. The length of each interval is invariant. They're just different.

No. I am talking about an invariant spacetime interval viewed from two different reference fame which have a relative velocity with respect to each other.

 

[PeterDonis]

I am talking about an invariant spacetime interval viewed from two different reference fame which have a relative velocity with respect to each other.

If you are, then much of what you have said about that invariant spacetime interval is simply false, since much of what you have said asserts that that interval is frame-dependent, and invariants are not frame-dependent.

I strongly suggest that you take a step back and think very carefully about what you are saying, and re-read very carefully what others have said to you in this thread.

 

[hnaghieh]

No. I am talking about an invariant spacetime interval viewed from two different reference fame which have a relative velocity with respect to each other.

The observer in each frame will determine that interval or “distance “ if you prefer, in their rest frame using rods and clocks of their frame (propertime and proper length). When they compare they don’t agree what the length of that distance is unless they know the relationship between their rods and clocks. (Transform laws)

 

[hnaghieh]

If you are, then much of what you have said about that invariant spacetime interval is simply false, since much of what you have said asserts that that interval is frame-dependent, and invariants are not frame-dependent.

I strongly suggest that you take a step back and think very carefully about what you are saying, and re-read very carefully what others have said to you in this thread.

The interval is invariant “same for all observers “ only with the appropriate transformation equations relating the frame depandant components of one frame with another frame with relative velocity.Under these appropriate transformation equations all observers agree. Because they know it was their units of measurements which were affected by their relative motion and they would need to account of these relativistic effects. Once they “transform “ their frame depending measurements all will be happy and all spacetime intervals will be the same for all observers but only if they transform their unit coordinate basis or unit vectors.

 

[hnaghieh]

two different spacelike intervals. Yes, these two intervals will have different lengths. But that doesn't mean the intervals themselves are not invariant. The length of each interval is invariant. They're just different.

Two different “ spacelike “ intervals can never be “invariant” just as two “ timelike “ intervals can never be invariant. It is only “spacetime “ intervals that are invariant under approximate transformation equations

 

[PeroK]

The observer in each frame will determine that interval or “distance “ if you prefer, in their rest frame using rods and clocks of their frame (propertime and proper length). When they compare they don’t agree what the length of that distance is unless they know the relationship between their rods and clocks. (Transform laws)

If someone measures a physical quantity they may do so without any knowledge of any other observers. They get a single answer.

An invariant quantity is one where everyone gets the same answer. No one need be aware of anyone else's measurements nor transform their data in any way.

 

[hnaghieh]

If you are, then much of what you have said about that invariant spacetime interval is simply false, since much of what you have said asserts that that interval is frame-dependent, and invariants are not frame-dependent.

I strongly suggest that you take a step back and think very carefully about what you are saying, and re-read very carefully what others have said to you in this thread.

Ok thank you all for your patience . I think I have my answer.

 

[hnaghieh]

If someone measures a physical quantity they may do so without any knowledge of any other observers. They get a single answer.

That is exactly what Newton said.

 

[hnaghieh]

That is exactly what Newton said.

Absolute space and absolute time.