B Why do we use expanding metric?

[stoper]

TL;DR

Why do we use spatially expanding metric to measure the size of the expanding universe?

Being material observers, we do not expand with the universe. Our ruler for measuring its increasing size does not expand either - its scale does not change. If I identify the ruler with a metric, then from my perspective it should be invariant both spatially and temporally. If it expanded with the universe, then its size measured with this ruler would be constant.

Why then do we use a metric with the spatial scale expanding with the universe and constant temporal scale to measure the increasing size of the universe?

 

[jbriggs444]

TL;DR: Why do we use spatially expanding metric to measure the size of the expanding universe?

Being material observers, we do not expand with the universe. Our ruler for measuring its increasing size does not expand either - its scale does not change. If I identify the ruler with a metric, then from my perspective it should be invariant both spatially and temporally. If it expanded with the universe, then its size, as measured by this ruler, would be constant.

Why then do we use a metric with the spatial scale expanding with the universe and constant temporal scale to measure the increasing size of the universe?

What do you mean when you say that the "spatial scale" expands with the universe?

 

[stoper]

What do you mean when you say that the "spatial scale" expands with the universe?

Scale factor.

 

[Ibix]

Why then do we use a metric with the spatial scale expanding with the universe and constant temporal scale to measure the increasing size of the universe?

You don't have to. You can rescale your length scale to match the expansion - this is called co-moving coordinates. And then you can further rescale your time coordinate to make light have constant coordinate velocity - this is called conformal time.

However, it's very common to want to identify spatial coordinate differences with ruler measures and temporal coordinate differences with clock measures, so the default description used in most texts is the one where we measure time with our clocks and distance with our rulers.

 

[renormalize]

Do values of the metric tensor elements depend on the coordinate system of our choosing, but stress-energy tensor elements don't? If that's the case, how is the Einstein field equation satisfied for different coordinate systems of our choosing?

The individual components of any tensor (except the single component of a scalar) vary with the choice of coordinate system. This is true for both the metric and the stress-energy-momentum tensors.

 

[stoper]

The individual components of any tensor (except the single component of a scalar) vary with the choice of coordinate system. This is true for both the metric and the stress-energy-momentum tensors.

Right. If there is a coordinate system in which the universe is not expanding, then the energy density and pressure in the stress-energy tensor should be constant, right? Doesn't it invalidate this coordinate system physically?

 

[Nugatory]

Right. If there is a coordinate system in which the universe is not expanding, then the energy density and pressure in the stress-energy tensor should be constant, right? Doesn't it invalidate this coordinate system physically?

There's no such thing as "invalidating a coordinate system physically". We choose a coordinate system to make it easy to calculate quantities of interest. For example, if we are calculating the damage done to a car colliding witha tree we will likely choose coordinates in which the tree and the surface of the earth are at rest, although in those coordinates we find that the nearby stars are moving at speeds greater than light. That doesn't make these coordinates physically invalid, it makes them more suitable for calculating the trajectories of objects on the surface of the earth than the trajectories of nearby stars.

It should not surprise you that energies change with the coordinate system. Consider shooting an elephant with an elephant gun: Choose coordinates in which the elephant is at rest and we have a fast-moving bullet, choose coordinates in which the bullet is at rest we have a fast-moving elephant, clearly the total energy in the bullet-elephant system will be different. However, the useful and physically meaningful thing is the damage the unfortunate elephant suffers, which is determined by the amount of energy transferred between bullet and elephant. When we use tensor methods to calculate this quantity, we find that it comes out the same using either coordinates.

 

[stoper]

For example, if we are calculating the damage done to a car colliding witha tree we will likely choose coordinates in which the tree and the surface of the earth are at rest, although in those coordinates we find that the nearby stars are moving at speeds greater than light.

It should not surprise you that energies change with the coordinate system. (...) When we use tensor methods to calculate this quantity, we find that it comes out the same using either coordinates.

So my feeling that something is wrong with such coordinate system in which the density and pressure in the SEM tensor are constant despite the actual expansion and the actual decrease of their values, is just a feeling...

 

[Ibix]

If there is a coordinate system in which the universe is not expanding, then the energy density and pressure in the stress-energy tensor should be constant, right?

