# Thread: GR without spacetime manifold

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Originally Posted by Reality Check
It is not true that Minkowski introduced a Lorentzian manifold from the postulates of SR which is your assertion. He noted that space + time = space rime and that it has to be Lorenz invariant thus what we call a Minkowski spacetime.
Minkowski spacetime is a Lorentzian "manifold".

Your "vacuum" ignorance again. It is persisting with world lines along with "vacuum" that suggests you think that these are the entire manifolds. You are making obvious errors about manifolds.
I never said that worldlines are manifolds, Reality Check, I said that they are continuous, but I would not be interested in onedimensional manifolds. Threedimensional space is a manifold.

Think about Euclidean space in math and used in classical physics. This is a continuous space with an infinite number of points between each point. No vacuum there! Manifolds are a generalization of Euclidean space. They are mathematical. They are defined to be continuous. They do not have vacuum.
You are right, Newtonian spacetime with its standard topology may be understood as a manifold. There is also vacuum between worldlines in Newtonian spacetime, but these points are part of the standard topology. In contrast, Lorentzian manifolds cannot be provided with a standard topology, their topology is rather a whole mess.

IF03: Give your sources that say that a Lorentzian manifold must have "vacuum" or "void(s)", Rinaldo.
This is a physical reality, nothing mathematical. Space is simply not filled entirely with particle worldlines which would not leave any void. By consequence, there is void between the particles.

IF04: State where the 2 postures of SR exclude ("do not apply to") "vacuum points" (presumably points in physical vacuums), Rinaldo.
Special relativity
What are the postulates talking about? Postulate 1 talks about IRF which must be mass particles, postulate 2 talks of observers and light sources which must be mass particles. Then, postulate 2 talks about c, the speed of light in a vacuum. "in a vacuum" may not have any other sense than "without interaction with particles (or possibly: fields). We can extend this postulate to fields such like an electromagnetic field or (in GR) gravity fields which are propagating at speed of light. Laws of physics of vacuum are not mentioned, the postulates do not care about vacuum.

The same is true for the whole physics which is derived from these postulates: Lorentz transformation does not talk about the vacuum. The Lorentz factor depends on the relative velocity between two reference frames (=particles). This is why the physics of SR do not care about the vacuum.

You persist with a blatant error that a Lorentzian manifold is not compatible with GR when it is easy see that it is from reading about GR. Differentiable manifolds in physics have to be continuous because we want to apply calculus and calculus needs a continuous space. A simply put example: a speed dx/dt is a limit as finite intervals in x and t tend to 0. Those intervals cannot come to a screeching halt at a finite value! GR has the additional requirement that its manifold be Lorentz invariant.
Lorentzian spacetimes are no manifolds, they are not continuous and differentiable, in particular not in spacelike direction.
How can differential geometry be possible? The answer is simple: each of the worldlines which are parameterized by their respective proper time is perfectly continuous, differentiable and Lorentz invariant! As long as differential geometry treats worldlines of particles (including hypothetical worldlines) it works.
One counter-example: The spacelike foliation of spacetime into hypersurfaces is part of numerous theories of quantum gravity, and the corresponding application of differential geometry does not lead to any reasonable results here.

IF05b: Show that "particle worldlines are Lorentz invariant if each of them is parameterized by its respective proper time", Rinaldo.
Proper time is the "time measured by a clock following a given particle". All observers agree on the proper time of the particle. Every observer can take the observed worldline of a particle (which is parameterized by the observer's coordinate time and coordinate space), and by calculation he can recover the original worldline of the particle, that is the worldline parameterized by its proper time. The resulting particle worldline will be identical for all observers.

IF05a: Show that "A threedimensional space manifold is Lorentz invariant" for example the Euclidean space used in classical physics, Rinaldo.
Here is one try (there might be simpler): In order to measure an arbitrary distance from A to B in Euclidean threedimensional space, any observer in a spaceship may travel to A and then from A to B. That means, even if he does not make this travel he can at least calculate the space interval between A and B. Any observer who travels from A to B will get the same result, that means that any observer will get the same result of calculation, that means that a Euclidean threedimensional space manifold is observer-independent and Lorentz-invariant.
Note: The Euclidian space manifold used in classical physics is perfectly appropriate for modern quantum physics, and I assume that there will be also a way to express gravity in flat space (in the form of gravitational time dilation) instead spacetime curvature.

IF05c: Give your definition of calculus in your "world line only manifold" and show that it works, Rinaldo.
As I said, the worldlines do not form any manifold, they are solipsistic worldlines parameterized by their respective proper time, and all this within a 3D space manifold. What do you mean exactly by "definition of calculus"? See also above "How can differential geometry be possible?" in this post.

... look at the definition of spacetime and see it has no complex numbers!
Yes, but it has negative squares which leads to an equivalent contradiction.

Spacelike spacetime intervals are defined as being negative.
This is not so clear, Reality Check. It is an undeniable fact that physicists obscured the subject by the introduction and the maintenance of four different conventions (+,-,-,- and -,+,+,+, extraction of the root or not), this fact is not very reassuring. Has anybody reflected on this point? I studied as much literature as I could get with my limited means, but the more I read, the more I get the impression, that in 1908, people were too much absorbed by the idea of fourdimensional spacetime, and I fear that this issue has never been treated by Mainstream.
Last edited by Rinaldo; 2019-Dec-12 at 02:49 PM.

