Time and Lorentz transformations

[trees and plants]

Hello, why time is the fourth dimension and not another quantity or variable? General relativity has as a special case the special relativity, so Lorentz transformations are contained in general relativity but are they in a more general form than that of special relativity generally? If they are, what does that have as a consequence for the effects of gravity on matter or energy? Thank you.

 

[Orodruin]

Using time and position to describe events long predates special relativity. The new insight in special relativity is that time and space are much more intimately connected than previously believed.

Lorentz transformations are a special case of general coordinate transformations on Minkowski space, i.e., the flat Lorentzian space that describes special relativity. In general relativity, spacetime is instead a general (possibly curved) Lorentzian manifold. Gravity is in essence described by the curvature of the spacetime and this is generated by the stress-energy tensor of the matter in the spacetime.

 

[trees and plants]

So, why time transforms and not another quantity or variable? What are the connections of time and space in special relativity? Another question i have is: is curvature in general relativity described by the Ricci tensor, the Riemann curvature tensor or the Einstein tensor?

I think that in a vacuum the Ricci tensor is equal to 0 and this is where Einstein started generating his derivation of his field equations?

 

[trees and plants]

Lorentz transformations have their proof for their derivation? I think i read somewhere about this.

 

[Orodruin]

So, why time transforms and not another quantity or variable?

You have to define what kind of answer you are looking for or you can go forever down the rabbit hole of asking why not this? The ultimate answer is that it gives a good description of nature as far as we are aware and other variables do not. Spacetime is a thing already from Galilean relativity and is ultimately the result of wanting to describe where and when an event occurs.

Another question i have is: is curvature in general relativity described by the Ricci tensor, the Riemann curvature tensor or the Einstein tensor?

All of those are to some extent related to the curvature of the spacetime so the question is a bit semantical. If you know what GR says, you really do not need to ask this question. If you would just say "curvature" I would take it to refer to the curvature tensor itself, but the Ricci and Einstein tensors are of course both related to it and what appears in the Einstein field equations is the Einstein tensor. However, it should then mostly be seen a second order non-linear differential equation for the metric tensor, which in turn determines the curvature tensor and all related quantities.

 

[trees and plants]

You have to define what kind of answer you are looking for or you can go forever down the rabbit hole of asking why not this? The ultimate answer is that it gives a good description of nature as far as we are aware and other variables do not.

I think that sometimes to better understand something in math a proof that shows how something is derived is what is needed. You can see and answer if you want post #4.

 

[Orodruin]

I think that sometimes to better understand something in math a proof that shows how something is derived is what is needed. You can see and answer if you want post #4.

But GR is not math. It is physics. You do not "prove" things in physics, you define a model and make experimental observations that determine whether or not the model is an accurate description. Sure, you can derive the Lorentz transformations based on a number of postulates, but that in no way "proves" them. They are not even central to special relativity in my mind (what is central is the description of spacetime as a 4-dimensional flat Lorentzian manifold, i.e., Minkowski space). Lorentz transformations are to Minkowski space what rotations are to Euclidean space - just coordinate transformations between different orthonormal coordinate systems.

 

[trees and plants]

But GR is not math. It is physics. You do not "prove" things in physics, you define a model and make experimental observations that determine whether or not the model is an accurate description.

My personal opinion:

Proofs is one of sciences main and fundamental element among others. Proofs give answers to questions or problems in sciences. With proofs understanding comes in sciences. This applies in physics too. If a person can not prove something like an axiom he accepts it as an axiom. How can someone understand a statement in physics without proof? He can learn it and accept it as proved if it has a proof, but can he else understand it?

 

[Orodruin]

My personal opinion:

Proofs is one of sciences main and fundamental element among others. Proofs give answers to questions or problems in sciences. With proofs understanding comes in sciences. This applies in physics too. If a person can not prove something like an axiom he accepts it as an axiom. How can someone understand a statement in physics without proof? He can learn it and accept it as proved if it has a proof, but can he else understand it?

But personal opinion does not matter here. It is not how science works. You simply cannot prove things in empirical science. You can collect evidence that either falsifies a theory or not. This is not the same as proving that it is ultimately a true theory. This is a fundamental cornerstone of empirical science.

 

[trees and plants]

What about the other question i made about the Lorentz transformations and if they have a more general form in general relativity? Do they have?

