B Can you explain GR without math?

[alba]

Is it possible to explain themain ideas of GR without complicated math?
In particular,
what is the relation of current model of universe (FLRW) and GR?
what is the difference between spacetime and universe, spacetime is surely curved and universe can by flat at same time?
both mass and energy bend spacetime, in what proportion? what is the max quantity of mass allowed, due to what? I read that neutrinos can't have more than 2 eV of mass-energy because there would be too mauch mass in the universe, it that is true, why so?

 

[Andrew Kim]

Sure, it's possible to explain GR in words, but the words will essentially be replacements for or simplifications of the math.

1. You mentioned the acronym FLRW, which is a solution of the principle equations of GR that holds for an isotropic expanding fluid. That essentially means that the FLRW solution describes the universe using the framework of GR.
2. Spacetime is a well defined concept: in simplified form, it's the fabric of the universe on which matter exists. The universe is a more general concept. It's unconventional to describe the universe as curved, because we normally simply replace the word 'universe' with 'spacetime' when talking about GR and curvature. However, I guess you could say the universe is curved.
3. Any quantity of mass can technically exist in the universe. The Einstein equations read: $$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{{c}^4}T_{\mu\nu}$$ This essentially means:
curvature= $\frac{8\pi G}{{c}^4}$ matter/energy density
So you could loosely say that $\frac{8\pi G}{{c}^4}$ is the constant relating matter and curvature. Our description of spacetime runs into problems when the curvature is infinite, and if you want the curvature to be infinite, the mass/energy density must be infinite. Therefore, there is no 'max quantity' of mass or energy that a universe simply can't hold. What you read about neutrinos is probably referencing the fact that, if neutrinos were heavier, then perhaps the universe could have recollapsed a short time after the big bang. I have no knowledge whether that is accurate or what motivated the statement about neutrinos.

 

[alba]

3. Any quantity of mass can technically exist in the universe. The Einstein equations read:
So you could loosely say that $\frac{8\pi G}{{c}^4}$ is the constant relating matter and curvature. .

Thanks, that is wery helpful, can you expand on that, which is the core of GR?
Can you describe what the individual tensor mean, and how/why are they related?
What does the constant (roughly 10^-44) mean?
Why do they currently say universe is flat, if it is surely curved?

What is the expanding fluid, mass-energy or spacetime?

 

[Andrew Kim]

The Ricci Tensor ($R_{\mu\nu}$) is a tensor that represents curvature. It was invented by mathematicians, and satisfies a number of useful properties. The tensor $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is called the trace reverse of the Ricci Tensor, and satisfies the following very important property: It's components represent the same geometric object all over spacetime. It's like having a vector field whose vectors are parallel everywhere.

The Stress Energy Tensor ($T_{\mu\nu}$) is a tensor whose components represent energy and momentum in spacetime. Roughly speaking,
$T_{00}$ = Mass/Energy
$T_{01}, T_{02}, T_{03}$ = Momentum
$T_{11}, T_{12}, T_{13},T_{22},T_{23},T_{33}$ = Stress
(The Stress Energy Tensor is symmetric, which means that $T_{\mu\nu} = T_{\nu\mu}$, and so $T_{10}$ is the same as $T_{01}$.)
The generalization of conservation of Energy and Momentum into General Relativity is the fact that this tensor is also geometrically invariant, just like the Ricci Tensor. Einstein proved that this means we can say:
$$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu} = kT_{\mu\nu}$$
where $k$ is some constant. If you solve these equations for the case of a weak gravitation field, you retain the Newtonian equations of gravity if and only if $k=\frac{8\pi G}{c^4}$. However, physicists can perform special tricks that get rid of the $G$ and the $c$, so long as they transform all measured quantities into new units before plugging them into equations. This means that the only part of the constant that is fundamental to General Relativity is $8\pi$, the fact that the constant is (roughly) $10^{-44}$ and not the fundamental $8\pi$ is a result of poorly chosen units for measuring distance and mass.

So, to answer your question directly, $10^{-44}$ doesn't have much real significance in relativity, it simply pops up because we chose to work with unnatural units. $8\pi$ has the real significance, and there is probably no simple physical reason that $8\pi$ shows up in our equations, though I would encourage someone with more knowledge on the topic to chime in if they have a purely physical explanation.