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Originally Posted by Rinaldo
Minkowski spacetime is a Lorentzian "manifold". ...
Yes but: IF06: Give your evidence that states that Minkowski introduced a Lorentzian manifold from the postulates of SR, Rinaldo.
I wrote what the textbooks say. First came SR which showed that space and time were connected by the Lorentz transformation. Minkowski took the next step of uniting space and time into Minkowski spacetime.

You persist with a blatant error that a Lorentzian manifold is not compatible with GR when it is easy see that it is from reading about GR. Differentiable manifolds in physics have to be continuous because we want to apply calculus and calculus needs a continuous space. A simply put example: a speed dx/dt is a limit as finite intervals in x and t tend to 0. Those intervals cannot come to a screeching halt at a finite value! GR has the additional requirement that its manifold be Lorentz invariant.
I did not write not about your still unsupported world line idea. It is the mathematics and physics that Lorentzian manifolds are compatible with GR. That statement in your OP is blatantly wrong. Physics needs differentiable manifolds. GR needs its differentiable manifolds to be Lorentz invariant. Thus Lorentzian manifolds are compatible with GR.
But you may have a math and physics derivation that have falsified a century of physics so I asked:
IF01: What happens to the Lorentzian manifold according to the definition of a Lorentzian manifold, Rinaldo?
IF02: Give your derivation that Lorentzian manifolds are not compatible with GR using valid math and physics, Rinaldo.

More "vacuum" fantasy: IF03: Give your sources that say that a Lorentzian manifold must have "vacuum" or "void(s)", Rinaldo
Some "mass particle" stuff seems not about SR and its postulates: IF04: State where the 2 postures of SR exclude ("do not apply to") "vacuum points" (presumably points in physical vacuums), Rinaldo.
IF04b: Give the textbook definition of the laws of physics that states they are or must be about "mass particles", Rinaldo.
IF04c: Give the textbook definition of an observer that states they are "mass particles", Rinaldo.

IF05a: Show that "A threedimensional space manifold is Lorentz invariant" for example the Euclidean space used in classical physics, Rinaldo.
IF05b: Show that "particle worldlines are Lorentz invariant if each of them is parameterized by its respective proper time", Rinaldo.
IF05c: Give your definition of calculus in your "world line only manifold" and show that it works, Rinaldo.
Last edited by Reality Check; 2019-Dec-12 at 08:16 PM.

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Originally Posted by Rinaldo
Lorentzian spacetimes are no manifolds, they are not continuous and differentiable, in particular not in spacelike direction.
You are still ignoring my sources that state the textbook physics that Lorentzian manifolds are continuous and differentiable (as required for physics such as GR to work) so:
IF07: Give your sources that state "Lorentzian spacetimes are no manifolds, they are not continuous and differentiable, in particular not in spacelike direction", Rinaldo.
Otherwise show that textbooks are wrong.
• Give a clear mathematical derivation that Lorentzian spacetimes do not fit the definition of a manifold. Make sure you cite the definition you use.
• Give a clear mathematical derivation that Lorentzian spacetimes are not continuous and differentiable.
• Give a clear mathematical derivation that Lorentzian spacetimes are not continuous and differentiable in a "spacelike direction".
Last edited by Reality Check; 2019-Dec-12 at 08:41 PM.

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Originally Posted by Rinaldo
There are both conventions, Shaula. Some authors extract the root of the square and some don't.
The latter are fearing the imaginary result, but the physical issue of spacelike spacetime intervals persists.

In particular it says that spacelike spacetime intervals are imaginary (or negative squares).
Congratulations on ignoring the important part of the post. I assume that means you are going to continue to pretend that your misconceptions say something profound rather than just being because of your mischaracterisation of SR/GR.

No one is afraid of negative spacetime intervals, that is how we define the interval between two events as spacelike.

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Originally Posted by Shaula
Congratulations on ignoring the important part of the post. I assume that means you are going to continue to pretend that your misconceptions say something profound rather than just being because of your mischaracterisation of SR/GR.
Sorry, Shaula, I rechecked, but I don't know which part of your last post I had missed. Could you specify please?

No one is afraid of negative spacetime intervals, that is how we define the interval between two events as spacelike.
I claim that the replacement of the spacetime interval ds with the spacetime interval ds2 is a mathematical modification whose only visible purpose is avoiding imaginary results.

Moreover, I claim that this mathematical transformation of imaginary spacetime intervals into real intervals is not sufficiently backed by physical considerations. Even if you call the spacelike spacetime interval real, it will not satisfy your physical needs. This is the same as if I take 1 shoe, and I consider that this 1 shoe is 2 shoes because I need a pair.