In such systems the components of the stress-energy tensor correspond to things like the energy per unit co-moving volume, not the energy per unit volume-measured-with-a-ruler. Since a fixed co-moving volume is growing compared to a fixed volume measured with a ruler, a fixed energy density per unit co-moving volume is decreasing when translated to energy density per unit volume measured with a ruler.

There's considerable flexibility to move things around the representation. As @Nugatory says, carefully using tensor methods is pretty much a necessity here because it keeps track of where in the representation you put information like "what is the coordinate length of a ruler".

 

[stoper]

Magic :)

Since a fixed co-moving volume is growing compared to a fixed volume measured with a ruler, a fixed energy density per unit co-moving volume is decreasing when translated to energy density per unit volume measured with a ruler.

In what way a growing comoving volume is fixed?

 

[PAllen]

I should note that there are invariant (coordinate independent) ways to say the universe is expanding. They need some math background, but here are some words: “if there exists a global everywhere expanding congruence, then the universe is expanding”.

 

[Ibix]

In what way a growing comoving volume is fixed?

A fixed co-moving volume is a growing volume in the usual sense of the word volume. A growing co-moving volume would also be a growing volume in the usual sense, just growing faster than a fixed co-moving volume.

 

[stoper]

I was asking about the meaning of the word "fixed" describing a growing, comoving volume. What does it mean in this case?

 

[stoper]

I should note that there are invariant (coordinate independent) ways to say the universe is expanding. They need some math background, but here are some words: “if there exists a global everywhere expanding congruence, then the universe is expanding”.

This one? https://en.wikipedia.org/wiki/Congruence_(general_relativity)

 

[PeterDonis]

a global everywhere expanding congruence

Where here "expanding" means "has a positive expansion scalar", which is an invariant property of a congruence.

 

[stoper]

In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation.

I don't want to be so picky, but don't we need coordinates to describe a curve?

 

[Ibix]

I was asking about the meaning of the word "fixed" describing a growing, comoving volume. What does it mean in this case?

A fixed co-moving volume is one that does not change over time. In terms of the usual meaning of volume, a fixed co-moving volume would be growing.

An analogy: your parents' height measured with a ruler probably hasn't changed much over your lifetime. Your parents' height as a multiple of your height has changed a lot. On the other hand your twin brother's height measured with a ruler has changed, but as a multiple of your height it has remained constant. There is absolutely no problem with measuring height relative to yourself (useful for answering the question "can I reach that shelf") or relative to a ruler, as long as you keep track of whether a length of 1 means 1m or one times your height.

Similarly, expressing distance as a multiple of ruler lengths or as a multiple of distances between co-moving galaxies is fine as long as you keep track of which one you are doing. Used correctly, tensor methods do the keeping track for you.

 

[stoper]

A fixed co-moving volume is one that does not change over time. In terms of the usual meaning of volume, a fixed co-moving volume would be growing.

(...)

Similarly, expressing distance as a multiple of ruler lengths or as a multiple of distances between co-moving galaxies is fine as long as you keep track of which one you are doing. Used correctly, tensor methods do the keeping track for you.

If I understand correctly, in comoving coordinates a ruler expands with the universe, yes?

 

[Ibix]

I don't want to be so picky, but don't we need coordinates to describe a curve?

Of course not. Do you tell someone how to get to the shops by giving them a sequence of latitude and longitude pairs? No - you say "go down the street and turn left at the second junction". In other words, you describe the path with reference to other physical objects.

You can specify a timelike congruence in terms of direct physical observables and the relationships between members of the congruence. In this case you would say that the congruence is inertial and some things about the radar flight time between pairs of members of the congruence that would boil down to homogeneity, isotropy, and expansion.

 

[stoper]

Of course not. Do you tell someone how to get to the shops by giving them a sequence of latitude and longitude pairs? No - you say "go down the street and turn left at the second junction". In other words, you describe the path with reference to other physical objects.

That's a verbal description of the path relative to static objects. Fixed positions of static objects are exactly like fixed points in the coordinate system.

 

[Ibix]

If I understand correctly, in comoving coordinates a ruler expands with the universe, yes?

Depends what you mean by "ruler". Do you mean the physical object on your desk, or do you mean the agreed standard length?

The physical object doesn't care what coordinates you use. If you choose to use its length as your distance standard then it has constant length and the distance between galaxies is growing. If you choose to use the distance between co-moving galaxies as the standard length then they are a constant distance apart and the ruler's length is shrinking. The latter choice would have quite a range of impacts on the maths used in physical models, but no impact on measurable quantities.