 

[alba]

T

The Stress Energy Tensor ($T_{\mu\nu}$) is a tensor whose components represent energy and momentum in spacetime. Roughly speaking,
$T_{00}$ = Mass/Energy
$T_{01}, T_{02}, T_{03}$ = Momentum
$T_{11}, T_{12}, T_{13},T_{22},T_{23},T_{33}$ = Stress
.

Thanks Andrew, that is awesome.
Can you expand on this tensor? I understand that energy in the form of mass can curve spacetime, but why momentum, and momentum of what, of bodies expanding at more than c? and also momentumof photons?
what evidence is there tha momentum has any influence on the structure of universe?
and, lastly , what about stress? whay is it? I thought that stress is a tensor which is made up of individual vectors of force/momentum , isn't it same as momentum on a surface?

 

[haushofer]

The 8pi is there because of the correspondence principle: the einstein eqns should reproduce the Poisson eqn of Newtonian gravity in a certain limit.

 

[alba]

The 8pi is there because of the correspondence principle: the einstein eqns should reproduce the Poisson eqn of Newtonian gravity in a certain limit.

that's too tall for me

 

[Ibix]

that's too tall for me

You need $G/c^4$ to make the units work out. Then you insist that Einstein's gravity look like Newton's gravity in the cases where Newton's gravity is a good description of the world. That turns out to mean you need an $8\pi$ too.

This is an application of the correspondence principle - a new theory must look like an accepted theory where the accepted theory has been tested. Otherwise it can't explain existing experiments and must be wrong.

 

[Vanadium 50]

For "without math", there sure is a lot of math in this thread.

 

[pervect]

Is it possible to explain themain ideas of GR without complicated math?

To a limited extent. For instance, Wheeler's remark "Matter tells space how to curve, space tells matter how to move" is an "explanation" of General Relativity, but it omits a lot of the details. I don't think there's a way to understand all the details without understanding the math.

In particular,
what is the relation of current model of universe (FLRW) and GR?

FLRW (Friedmann, Lematire, Robertson, Walker) is a specific solution of the GR field equations that applies to cosmology, i.e. the universe.

what is the difference between spacetime and universe, spacetime is surely curved and universe can by flat at same time?

When people talk about a "flat" universe, they're talking about the spatial slice of the universe, rather than the space-time. So you can have flat spatial slices of a curved space-time, this is commonly called a flat universe. The word "universe" can mean other things than spatial slice, so the usage "flat universe" is a bit imprecise.

both mass and energy bend spacetime, in what proportion?

Not only mass and energy curve space time, but so do momentum and pressure. But it takes some math to talk about the ratios, and more math to talk about how curvature is described.

what is the max quantity of mass allowed, due to what? I read that neutrinos can't have more than 2 eV of mass-energy because there would be too mauch mass in the universe, it that is true, why so?

I'm not sure how to answer this. Might you be asking about "the critical density" in cosmology?

 

[alba]

T
Not only mass and energy curve space time, but so do momentum and pressure. But it takes some math to talk about the ratios, and more math to talk about how curvature is described.

I'm not sure how to answer this. Might you be asking about "the critical density" in cosmology?

Thanks, that was very helpful, how does momentum curve spacetime, what is the evidence, and what is pressure of what on what?
I suppose that is the right formulation, what is critical density? why neutrinos can't have more than 2 eV rest mass?

 

[Andrew Kim]

Much of General Relativity is devoted to the discovery of post-Newtonian effects, or effects that don't exist in the Newtonian approximation. A prime example is that of active gravitational mass. In general relativity, $(\rho+3p)$ replaces $\rho$ in the Newtonian equations. The $\rho$ part comes from the $00$ part of the Stress Energy Tensor. The $3p$ part comes from the other terms of the Stress Energy tensor. As I mentioned, physicists change the scale of quantities to eliminate $G$ and $c$ in the Einstein Equations. Because of this change of scale, $\rho>>p$ in normal pressure situations, and so the active gravitational mass $(\rho+3p)$ reduced to $\rho$. However, in very high pressure situations, $(\rho+3p)$ takes that high pressure into account.

So, to summarize, in normal situations only mass/energy affects the gravitational field. Only when stress and momentum are extremely large do they affect the equations of general relativity. This is a remarkable effect that wasn't present in general relativity, so we consider it a post-Newtonian effect.