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Originally Posted by Reality Check
Yes but: IF06: Give your evidence that states that Minkowski introduced a Lorentzian manifold from the postulates of SR, Rinaldo.
I wrote what the textbooks say. First came SR which showed that space and time were connected by the Lorentz transformation. Minkowski took the next step of uniting space and time into Minkowski spacetime.
(Sorry, up to now I did not see the highlighting, Reality Check): I can agree with your last sentence, I have not analyzed this historical question. In fact, I lose a bit the comprehension what you want to say. I found the following in ##17,18, but I don't understand what you want to say. If there is an important open issue left, please formulate again.
Reality Check said in #17: 1 a) Lorentzian manifolds are nothing to do with the postulates of SR.
Rinaldo answered in #18: What do you mean? Isn't it true that Minkowski introduced a Lorentzian manifold for the description of SR?

I did not write not about your still unsupported world line idea.
What is unsupported?
The existence of continuous worldlines is mainstream.
The existence of Lorentz-invariant proper time is mainstream.
The possibility of different parameters for worldlines is mainstream.
The existence of a threedimensional space manifold is denied by GR mainstream, but it is conform with quantum mechanics.

But you may have a math and physics derivation that have falsified a century of physics so I asked:
There is no mathematical falsification at all. Mathematically, I simply state that spacelike spacetime intervals are imaginary - Period (your only answer is that there are other conventions where the root is not extracted). Saying that "the emperor is naked" is no falsification, it is a statement that everybody can see.

IF04b: Give the textbook definition of the laws of physics that states they are or must be about "mass particles", Rinaldo.
No, the laws of quantum physics are also about vacuum

IF04c: Give the textbook definition of an observer that states they are "mass particles", Rinaldo.
I don't know, but I cannot imagine massless observers.

You are still ignoring my sources that state the textbook physics that Lorentzian manifolds are continuous and differentiable (as required for physics such as GR to work) so:
No need to provide sources, Reality check, according to mainstream Lorentzian "manifolds" are continuous and differentiable, but I am here to show that this assumption is wrong.

IF07:...
Give a clear mathematical derivation that Lorentzian spacetimes do not fit the definition of a manifold. Make sure you cite the definition you use.
Give a clear mathematical derivation that Lorentzian spacetimes are not continuous and differentiable.
Give a clear mathematical derivation that Lorentzian spacetimes are not continuous and differentiable in a "spacelike direction".
Mathematical derivation:
For simplicity, I limit my definition of Lorentzian spacetime to all (flat or curved) spacetimes to which the following squared interval equation does apply:

It is easy to notice that for real spacetimes, spacetime intervals ds are defined only in timelike and lightlike direction, but not in spacelike direction. That implies the lack of spacelike continuity and differentiability, and by consequence, the lack of manifold character.

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Originally Posted by Rinaldo
Sorry, Shaula, I rechecked, but I don't know which part of your last post I had missed. Could you specify please?
Well, lets see. The post was 5 sentences long. One of these was a summary. Two you responded two. The remaining two said:
"This example neatly sums up what you don't seem to understand about a spacetime interval. Spacetime intervals are defined between events."
So that would be a good place to start. Your example of the ruler, which I was replying to, highlights the problem. A ruler doesn't have a spacetime interval of 400. A ruler is not two events. Pretty much everything you have tried to use to claim inconsistencies stems from this misunderstanding. You are treating spacetime intervals as space and time intervals separately by setting dt or dr as 0 and equating the resultant spacetime interval as a space or time interval. That's not how it works.

Originally Posted by Rinaldo
I claim that the replacement of the spacetime interval ds with the spacetime interval ds2 is a mathematical modification whose only visible purpose is avoiding imaginary results.
And I claim it isn't. Not sure why your beliefs about this are relevant, but while we are sharing unsupported claims this is mine.

Originally Posted by Rinaldo
Moreover, I claim that this mathematical transformation of imaginary spacetime intervals into real intervals is not sufficiently backed by physical considerations. Even if you call the spacelike spacetime interval real, it will not satisfy your physical needs. This is the same as if I take 1 shoe, and I consider that this 1 shoe is 2 shoes because I need a pair.
And I claim that all of this is derived from a lack of understanding of what a spacetime interval is and how to apply it. Luckily in this case I actually do have some evidence to back this up in the form of your ruler example and reasoning associated with it.

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Originally Posted by Shaula
A ruler is not two events.
Thank you for your explications, Shaula, for me it was not so clear as you suppose.
I provide the following explications in order to make visible the two ruler events:

An observer is observing a ruler at time t0, he measures a distance between two events: the beginning and the end of the ruler of 20cm at t0.

Pretty much everything you have tried to use to claim inconsistencies stems from this misunderstanding. You are treating spacetime intervals as space and time intervals separately by setting dt or dr as 0 and equating the resultant spacetime interval as a space or time interval. That's not how it works.
Oh no, I wonder what makes you believe this, Shaula. I talk about spacelike and timelike intervals and about the equation

Accordingly, the banal "misunderstanding" you are claiming does not exist at all.

It is true that I used the example of the ruler to show the extreme case dt=0 where the spacetime interval corresponds to a space interval.

