B Does Spacetime Have Physical Existence?

[Uncle Thi]

His "explanation" is wrong and should be ignored.

There is no such thing.

Not surprising since there is no such thing.

Here is my question:

In Einstein's field equation:

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$

where is the energy-momentum tensor representing matter.
⟹ Spacetime is curved by matter (), but...

My question: If spacetime is not matter, how can this equation be valid when it requires both sides to be mathematically consistent in their physical nature?
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Is the above question considered a popular science question?

 

[PeterDonis]

Spacetime is curved by matter

Or radiation, or anything else that is included in the stress-energy tensor. Yes, that's what the EFE describes: how spacetime is curved by stress-energy. The LHS is the Einstein tensor, which describes spacetime curvature. The RHS is the stress-energy tensor, which describes the matter and other stuff that causes the curvature.

If spacetime is not matter, how can this equation be valid when it requires both sides to be mathematically consistent in their physical nature?

What does "mathematically consistent in their physical nature" even mean?

Is the above question considered a popular science question?

No, it's considered a vague and ill-defined question.

 

[Orodruin]

My question: If spacetime is not matter, how can this equation be valid when it requires both sides to be mathematically consistent in their physical nature?

If electric field is not charge, how can Maxwell’s equations be valid if they require both sides to be mathematically consistent in their physical nature?

The above is the corresponding question for electromagnetism. I hope you are not suggesting electric fields are made of charge.

Is the above question considered a popular science question?

No. It just does not make sense. Both sides of an equation in physics need to be dimensionally consistent. That does not mean they represent the same thing.

 

[Uncle Thi]

Your question is not relevant to the current discussion. Often, as here, when a post contains multiple questions people will naturally focus on the more important ones, leaving less useful ones unanswered.

However, in the interest of directness:That is correct.

It is irrelevant since we are not discussing a “theoretical model without any physical quantities”.

Let me also ask you a direct question: do you believe that the statement “the table top is flat and the legs are perpendicular to the table top” is a physical statement which could be experimentally observed and measured?

Uncle Thi: A theoretical model without any physical quantities cannot be considered a physical model. Is that correct?

Dale: That is correct.

But it is irrelevant since we are not discussing a “theoretical model without any physical quantities.”
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I think it is relevant: When you confirm that a theoretical model without any physical quantities cannot be considered a physical model.

Does this mean that spacetime is merely a mathematical model? Because I also cannot find any physical quantities in Einstein’s field equations.

Is that really the case? Am I mistaken?

 

[Ibix]

Because I also cannot find any physical quantities in Einstein’s field equations.

They constrain the metric, which is the quantity that defines distances and angles, to depend on the stress-energy tensor, which describes the matter and radiation content of a region. The angle between two lines or the length of a line is certainly relevant to physics, dictating things like whether two things will meet, or whether one needs to apply a force to stop them separating.

You seem to have categorised parts of the field equations in an arbitrary manner and are now finding that your arbitrary classification leads you into problems. The problem is your expectation that your arbitrary classification should have any consequence.

 

[martinbn]

The problem is that you think you are having a scientific discussion, but you are not. You are trying to have a discussion about the philisophy of science. The problem with that is that one needs to be familiar with the science itself first.

 

[Uncle Thi]

I have no idea how you think you can tell if something "objectively exists", especially via experiment (which is what science is primarily about), so I won't address this part of your question. You're on your own, enjoy.

I think things might go better if we discuss Euclidean geometry for a moment, and once we have a common understanding or an agreement to disagree about that, we can revisit relativity. At least I'll attempt to -as I wrote this, things started to branch out a bit.

One can measure distance with a ruler. So, I'd assume you would share the belief that it's a physical quantity. "Entity" seems to be a bit on the wrong track to me, I don't usually think of distance as an "entity". But that may be just the way I use the words.

I would say that geometry, which I think of as the study of distance, is more of a mathematical structure. So, we have "physical" distance, and the theoretical structure of Euclidean geometry that organizes it. Note that we have some closely related alternatives to Euclidean geometry, for instance spherical and hyperbolic geometries. And, somewhat importantly for later on, we have Riemannian geometry.

Now, let's move on to relativity.

The organization principle of special (and general) relativity basically replaces "distance" with the "Lorentz interval".

So, the obvious the next question would be - is the Lorentz interval real? An argument for, is that it's the same for all observers. Distance, in special relativity, is NOT the same for all observers. Proper time, the sort of time one measures with a clock, falls into the category of a Lorentz interval, and proper distance, the sort of distance one measures with a ruler, also falls in the category of the Lorentz interval. So I would put proper distance, proper time, and the Lorentz interval all into the "physical" category.