 

[alba]

. Only when stress and momentum are extremely large do they affect the equations of general relativity. This is a remarkable effect that wasn't present in general relativity, so we consider it a post-Newtonian effect.

when is momentum extremely large? what is stress what produces stress on what?

 

[Andrew Kim]

Sorry, a more clear statement is:
Pressure affects a gravitational field, but only when it is extremely large. An example is a neutron star, which is a star so dense that the atoms get mushed together. High pressure would have effects on space time within the boundaries of the star.

 

[alba]

A prime example is that of active gravitational mass.ct.

Why do you say 'active' is there a 'passive' gravitational mass? a mass which has weight but does not exercize a pull on other mass? and what is it, relativistic mass?

 

[PeterDonis]

the active gravitational mass $(\rho+3p)$

Do you have a reference for the term "active gravitational mass" being used to refer to this quantity? It doesn't seem to me to be a standard usage. Standard usage of the term "active gravitational mass" as I understand it is as described, for example, here:

https://en.wikipedia.org/wiki/Mass#Definitions_of_mass

 

[alba]

Do you have a reference for the term "active gravitational mass" being used to refer to this quantity? It doesn't seem to me to be a standard usage. Standard usage of the term "active gravitational mass" as I understand it is as described, for example, here:

https://en.wikipedia.org/wiki/Mass#Definitions_of_mass

can you give an example where active and passive mass do not coincide? a proton has1836 electron masses, is that active or passive? I only heard that relativistic KE-mass has weight but produces no pull. what about thermal mass? has it active mass?

 

[PeterDonis]

can you give an example where active and passive mass do not coincide?

As the Wikipedia article says, nobody has ever found such an example.

a proton has1836 electron masses, is that active or passive?

Both.

I only heard that relativistic KE-mass has weight but produces no pull.

Where did you hear that? Please give a reference.

what about thermal mass? has it active mass?

Both.

 

[alba]

Where did you hear that? Please give a reference.
.

In several forums, they said that's why they decided to replace *relativistic mass* with energy.

 

[PeterDonis]

In several forums, they said that's why they decided to replace *relativistic mass* with energy.

This is still not a reference. The statement as you give it looks wrong to me, so I'm trying to figure out where you got it from so I can see if it's your misunderstanding or theirs. I need an actual link, not just vague allusions.

 

[alba]

The Ricci Tensor ($R_{\mu\nu}$) is a tensor that represents curvature. ...

The Stress Energy Tensor ($T_{\mu\nu}$) is a tensor whose components represent energy and momentum in spacetime...

This means that the only part of the constant that is fundamental to General Relativity is $8\pi$,.

If I got it right, then, the ( TRRT; TraceReverseRicciTensor) curvature of spacetime (and not universe, right) is rougly 25 times the SET(pressure of mass and momentum.on spacetime, right)
Now what happens if TRRT is smaller or greater than 25 SET? Since this regards only the local curvature of spacetimeand not the universe, I do not understan what is the problem. How do you calculate the value of SET?

 

[MeJennifer]

that's too tall for me

I basically means it has to fit with the Newtonian limit.

 

[PeterDonis]

If I got it right, then, the ( TRRT; TraceReverseRicciTensor) curvature of spacetime (and not universe, right) is rougly 25 times the SET(pressure of mass and momentum.on spacetime, right)

No. The constant $8 \pi$ does not mean the Einstein tensor (what you are calling the TRRT) is "bigger" than the SET. It is just a numerical factor that is there so that, with the usual definitions of those tensors, the EFE reduces to the appropriate equations for Newtonian gravity in the limit of weak fields and slow motion.

what happens if TRRT is smaller or greater than 25 SET?

This question makes no sense. The Einstein Field Equation, which says that $G = 8 \pi T$, always applies.

this regards only the local curvature of spacetimeand not the universe

I don't know what you mean by this. GR describes the spacetime geometry of the universe perfectly well.

How do you calculate the value of SET?

You don't calculate it, you measure it, by measuring things like energy density and pressure.

 

[alba]

This question makes no sense. The Einstein Field Equation, which says that $G = 8 \pi T$, always applies.
I don't know what you mean by this. GR describes the spacetime geometry of the universe perfectly well.
You don't calculate it, you measure it, by measuring things like energy density and pressure.