And I claim it isn't. Not sure why your beliefs about this are relevant, but while we are sharing unsupported claims this is mine. (...) And I claim that all of this is derived from a lack of understanding of what a spacetime interval is and how to apply it. Luckily in this case I actually do have some evidence to back this up in the form of your ruler example and reasoning associated with it.
My claim is supported
-by the fact that the non-extraction of the root is a convention only - some authors don't extract the root, but some extract the root - example: Landau-Lifschitz, The classical Theory of Fields, the comment to equation 2.4.
-that according to my knowledge, there is no other physical reason provided by literature for not extracting the root.

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Originally Posted by Rinaldo
An observer is observing a ruler at time t0, he measures a distance between two events: the beginning and the end of the ruler of 20cm at t0.
That is not what an event is.

Originally Posted by Rinaldo
Oh no, I wonder what makes you believe this, Shaula. I talk about spacelike and timelike intervals and about the equation
The equation you have made unbalanced by losing a factor of c on the left hand side.

Originally Posted by Rinaldo
It is true that I used the example of the ruler to show the extreme case dt=0 where the spacetime interval corresponds to a space interval.
Which is wrong. Setting dt to zero doesn't make it a spatial interval, it makes it spacetime interval for simultaneous events in a given frame. If you want to calculate the spatial interval then use the geometric distance, or integrate to get a path length in more complex geometries.

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Originally Posted by Shaula
That is not what an event is.
I refer to the definition of Wikipedia, Shaula:
"In physics, and in particular relativity, an event is the instantaneous physical situation or occurrence associated with a point in spacetime (that is, a specific place and time)."

I defined spacetime (the observer's Minkowski diagram with the corresponding simultaneity lines of the observer), the time coordinate (t0) and the space coordinates. What have I wrong?

The equation you have made unbalanced by losing a factor of c on the left hand side.
No c is missing. Landau Lifschitz is wrong in this point because ds has a time unit, there are no spacetime intervals with a space distance unit. This is on the one hand part of my claim of this thread, on the other hand it is supported e.g. by Sexl/ Urbantke: Relativity, Groups, Particles where ds is the proper time.

Which is wrong. Setting dt to zero doesn't make it a spatial interval, it makes it spacetime interval for simultaneous events in a given frame. If you want to calculate the spatial interval then use the geometric distance, or integrate to get a path length in more complex geometries.
I said "where the spacetime interval corresponds to a space interval". I did not make of the spacetime interval a space interval.

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## Differential geometry without manifold

Originally Posted by Reality Check
Physics needs differentiable manifolds. GR needs its differentiable manifolds to be Lorentz invariant. Thus Lorentzian manifolds are compatible with GR.
I found an answer to your question how to apply differential geometry without any manifold, Reality Check:

Differential geometry may apply within Lorentzian spacetime, even if Lorentzian spacetime is no manifold, but it may apply only in timelike direction (!)
There must be cleared two obstacles: my mathematical and my physical derivation.

First obstacle: My mathematical derivation
Mathematically, we saw that there is no spacelike continuity. However, timelike particle worldlines are continuous, and we could fill the whole lightcone with hypothetical worldlines. That means that we can construct a general continuity/ differentiability in timelike direction. You may check by yourself: In a general manner, differential geometry does rarely include spacelike movements which would require speed beyond the speed limit. As soon as we try to include spacelike movements/ spacelike spacetime intervals into our differential geometry, we get trouble (e.g. when we try to foliate spacetime into spacelike hypersurfaces).

Second obstacle: My physical derivation
Physically, we saw that the vacuum between worldlines is not defined by the theory of gravity of GR. How can differential geometry work if there are only worldlines?

For the answer an example: imagine that a particle A is generating a spherical field of a force which is acting on other particles, which is inversely proportional to the square of the distance between the particle and the particle A, and which is propagating with light speed. Suppose that the force acts only in one direction, for simplification. In Newtonian spacetime, the spherical, lightlike field may be described by differential geometry on a continuous manifold. In a Lorentzian spacetime this is not possible because vacuum between worldlines is not defined. However, wherever a particle is found, the action of the force on this particle may be described, and even worldlines of hypothetical particles may be described. Wherever around particle A there are particles, the force may be described.
As a result, by taking into account all worldlines and all possible hypothetical worldlines, we get again a space which is continuous in timelike direction only, and where we can apply the differential geometry.

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Originally Posted by Rinaldo
I refer to the definition of Wikipedia, Shaula:
"In physics, and in particular relativity, an event is the instantaneous physical situation or occurrence associated with a point in spacetime (that is, a specific place and time)."

I defined spacetime (the observer's Minkowski diagram with the corresponding simultaneity lines of the observer), the time coordinate (t0) and the space coordinates. What have I wrong?
You defined two points in space. Your example treated them as one event. What you actually have is "The observer makes a measurement of the location of the start of the ruler at t0 and the observer makes an observation of the location of the end point of the ruler at t0". Two events. The spacetime interval separating them is spacelike, 400cm2. The distance between them is 20cm in their own frame, velocity dependent in the observer frame. No contradictions or problems. Your claimed problem happens because you equate the spacetime interval to a spatial interval.

