Did Einstein prove the existence of a fourth spatial dimension?

I'm no expert and still trying to understand GR. So, take what I say with a grain or two of sodium chloride.

With inertial frames of reference, it is obvious that momentum and energy have to be the same thing, presented in different ways relative to which frame you reference. And you probably know that stress, energy, and momentum are basically all the same thing, presented differently depending on which frame of reference (inertial or non-inertial) we happen to be using.

This is all fairly straightforward because of the necessary symmetry and requirement that the results do not depend on the frame you choose.

And if you need a lecture on the difference between mass and rest mass, then you really need to go back to Special Relativity and review it.

We talk about "shape" in ordinary conversation to describe the external surface of an object. This is completely wrong when thinking about fields and space and space-time and is useless. Rather in these areas "shape" is the infinitesimal curvature of the fabric we call space-time. (with obvious classical analogies for fields). So, from the most crude way of looking at it we have completely flipped the meaning of 'shape; from the external and superficial to the intrinsic internal structure.

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You claim that a mass, with a given rest mass, m° will be larger as the space-time curvature increases (under a higher gravitational "influence"). Hmmm. Well mass, energy, momentum and stress are conserved and since stress is HIGHER under higher curvature, then mass must be less for the sum to be conserved (everything else being equal)...What am I missing? I believe you are wrong, but I am not conversant with the tensoral field equations to have much confidence in my belief/suspicion. Perhaps you can cite the source of your claim?

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Now on to your question about spatial dimensionality. It IS possible to envision a 2-dimensional surface embedded in a 3-dimensional space, and by analogy to envision a 3-space embedded in a 4-dimensional hyperspace. You may know, probably do know, that rather than using all sorts of different names/descriptions for these things, we call them n-spaces, ie 2-space, 3-space, 4-space.

So, one way to describe the surface of a sphere (2-space) is that it is embedded in a 3-space and defined by the points which obey the equation K = x² + y² + z². It is easy to understand this and convenient. BUT. It is NOT necessary to use a 3-vector to describe the properties of this surface (this 2-space), a completely equivalent description is to define the curvature at each point as constant and positive and of a magnitude which is of course related to K. it seems to be a bit more abstract to do it this way, but it has the significant advantage that we only need to consider 2-vectors. There is no need to invoke higher dimensional spaces.

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I am a bit puzzled by your apparent confusion about 4-D space-time. The geodesic ds² = -c²dt² + dx² + dy² + dz² is the definition of distance in 4-D space-time (in our pseudo-Riemannian space-time) and is the shortest line between two events (two points). This basic equation displays the EQUIVALENCE of time and space. The four dimensions are all space-time dimensions. Two observers may differ in their frames of reference in one or more of them, but in an absolute way, there is no difference between them. The units of "distance" can be miles (or megalightyears) or can be seconds. They are just different units of the same underlying quantity.

IF the metric were ds² = +c²dt² + dx² + dy² + dz² then space-time would be Euclidean and flat. How boring.

Specifically answering your question: there is no need to invoke another spatial dimension. The four dimensions of space-time are all that are needed. But because of the curvature, we can add another dimension to account for it, I do not think that that dimension would be spatial, it would be spatial-temporal just like the other four dimensions of our Universe.

" I can't quite wrap my head around an increase in mass under the same influence, but I can take it as a given."

You shouldn't. It doesn't.

"So am I correct in assuming that if a 2D spacetime is bent in a third dimension, our 3D spacetime is bent in a fourth spacial dimension?"

No. They presented you with the rubber sheet analogy, which uses gravity to explain gravity. There is no 4th spatial dimension. That analogy needs to be forgotten just shortly after it being presented.

Near mass, there are more vectors that point towards the mass center, than away from it. That is the spatially curved part of curved spacetime.

"When I fall while skydiving, am I really gliding down fourth-dimensional chute?"

Without air, you would would be confined to a 4D spacetime geodesic, that any mass with your momentum would follow, and without external force would follow.

"If that is indeed the conclusion to be drawn from Einstein's theory, why is he commonly just credited with discovery of the fourth dimension as time?"

That is not the conclusion.

nicely, in case you outline the universe as (a pantheistic) God, why difficulty including the more suitable note? Neither Einstein nor his discoveries gave any info of any god. Your note video games do not create any info both. only because matter and ability could be interchanged would not make any god actual. For thousands of years, human beings have pronounced that their gods were behind what they did not comprehend -- existence, lightning, stars, earthquakes, the muse of existence, the international or the universe, and so on. Positing a god to supposedly answer a question solves no longer some thing. It only delivers an unwarranted element of complexity and forestalls you from asking more suitable questions. "It grow to be, for sure, a lie what you analyze my non secular convictions, a lie it fairly is being systematically repeated. i do no longer have self assurance in a own God and that i have never denied this yet have expressed it obviously. If some thing is in me that could be talked about as non secular then it truly is the unbounded admiration for the kind of the international so some distance as our technological know-how can teach it." — Albert Einstein, 1954, Albert Einstein: The Human section

Mass does not increase with increasing speed. Nor does it increase as it approaches the mass, energy or momentum (or stress) that is the source of space time effects that result in those properties we attribute to gravity force: acceleration and attraction.

Inertia increases with relative speed, v, as in M = m/sqrt(1 - (v/c)^2) = m/L(v/c). Mass, rest mass m, stays the same as it was when at rest. Thus the name...rest mass. 1/L(v/c) is called the Lorentz Transformation.

But as a platform increases its speed v --> c, M --> infinity. And from A = F/M that means it takes incrementally more and more force to keep the mass m accelerating as its inertia M grows. Eventually as the speed increases, there will come a point where the inertia is so great that there is insufficient energy to accelerate the rest mass one iota more to reach v = c. And thus the speed limit.

Time slows down as we approach a gravity source. And space becomes extremely warped as well. But mass, in the GTOR tensor relationship is an independent variable, along with the energy and momentum. And the dependent results are spatial vectors (non linear unfortunately), adjusted time, and gradients. No mass or inertia that I know of.

As to your 2D projection of the 3D object, I have no clue what you are talking about. How does the 2D projection lend itself to a 3D bent space? I'd have to see the documentary; perhaps you can cite it.

I guess U can say that, if U wanna sound fancy... But an answer is simple, Gravity is a ***** !

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hope it helps

/Uniontera Ja