# B Can you explain GR without math?

1. Jul 8, 2016

### 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?

2. Jul 9, 2016

### 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.

3. Jul 9, 2016

### alba

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?

Last edited: Jul 9, 2016
4. Jul 9, 2016

### 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.

5. Jul 9, 2016

### alba

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?

6. Jul 9, 2016

### 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.

7. Jul 9, 2016

### alba

that's too tall for me

8. Jul 9, 2016

### Ibix

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.

9. Jul 9, 2016

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

10. Jul 9, 2016

### pervect

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

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?

11. Jul 9, 2016

### alba

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?

12. Jul 9, 2016

### 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.

13. Jul 9, 2016

### alba

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

14. Jul 9, 2016

### 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.

15. Jul 9, 2016

### alba

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?

16. Jul 9, 2016

### Staff: Mentor

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

17. Jul 9, 2016

### alba

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?

18. Jul 9, 2016

### Staff: Mentor

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

Both.

Where did you hear that? Please give a reference.

Both.

19. Jul 9, 2016

### alba

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

20. Jul 9, 2016

### Staff: Mentor

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.