Originally Posted by Rinaldo
No c is missing. Landau Lifschitz is wrong in this point because ds has a time unit, there are no spacetime intervals with a space distance unit. This is on the one hand part of my claim of this thread, on the other hand it is supported e.g. by Sexl/ Urbantke: Relativity, Groups, Particles where ds is the proper time.
OK, so now you are using non-standard definitions of the spacetime interval too. I'm not going to address that - there's too much else wrong with your claims so far for me to want to delve into this as well.

Originally Posted by Rinaldo
I said "where the spacetime interval corresponds to a space interval". I did not make of the spacetime interval a space interval.
I'm not sure how you think this is logical. Either the spacetime interval is always a spacetime interval and your argument fails or you equate spacetime intervals to spatial intervals and your argument is wrong. Sorry, but however you slice it your argument is either founded on a misunderstanding or flawed logic.
Last edited by tusenfem; 2019-Dec-15 at 11:13 AM. Reason: added missing unquote tag

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Originally Posted by Shaula
You defined two points in space. Your example treated them as one event.
You misunderstood me, I never claimed that this is one event, Shaula, I defined 2 events as the limits of the spacetime interval. What I mean, is simply what you are saying:

Two events. The spacetime interval separating them is spacelike, 400cm2. The distance between them is 20cm in their own frame, velocity dependent in the observer frame.
But its not 400cm2 and 20 cm, its -400cm2 and 20 cm.

OK, so now you are using non-standard definitions of the spacetime interval too.
Sexl/ Urbantke is no non-standard, it is highly appreciated standard literature. - The different spacetime interval of Sexl/ Urbantke and Landau/ Lifschitz shows that there are even more than 4 different conventions for the spacetime interval - it seems that every author has his own concept for the spacetime interval, and we must remember that the spacetime interval is a key element of relativity. Our discussion is strongly suffering from this confusion.

I'm not sure how you think this is logical. Either the spacetime interval is always a spacetime interval and your argument fails or you equate spacetime intervals to spatial intervals and your argument is wrong. Sorry, but however you slice it your argument is either founded on a misunderstanding or flawed logic.
It seems that you misunderstand me. Again, it is simply founded on what you are saying:

"The spacetime interval separating them is spacelike, [I read: minus]400cm2. The distance between them is 20cm"

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Originally Posted by Rinaldo
"The spacetime interval separating them is spacelike, [I read: minus]400cm2. The distance between them is 20cm"
You are right - in the convention you are using I left out the minus sign. My bad.

So if you are happy with this statement then there is no contradiction. Spacetime intervals can be negative. In your ruler example there is no equality between the intervals you mention so no issue there. You have defined the spacetime interval so it is not undefined as claimed in your post 15.

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Originally Posted by Shaula
The equation you have made unbalanced by losing a factor of c on the left hand side.
Here is a simple reasoning why the unit of spacetime intervals is a time unit, not a space distance unit, Shaula:

As an example, we may consider a car traveling at non-relativistic velocity. According to our reference frame, in 1 second, the car is generating a proper time of about 1 second. If we assume that "proper time = spacetime interval", we get a spacetime interval of 1 second. If we follow the equation 2.4 of Landau/Lifschitz, we get a spacetime interval of 300.000 km for every second the car is traveling. According to the convention which does not extract the root of the squared interval, we get 90 billion squared kilometers every second which is not a very intuitive number.

It is obvious that proper time may be considered as the physical interpretation of the line element (supported by Sexl/Urbantke 2.6.), and that the solution "proper time = spacetime interval" provides natural, physical results whatever may be the constellation, in contrast, it seems incoherent to refer to speed of light c when talking about the speed of a mass particle or the speed of a car.

In timelike directions, proper time may be considered as the spacetime interval. This is a very natural and physical concept of the spacetime interval: The interval corresponds to the clock of the particle traveling the concerned interval. Multiplying by c gives a space distance unit, it is the distance light would travel in the corresponding proper time. If the proper time of the spacetime interval is 1 second, the spacetime interval with space distance unit would be 300.000 km. This does not make much sense because a timelike interval describes the path of the particle, not the path of light.

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Originally Posted by Rinaldo
What is unsupported?
What you have asserted about your world line idea is unsupported, thus my questions. You have no evidence whether any of physics works with your idea. It is obvious that your idea throws away all of physics . GR and QM are field theories with quanitities defined at ever point in spacetime. A gravitational field exists everywhere and an object's world line (such as a orbiting body) is its response to the field.

Originally Posted by Rinaldo
The existence of a threedimensional space manifold is denied by GR mainstream, but it is conform with quantum mechanics.
That is wrong.
The existence of a three-dimensional space manifold is acknowledged by the physics mainstream - that is the mathematics basis of classical physics! The facts are that SR and GR need a 4D spacetime manifold.
QM includes SR. QM needs a 4D spacetime manifold.