So you measure T, multiply it by 25 and you get the universe curvature, is that right?
how do you meausre T? you can have an educated guess about mass-energy in the universe, and how do you measure momentum and stress? what is the current value of T?does it change? can it get any value?

I asked this before, but still it is not clear: if the current curvature of the universe is 25(/10^44) T, why do they say the universe is flat? Kim said : a slice of the universe canbe flat, can you clarify that?

 

[martinbn]

@alba you talk about tensors as if they are scalars. It is not meaningfull to say "the value of the tensor".

 

[martinbn]

http://math.ucr.edu/home/baez/einstein/einstein.pdf

This isn't without maths, but this thread isn't either.

 

[alba]

@alba you talk about tensors as if they are scalars. It is not meaningfull to say "the value of the tensor".

Thanks for the precious link, bu it'll take a long time to digest that!

So how do you manage T? can you help me understand how it works showing how you apply it to Earth's gravitational field = 980 cm/s^2?

 

[m4r35n357]

Is it possible to explain themain ideas of GR without complicated math?
In particular,
what is the relation of current model of universe (FLRW) and GR?
what is the difference between spacetime and universe, spacetime is surely curved and universe can by flat at same time?
both mass and energy bend spacetime, in what proportion? what is the max quantity of mass allowed, due to what? I read that neutrinos can't have more than 2 eV of mass-energy because there would be too mauch mass in the universe, it that is true, why so?

From the answers and your responses to them so far, I'd say the answer is a resounding no. I don't want to discourage you though, this applies to all of us.

There are so many layers of concept and complexity buried in your question that you cannot even be aware of without doing acres and acres of difficult and crunching mathematics.

You will need to choose exactly how deep you want to descend into that rabbit hole, and stick to your decision. I have made my personal choice, and am not likely to exhaust the possibilities for study within that level. I can tell when my knowledge is insufficient and only marvel at the symbols flying by when more advanced subjects are being discussed.

I would heartily recommend looking at this story, though, to give you some idea of what you are asking for. It might help you decide how much you really want to know.

[EDIT] fixed link to Oz & the wizard.

 

[vanhees71]

Well, there's one lesson learned in this thread: The answer to the question in it's title is a clear "no". There's no way to adequately describe any theory in physics with the appropriate language, which is math (differentiable pseudo-Riemannian manifolds, Lie groups and algebras in the case of GR).

All the many words made in this thread and obviously in the popular-science sources the OP has read, lead to nothing else than utter confusion. First of all there's no relativistic mass, and the gravitational field doesn't couple only to the mass terms in the energy-momentum-stress tensor of matter and radiation but the the entire energy-momentum-stress tensors. That's what's very clearly said by the Einstein field equation, which follows under quite general assumptions from symmetry principles, leading to the Einstein-Hilbert action. This derivation also implies the universality of the coupling between matter+radiation to the gravitational field, which is the realization of the (strong) equivalence principle which is translated to a much clearer principle (thanks to math!) of general covariance.

 

[PeterDonis]

So you measure T, multiply it by 25 and you get the universe curvature, is that right?

No, because "T" and "universe curvature" are not single numbers. The stress-energy tensor and Einstein tensor, which are equated by the Einstein Field Equation, each have ten independent components--ten numbers, not one number. (And the numbers change when you change coordinates.) Also, the Einstein tensor is not all of spacetime curvature; it is only part of it. The full description of spacetime curvature is the Riemann tensor, which has twenty independent components (which change when you change coordinates).

how do you meausre T?

I already told you. Do you really not understand how things like energy density, momentum density, and pressure are measured? If you don't, this is not the place to fix that; you need to spend some time either doing experimental physics or at least reading about it.

if the current curvature of the universe is 25(/10^44) T, why do they say the universe is flat?

Because the universe is spatially flat (at least, in the coordinates usually used in cosmology). The curvature being referred to is spacetime curvature. Spacetime can be curved even if space is flat.

 

[PeterDonis]

how do you manage T? can you help me understand how it works showing how you apply it to Earth's gravitational field = 980 cm/s^2?

This is going beyond what can usefully be addressed in a PF thread. You need to consult a GR textbook. (Even before you do that, you probably need a good grounding in Newtonian physics and Newtonian gravity.)

This thread has gone as far as it can usefully go. Thread closed.