As an aside, I view most tensor quantities as "existing". This may not help if you are not familiar with tensor quantites. The key point of tensor quantities is that while they may have components that depend on the observer, the structure as a whole has rules that allow these components to be transformed between observers. Sorry if you're not familiar with tensors, this may be moot. If you are familiar and have some thoughts about whether tensors are "real", you could consider sharing these thoughts in the intersts of communication and discussion.

Going back to the Lorentz interval - I imagine one could come up with an argument that The Lorentz interval is a mathematical construct of some sort, but that's not my view. While one could argue about that, my personal reaction is that it is the sort of argument that doesn't go anywhere I find interesting, it's mostly about semantics.

This discussion may not help if you're not familiar with the Lorentz interval. Sorry about that if that's the case. Taylor & Wheeler's "space-time physics" talks about them a lot, I have always found "The Prable of the Surveyor" to be particular helpful, for whatever that's worth. Realistically, though, if you're not already familiar with the term and its implications, it would probably be a huge digression to talk about it. If you're interested in relativity, it'd be worth your time, it's just wouldn't be appropriate for this thread.

We can regard the organizational principles of the Lorentzian geometry, the geometry of the Lorentz interval, in the same category that we put Euclidean geometry, the study of distances. Special relativity is then a particular case of the geometry of the Lorentz interval, one that applies to "flat" spacetimes. General relativity opens things up to different geometries. General relativity basically applies Riemannian geometry to the Lorentz interval. (Some purists call it pseudo-Riemannian geometry for technical reasons).

The fact that one can apply the structure of a geometry (Euclidean or Riemannian) to multiple physical concepts is one argument for why I think of them as being organizational structures rather than physical entities.

So to recap the points I think are most important. "Distance" and "Lorentz intervals" are IMO physically measurable quantities which exist independent of the observer. The organization of either into a geometry is more mathematical. The same math can be used to organize different things, the math assumes some basic axioms, if the axioms fit the physics, we apply the structure of the math to the physical objects to draw conclusions.

Did Newton need to borrow mathematical formulas from others to explain the Law of Gravitation? No! I don't think so. Newton did not borrow mathematics from others to explain the Law of Gravitation in the way that Einstein borrowed Riemannian geometry to develop General Relativity.

1. Newton developed his own mathematics to describe gravity

When formulating the Law of Gravitation, Newton did not rely on any pre-existing geometric system.

He independently developed an entirely new mathematical framework: Calculus, along with the principles of differentiation and integration to describe motion.

Newton’s law required only basic arithmetic, Euclidean geometry, and algebra, all of which have a direct relationship with physical reality.


> Newton did not impose a mathematical model onto reality; he used mathematics to describe a real phenomenon in the physical world.


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2. Newton introduced gravity as a physical entity that could be tested

Newton's Law of Gravitation is not just a theoretical mathematical model. It can be directly tested by measuring the force between two masses.

The formula applies to all massive objects, and this force can be measured experimentally.

He did not need curved space geometry or spacetime; he only required basic concepts of force and mass.


Newtonian gravity has a clear physical foundation and is measurable, unlike Einstein’s gravitational model, which is purely geometric and lacks a direct physical entity representing spacetime.


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3. Einstein had to rely on pre-existing mathematics

Einstein did not develop Riemannian geometry; instead, he used it as the foundation for General Relativity.

In reality, Einstein did not derive an independent gravitational formula; he merely modified Riemann’s equations to fit his postulates.

There is no experimental tool to measure the curvature of spacetime directly; it is only interpreted indirectly through mathematical models.


General Relativity depends on an existing mathematical framework and cannot independently verify its own physical reality.


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✔ Newton: Developed his own mathematics to describe a real phenomenon.
✔ Einstein: Used pre-existing mathematics to create a theoretical model without a physical entity.
✔ Newtonian gravity is measurable, whereas Einstein’s spacetime is not directly measurable.

Newton did not need to borrow mathematical formulas from others to explain gravity, but Einstein needed Riemannian geometry to construct spacetime.

 

[Dale]

I think it is relevant: When you confirm that a theoretical model without any physical quantities cannot be considered a physical model.

We are not discussing such a model. Every tensor in the EFE is a physical quantity.

Does this mean that spacetime is merely a mathematical model? Because I also cannot find any physical quantities in Einstein’s field equations.

Is that really the case? Am I mistaken?

You are very mistaken. All of the tensors in the EFE are physical quantities.

You did not answer my direct question. So I will do so myself, not asserting that it is your answer, just the correct answer.

do you believe that the statement “the table top is flat and the legs are perpendicular to the table top” is a physical statement which could be experimentally observed and measured?

Yes, it is a physical statement. Geometry is part of physics. There is a physically measurable geometrical difference between a flat table top and a bumpy one. There is a physically measurable geometrical difference between perpendicular legs and non-perpendicular. Geometry is part of physics.

Spacetime is the geometry of physics.

 

[PeterDonis]

Since the OP's question has been thoroughly answered, this thread is now closed. Thanks to all who participated.