You seem to not even read the questions which remain as:
IF01: What happens to the Lorentzian manifold according to the definition of a Lorentzian manifold, Rinaldo?
IF02: Give your derivation that Lorentzian manifolds are not compatible with GR using valid math and physics, Rinaldo.
IF03: Give your sources that say that a Lorentzian manifold must have "vacuum" or "void(s)", Rinaldo
IF04b: Give the textbook definition of the laws of physics that states they are or must be about "mass particles", Rinaldo.
IF04c: Give the textbook definition of an observer that states they are "mass particles", Rinaldo.
IF05a: Show that "A threedimensional space manifold is Lorentz invariant" for example the Euclidean space used in classical physics, Rinaldo.
IF05b: Show that "particle worldlines are Lorentz invariant if each of them is parameterized by its respective proper time", Rinaldo.
IF05c: Give your definition of calculus in your "world line only manifold" and show that it works, Rinaldo.
IF07: Give your sources that state "Lorentzian spacetimes are no manifolds, they are not continuous and differentiable, in particular not in spacelike direction", Rinaldo (or give your mathematic derivation of this).
As soon as you wrote "implies" you were wrong!
FYI: Lorentzian spacetimes are defined to be manifolds. Lorentzian spacetimes are continuous and differentiable. This is a reason why Lorentzian manifolds are used in GR - the Einstein field equations are a set of partial differential equations!
Last edited by Reality Check; 2019-Dec-15 at 08:20 PM.

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## The textbook mathematics and physics that this thread denies

Originally Posted by Rinaldo
I found an answer to your question how to apply differential geometry without any manifold, Reality Check:
I did not ask a question . Read what you quote. That is the textbook mathematics and physics that this thread denies yet again.
Originally Posted by Reality Check
It is the mathematics and physics that Lorentzian manifolds are compatible with GR. That statement in your OP is blatantly wrong. Physics needs differentiable manifolds. GR needs its differentiable manifolds to be Lorentz invariant. Thus Lorentzian manifolds are compatible with GR.
The rest of the post is not even wrong.
Basically a "Mathematically, we saw that there is no spacelike continuity." lie when you did no such thing. Writing the wrong equation does not show anything about continuity (this is a spacetime interval). There is an actual mathematical definition of continuity that you have ignored or not learned (hint - it is something to so with differentiation).
Deep "spacetime intervals ds are defined only in timelike and lightlike direction" ignorance when you know what a spacetime interval is and that it is defined in spacelike, null and timelike directions. There are defined spacelike, null and timelike spacetime intervals.
A story about "constructing" continuity/ differentiability only in timelike directions.

ETA: If you have a "thing" that you can mathematically show to be differentiable and Lorentz invariant, Lorentzian manifolds are still compatible with GR. You have to mathematically show that Lorentzian manifolds are not differentiable or not Lorentz invariant. That is impossible because they are defined to be differentiable and Lorentzian. An analogy: a meter has a defined length. It is impossible to make a meter any other length. But this is essentially whet you are trying to do!
Last edited by Reality Check; 2019-Dec-15 at 10:29 PM.

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Originally Posted by Reality Check
What you have asserted about your world line idea is unsupported, thus my questions. You have no evidence whether any of physics works with your idea.
Evidence that physics work: I have the evidence that physics is not working without my idea, Reality Check: Theories of quantum gravity are attempting without success to quantize spacetime.

GR and QM are field theories with quanitities defined at ever point in spacetime.
This is mainstream. As it is not complying with my claim, I felt obliged to explain how lightlike fields may work without spacelike continuity of spacetime.

A gravitational field exists everywhere and an object's world line (such as a orbiting body) is its response to the field.
After further reflection, I think that I can accept the gravitational field (described by spacetime curvature) existing in every point of spacetime, because gravitation is propagating with speed of light. That means (also in compliance with mainstream) that the gravitational field exists everywhere within the lightcone. The geometry of lightlike fields is not spacelike, and I suppose that the whole description of lightlike fields by mainstream does not include any spacelike intervals.

That is wrong.
The existence of a three-dimensional space manifold is acknowledged by the physics mainstream - that is the mathematics basis of classical physics! The facts are that SR and GR need a 4D spacetime manifold.
I said this because I thought it was evident, but it is not directly part of my claim. I wonder how relative spacetime may be compatible with an absolute space manifold. But I don't insist.

QM includes SR.
This is QFT.

QM needs a 4D spacetime manifold.
This is no argument, a 4D spacetime manifold which cannot be curved is useless for QM!

You seem to not even read the questions which remain as:
I answered all these FI-questions. What do you want me to do?

As soon as you wrote "implies" you were wrong!
The lack of definition of spacelike space intervals implies the lack of continuity and differentiability in spacelike direction - what is wrong with that?

FYI: Lorentzian spacetimes are defined to be manifolds. Lorentzian spacetimes are continuous and differentiable. This is a reason why Lorentzian manifolds are used in GR
and
Quote Originally Posted by Reality Check View Post
It is the mathematics and physics that Lorentzian manifolds are compatible with GR. That statement in your OP is blatantly wrong. Physics needs differentiable manifolds. GR needs its differentiable manifolds to be Lorentz invariant. Thus Lorentzian manifolds are compatible with GR.
Mathematically, precisely, there are two possibilities: Either Lorentzian manifolds are not defined because they are imaginary in spacelike direction, or they are mathematically "overdefined": Overdefined means a definition of extreme abstractness and unintuitivity (which may be based on squared intervals avoiding imaginary intervals). Now, I have nothing against definitions of extreme abstractness and unintuitivity, but this would be in contrast to the completely carefree handling of such definition, because no explanation is given, and the issue is covered with a myriad of different conventions on signature, extraction of the root or not, units. This fact is supporting the first possibility that mathematicians are wrong, the Lorentzian manifold is not defined. But also in the second case, the Lorentzian manifold would be in contradiction to the principles of GR.