 

[Orodruin]

What about the other question i made about the Lorentz transformations and if they have a more general form in general relativity? Do they have?

Again, Lorentz transformations are not central to special relativity. You can in principle make any coordinate transformations you want in either SR or GR, it is just that Lorentz transformations will be transformations between particular coordinate systems on Minkowski space. Whether special transformations exist in the GR case depends on the spacetime symmetries.

 

[Dale]

why time is the fourth dimension and not another quantity or variable?

We have found that the quantity $ds^2= -c^2 dt^2+ dx^2 + dy^2 + dz^2$ is invariant in physics. Due to this formula’s close analogy to the standard Euclidean invariant length formula ($ds^2=dx^2+dy^2+dz^2$) we call time the 4th dimension. No other quantity or variable satisfies this or a similar relationship with the 3 dimensions of space.

 

[trees and plants]

We have found that the quantity $ds^2= -c^2 dt^2+ dx^2 + dy^2 + dz^2$ is invariant in physics. Due to this formula’s close analogy to the standard Euclidean invariant length formula ($ds^2=dx^2+dy^2+dz^2$) we call time the 4th dimension. No other quantity or variable satisfies this or a similar relationship with the 3 dimensions of space.

So, so far only time and space coordinates satisfy this or a similar relationship in physics. Thank you.

 

[trees and plants]

We have found that the quantity $ds^2= -c^2 dt^2+ dx^2 + dy^2 + dz^2$ is invariant in physics. Due to this formula’s close analogy to the standard Euclidean invariant length formula ($ds^2=dx^2+dy^2+dz^2$) we call time the 4th dimension. No other quantity or variable satisfies this or a similar relationship with the 3 dimensions of space.

Somewhere i read about the line element. Does this formula work like the norm squared? Perhaps they derived this formula from other results?

 

[Dale]

Somewhere i read about the line element. Does this formula work like the norm squared? Perhaps they derived this formula from other results?

I can’t help with “somewhere I read”. Can you cite the source?

 

[trees and plants]

I can’t help with “somewhere I read”. Can you cite the source?

These are some links https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity) , https://en.wikipedia.org/wiki/Line_element but i read it if i remember correctly from a pdf about general relativity, i think they were lecture notes.

 

[FactChecker]

Hello, why time is the fourth dimension and not another quantity or variable?

The short answer:
It all started with the speed of light and speed is distance divided by time. So there it is. Why should another variable be considered?

 

[Dale]

These are some links https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity) , https://en.wikipedia.org/wiki/Line_element but i read it if i remember correctly from a pdf about general relativity, i think they were lecture notes.

The first reference says “the metric determines the invariant square of an infinitesimal line element, often referred to as an interval.” And later in that article it used the formula I posted earlier. Did you have any questions about the article? I didn’t see any references to norms.

 

[Nugatory]

Somewhere i read about the line element. Does this formula work like the norm squared? Perhaps they derived this formula from other results?

This would be a good time for you to back up some and learn the minimum amount of differential geometry needed to take on General Relativity. Every serious textbook (including Sean Carroll, free online at https://arxiv.org/abs/gr-qc/9712019) will cover this material early on, and will rederive SR as a special case of GR.

 

[trees and plants]

The first reference says “the metric determines the invariant square of an infinitesimal line element, often referred to as an interval.” And later in that article it used the formula I posted earlier. Did you have any questions about the article? I didn’t see any references to norms.

I meant it is like the Euclidean norm.

 

[Dale]

I meant it is like the Euclidean norm.

It is indeed very much like it in some ways and substantially different in others. The biggest difference is that it is not positive definite. For physics that is useful because it indicates if an interval is measured by clocks or by rulers.

 

[trees and plants]

The short answer:
It all started with the speed of light and speed is distance divided by time. So there it is. Why should another variable be considered?

This is related to the phenomena near the speed of light? What do you mean? I am sorry because i do not know or i do not understand.

 

[FactChecker]

This is related to the phenomena near the speed of light? What do you mean? I am sorry because i do not know or i do not understand.

I am referring to SR. Under a few natural and desirable assumptions, there are only two mathematical possibilities.
A paper by Polash Pal, "Nothing but Relativity" shows that the assumption of relativity will only allow Galilean relativity (all inertial frames share a universal time) or Einseteinian relativity (constant speed of light in a vacuum).
(See https://arxiv.org/abs/physics/0302045 )