 

[stoper]

Depends what you mean by "ruler". Do you mean the physical object on your desk, or do you mean the agreed standard length?

The one you were talking about:

A fixed co-moving volume is one that does not change over time. In terms of the usual meaning of volume, a fixed co-moving volume would be growing.

(...)

Similarly, expressing distance as a multiple of ruler lengths or as a multiple of distances between co-moving galaxies is fine as long as you keep track of which one you are doing. Used correctly, tensor methods do the keeping track for you.

 

[Ibix]

That's a verbal description of the path relative to static objects. Fixed positions of static objects are exactly like a fixed points in the coordinate system.

Nope. A coordinate system is a smooth map between a region of an n-dimensional manifold and $\mathbb{R}^n$. The system of directions I described isn't even a map, and doesn't even require fixed points. For example, the instruction "walk along the road and cross over the next time there's a gap in the traffic" defines a path (uniquely so if we agree what gap in the traffic is sufficient to cross) but the gap is not a persistent entity - ideally it's a single event. It does not define a coordinate system.

 

[stoper]

The physical object doesn't care what coordinates you use. If you choose to use its length as your distance standard then it has constant length and the distance between galaxies is growing. If you choose to use the distance between co-moving galaxies as the standard length then they are a constant distance apart and the ruler's length is shrinking. The latter choice would have quite a range of impacts on the maths used in physical models, but no impact on measurable quantities.

What's the metric in case of the constant distance between the galaxies?

 

[Ibix]

The one you were talking about here:

Then a ruler is a physical object. In co-moving coordinates its coordinate length decreases as the universe expands.

Actual measurements (like "how many rulers can I fit end-to-end between these two galaxies at times A and B") are, of course, unaffected by whether you decide to regard the rulers as having shrunk or the distance between galaxies as having grown.

 

[stoper]

Then a ruler is a physical object. In co-moving coordinates its coordinate length decreases as the universe expands.

What's the metric in this case?

 

[Ibix]

What's the metric in case of the constant distance between the galaxies?

Start with the usual FLRW metric $g_{\mu\nu}=\mathrm{diag}(1,-a/(1-kr^2),-ar^2,-ar^2\sin^2\theta)$, then define $\chi=ar$. The transformation rule here is that $g'_{\rho\sigma}=\frac{\partial x^\mu}{\partial x'^\sigma}\frac{\partial x^\nu}{\partial x'^\rho}{g_{\mu\nu}}$, so note that $\frac{\partial t'}{\partial t}=\frac{\partial \theta'}{\partial \theta}=\frac{\partial \phi'}{\partial \phi}=1$, but that $\frac{\partial \chi}{\partial r}=a$ and $\frac{\partial\chi}{\partial t}=\dot{a}r$.

You can work through that if you want. I don't trust my ability to do it with only my phone and without pen and paper, which I don't have to hand right now. You'll get something messy with $g_{t\chi}$ and $g_{\chi t}$ off-diagonal terms as well as the diagonal terms, and it's the off-diagonal terms where you've hidden the expansion this time.

 

[stoper]

I prefer simplicity of Cartesian coordinates, so $g_{\mu\mu}=$diag$(1, -a^2, -a^2, -a^2)$, and I just want to know whether the spatial, diagonal components after the transformation will be decreasing or increasing with time due to $a=a(t)$.

 

[Ibix]

I prefer simplicity of Cartesian coordinates, so $g_{\mu\mu}=$diag$(1, -a^2, -a^2, -a^2)$, and I just want to know whether the spatial, diagonal components after the transformation will be decreasing or increasing with time due to $a=a(t)$.

In that case you are restricting yourself to the spatially flat solution. Fine. In that case you'll have $g_{tx}$, $g_{ty}$, and $g_{tz}$ off-diagonal terms (and the symmetric terms) which is where you've hidden the expansion.

Just looking at the diagonal components would be selectively ignoring derivatives of $a$. I feel like I've said that to someone recently...

 

[stoper]

I don't want to hide the expansion, I wrote what I want to know, and also $g_{\mu\nu}=0$ for $\mu\neq\nu$ in the FLRW metric in the Cartesian coordinates.