- the Einstein field equations are a set of partial differential equations!
After the result of the theories of QG it seems that the field equations are not working in spacelike direction. At minimum, concerning the theory of gravity of GR, they should not work as I explained in #36.

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Originally Posted by Reality Check
Basically a "Mathematically, we saw that there is no spacelike continuity." lie when you did no such thing. Writing the wrong equation does not show anything about continuity (this is a spacetime interval). There is an actual mathematical definition of continuity that you have ignored or not learned (hint - it is something to so with differentiation).
Deep "spacetime intervals ds are defined only in timelike and lightlike direction" ignorance when you know what a spacetime interval is and that it is defined in spacelike, null and timelike directions. There are defined spacelike, null and timelike spacetime intervals.
You are citing twice the Wikipedia "spacetime interval", Reality Check, so let's talk Wikipedia, I could not have explained it better:

Wikipedia says: The spacetime interval is the quantity s2 not s itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard s2 as a distinct symbol in itself, rather than the square of something.
This is the mainstream interpretation, but the interpretation is poor: "Rather deal with" imaginary numbers, physicists "customarily" are simply putting a big patch on the issue. I like science, I believe in physics, but this is simply very poor. And no one will contest what Wikipedia says here because it corresponds to the reality of our mainstream physics.

Mathematically, it is not possible to define on such a basis a Lorentzian manifold in compliance with GR.

An analogy: a meter has a defined length. It is impossible to make a meter any other length. But this is essentially whet you are trying to do!
I simply say that spacelike spacetime intervals are imaginary, I know that I am no mainstream, but I am simply leaving the result as it is while mainstream is using a trick to change it (see the Wikipedia citation above).

20. Rinaldo, it is about time that you start presenting something real. We have now two pages of discussion whether s or s2 should be used by mainstream, but there is NOTHING that you write about the rest YOUR theory. I see no real purpose in keeping this thread open. Please, start discussing someting significant, including the math, of your theory in your following posts. I think we have chewed enough on this s-s2 topic.

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Originally Posted by tusenfem

Rinaldo, it is about time that you start presenting something real. We have now two pages of discussion whether s or s2 should be used by mainstream, but there is NOTHING that you write about the rest YOUR theory. I see no real purpose in keeping this thread open. Please, start discussing someting significant, including the math, of your theory in your following posts. I think we have chewed enough on this s-s2 topic.
Thank you for your feedback, Tusenfem.
Concerning the s-s2 topic, I think that all has been said, but personally, it is not my impression that two pages have been wasted for the s-s2 conventions. All the contrary, I would like to thank here the questioners for their critical questions on a high level which are helping me enormously with the clarification of my theory. It is true that we have talked much about conventions in the mathematical part, and perhaps we can now talk about mathematical arguments which could be opposed to my theory.

Here is a new start of the thread, I hope that it is interesting enough for this forum.

At stake: This thread is about why theories of quantum gravity are not working. It is a quite important topic because much scientific manpower is currently bound by the attempts to quantize spacetime on the one hand and by the research for new spacetime structures for the modification of GR on the other hand. - According to my theory, all this is unnecessary because GR is harmonizing very well with quantum mechanics, it is only our concept of Lorentzian spacetime which is not entirely complying with GR.

It took me several years to come to a result because at first sight, everything seems perfectly without flaw, and the maths of GR are perfectly corresponding to experimental evidence. In this thread I show that nevertheless there is an incoherence of the maths of GR at a very precise point - that is my theory. The incoherence is hard to see because of the different conventions we discussed in extension, but once all conventions have been put aside it is hard to contest.

Mathematically, my derivation is very short and clear, as I have shown (see #15): Lorentzian spacetime is real, but spacelike spacetime intervals are imaginary, that means that they do not exist. My theory says that this is a clear mathematical flaw. This is the whole maths.

Wikipedia confirms indirectly that there is a problem, by saying that negative square roots are avoided by different conventions. And accordingly, the questioners put forward conventions, but I think that conventions cannot be used as the only arguments against my mathematical derivation. Now my theory should be appreciated with respect to mathematical arguments.

Physical arguments are corroborating the mathematical result, as shown in ##12 to 14, in particular, vacuum between worldlines is not defined by GR but by quantum physics.

22. this is nothing different and will lead to the exact same discussion about s-s2 and how to interpret imaginary (mathematical meaning, not general language meaning as "non existent/only in your mind") results.

closed for moderator discussion

23. Okay, after a bit of discussion, I think this is the best way to go here.
We start off from scratch and now ONLY the "real manifold" is going to be discussed, and looked at how this works for GR.
There will be no more discussions about the usual manifold, and the questions whether there are real or imaginary distances and all that stuff.
rinaldo will show how GR works with only real numbers.
Happy discussion!

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My thread is no theory in the sense of an alternative idea, it concerns the weak point of many current theories of quantum gravity.

My thread seems to concern the only domain of physics which seems to be governed by a bunch of different conventions and concepts without anybody worrying about possible consequences. I cannot explain why there is such a lack of interest among physicists, and in the same way I don't know why there is such a lack of interest in this forum.

Theories of quantum gravity are becoming more and more ingenious, but without success, and it is my prediction that there will be no theory of quantum gravity possible without clearing up the sign conventions and abandoning the concept of continuous Lorentzian manifolds.

If there is no further interest in discussion, I hereby ask to close this thread.

25. Originally Posted by Rinaldo
My thread seems to concern the only domain of physics which seems to be governed by a bunch of different conventions and concepts without anybody worrying about possible consequences. I cannot explain why there is such a lack of interest among physicists, and in the same way I don't know why there is such a lack of interest in this forum.
The trouble is that your thread is based on a series of misunderstandings of current theory.

There is a great deal of interest among physicists (and probably on this forum) in theories of quantum gravity. Some of which require a replacement of GR with something where spacetime is not continuous. Or, in some cases, doesn't exist at all as the background of the theory, but only emerges as a consequence of the theory.

So it appears you are making a strawman argument by insisting that you are the only person who is think that GR, with its continuous spacetime, may need to be replaced. In fact many people, with the requisite expertise, are thinking about this problem.

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The "lack of interest", Strange, is referring to the diverging sign conventions and the problems which are hidden by them (imaginary spacelike spacetime intervals), this is precisely what we talked about in this thread, physicists seem not to be interested to clear up this topic, and I learned with regret that also in this forum, this is not considered any longer to be an interesting topic.

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Originally Posted by Rinaldo
My thread seems to concern the only domain of physics which seems to be governed by a bunch of different conventions and concepts without anybody worrying about possible consequences. I cannot explain why there is such a lack of interest among physicists, and in the same way I don't know why there is such a lack of interest in this forum.
As Strange says - it is because most of your arguments are based either on misconceptions or unsupported assertions.

If GR and SR are so deficient as you claim then this should be an easy challenge:
Please show, in detail and in full, an SR or GR calculation using a Lorentz manifold that leads to a direct contradiction with observations that is due to one of your claims about the deficiencies of a Lorentz manifold. Not just with words, not just with arguments from incredulity - show us a calculation you have made that directly exposes this weakness by contradicting observations.
Please show, in detail and in full, how your assertion that worldlines are the only valid thing in GR or SR fixes this inconsistency.

28. Originally Posted by Shaula
As Strange says - it is because most of your arguments are based either on misconceptions or unsupported assertions.

If GR and SR are so deficient as you claim then this should be an easy challenge:
Please show, in detail and in full, an SR or GR calculation using a Lorentz manifold that leads to a direct contradiction with observations that is due to one of your claims about the deficiencies of a Lorentz manifold. Not just with words, not just with arguments from incredulity - show us a calculation you have made that directly exposes this weakness by contradicting observations.
Please show, in detail and in full, how your assertion that worldlines are the only valid thing in GR or SR fixes this inconsistency.

Okay, let me make that official. rinaldo, it is time to put your money where your mouth is, and this is a good start. Please proceed with an enlightening reply, including the math.

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Originally Posted by Shaula
Please show, in detail and in full, an SR or GR calculation using a Lorentz manifold that leads to a direct contradiction with observations that is due to one of your claims about the deficiencies of a Lorentz manifold. Not just with words, not just with arguments from incredulity
It is curved spacetime which is in direct contradiction with observations (of quantum mechanics), Shaula. This is the result of attempts to quantize curved spacetime, it is the current state of mainstream

The mathematical calculation leading to the direct contradiction is shown in my #15 - imaginary results in real spacetime are invalid - no additional calculation is required (if you do not agree please tell me what is missing, but an invalid result is an invalid result).

show us a calculation you have made that directly exposes this weakness by contradicting observations.

This is shown by the theories of quantum gravity which are trying to foliate spacetime into spacelike hypersurfaces, they are failing, and sometimes they claim that GR must be modified, in spite of the experimental confirmation of GR.

Please refer to these attempts of the theories of quantum gravity, I am not able to make similar sophisticated calculations which are leading nowhere.

Please show, in detail and in full, how your assertion that worldlines are the only valid thing in GR or SR fixes this inconsistency.
Worldlines of particles and fields are timelike and lightlike. The theory of gravity of GR refers only to worldlines (vacuum is not defined by it), that means that it is not based on spacelike intervals.

It seems that, at the exception of the theories of quantum gravity, the whole relevant physics of the theory of gravity of GR concern only timelike and lightlike worldlines such that there is no issue.
Last edited by Rinaldo; 2019-Dec-23 at 09:10 PM.

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Originally Posted by tusenfem

Okay, let me make that official. rinaldo, it is time to put your money where your mouth is, and this is a good start. Please proceed with an enlightening reply, including the math.
I tried my best, Tusenfem.

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