# Isotropic Observer

#### eoghan

It is said that in the case of a homogeneous and isotropic spacetime, the surfaces of homogeneity must be orthogonal to the tangents to the world lines of the isotropic observer.
Does this mean that the isotropic observers are always at rest in a surface of homogeneity?

Related Special and General Relativity News on Phys.org

#### nitsuj

Sounds like asking; do all observers measure proper time/length?

Is that similar to your question?

Always "at rest"? Always measure time/length properly; according to how they are defined.

Relative motion is (generally*) not part of measuring time/length, either is "at rest".

*those measurements taken while accelerating causes "issues", but given your choice of words I'd guess you are aware of this point...also.

Oh and is that a common term? "isotropic observer" never heard it before and can't picture what it would "mean".

Last edited:

#### alialice

No, it means that a given time the properties of the space are the same everywhere. At a given instant of course an observer is "at rest", but then it can move along its worldline!

#### nitsuj

oh yea isotropic is pretty commonly understood, especially in a physics forum. "Isotropic observer", given the definitions of "observer" & isotropic from a physics perspective, left me unsure if something besides plain ol' "observer" was implied.

"At a given instant of course an observer is "at rest"," relative to what?? the way time/length is measured?

anyhoo let's not carry on with this.

#### PeterDonis

Mentor
It is said that in the case of a homogeneous and isotropic spacetime, the surfaces of homogeneity must be orthogonal to the tangents to the world lines of the isotropic observer.
If by "isotropic observer" you mean "an observer that sees the spacetime as isotropic", then yes, this is true. At least, it's true of the standard example of a homogeneous and isotropic spacetime, FRW spacetime.

Does this mean that the isotropic observers are always at rest in a surface of homogeneity?
Yes. This is a bit of a strange way of stating it, though; the more standard way of stating it requires you to first set up the standard FRW coordinate chart, and then observe that what you are calling isotropic observers (the more usual term is "comoving" observers) remain at the same spatial coordinates in this chart for all time, hence are "at rest" in this chart.

It's also good to bear in mind that these comoving observers, although they are "at rest" in the standard coordinate chart, are *not* "at rest" relative to each other in other senses in an FRW spacetime. For example, if two comoving observers exchange light signals, they will see the round-trip travel times of the light signals continually increase or decrease (depending on whether they are in an expanding FRW spacetime or a contracting one).

#### Austin0

If by "isotropic observer" you mean "an observer that sees the spacetime as isotropic", then yes, this is true. At least, it's true of the standard example of a homogeneous and isotropic spacetime, FRW spacetime.

Yes. This is a bit of a strange way of stating it, though; the more standard way of stating it requires you to first set up the standard FRW coordinate chart, and then observe that what you are calling isotropic observers (the more usual term is "comoving" observers) remain at the same spatial coordinates in this chart for all time, hence are "at rest" in this chart.

It's also good to bear in mind that these comoving observers, although they are "at rest" in the standard coordinate chart, are *not* "at rest" relative to each other in other senses in an FRW spacetime. For example, if two comoving observers exchange light signals, they will see the round-trip travel times of the light signals continually increase or decrease (depending on whether they are in an expanding FRW spacetime or a contracting one).
From this it seems "isotropic " observer means measuring isotropic CMBR readings , is this right?

#### PeterDonis

Mentor
From this it seems "isotropic " observer means measuring isotropic CMBR readings , is this right?
In a universe with a CMBR, such as ours, yes. More generally, whatever matter/energy/radiation is present will appear isotropic to an "isotropic observer". (Obviously this only holds approximately in our actual universe.)

#### eoghan

oh yea isotropic is pretty commonly understood, especially in a physics forum. "Isotropic observer", given the definitions of "observer" & isotropic from a physics perspective, left me unsure if something besides plain ol' "observer" was implied.

I've read it in "Wald - General Relativity". It means an observer who sees the universe as isotropic.

alialice said:
At a given instant of course an observer is "at rest", but then it can move along its worldline!
The problem is that the worldline is always orthogonal, so the observer will always be at rest relative to the surface of homogeneity

PeterDonis said:
Yes. This is a bit of a strange way of stating it, though; the more standard way of stating it requires you to first set up the standard FRW coordinate chart, and then observe that what you are calling isotropic observers (the more usual term is "comoving" observers) remain at the same spatial coordinates in this chart for all time, hence are "at rest" in this chart.
Uhm.. if I take a picture of the universe at a frozen time t, then that is a surface of hoogeneity, is it right?

#### nitsuj

I've read it in "Wald - General Relativity". It means an observer who sees the universe as isotropic.

The problem is that the worldline is always orthogonal, so the observer will always be at rest relative to the surface of homogeneity

Uhm.. if I take a picture of the universe at a frozen time t, then that is a surface of hoogeneity, is it right?
I find the question as worded in the OP kind of awkward.

Personally I see it as asking about measurements, and the "consequence/effect" the invariance of c has on those measurements.

"The problem is that the worldline is always orthogonal, so the observer will always be at rest relative to the surface of homogeneity"

Like Peter said this is "coordinates".

As stated above I can't help but feel the train of thought would continue to "at rest" we are traveling at c relative to...., or some other similar "Greene" interpretation.

#### PeterDonis

Mentor
Uhm.. if I take a picture of the universe at a frozen time t, then that is a surface of hoogeneity, is it right?
If the time "t" is the time in the standard FRW chart (which is the same as proper time for "comoving" observers), then yes.

#### PeterDonis

Mentor
Personally I see it as asking about measurements, and the "consequence/effect" the invariance of c has on those measurements.
The reference to Wald makes it clear that the OP is talking about FRW spacetimes and their properties. I don't think the OP meant to raise any issues about measurements, the invariance of c, etc. (though he can of course correct me if I'm wrong).

Like Peter said this is "coordinates".
I did say that, but now I need to clarify that it's not *just* coordinates.

For the term "at rest" to be meaningful, one has to specify "at rest relative to what?" The standard FRW coordinate chart uses the worldlines of comoving observers as the standard of "rest"; each comoving worldline stays at the same spatial coordinates for all time. But the comoving worldlines themselves are invariant features of the spacetime; you can describe them in any coordinate chart you like, they just won't look as simple in any chart other than the standard FRW chart.

Similarly, the fact that the worldlines of comoving observers are orthogonal to the surfaces of homogeneity is an invariant; it does not depend on the coordinate chart. It's just easiest to illustrate this fact in the standard FRW chart, since the surfaces of homogeneity are just surfaces of constant time in this chart.

Sorry if the above wasn't clear from my previous posts.

#### nitsuj

Opps, then it's my mistake Peter.

I don't even know what FRW is.

#### eoghan

If the time "t" is the time in the standard FRW chart (which is the same as proper time for "comoving" observers), then yes.
So if I am an isotropic observer (then "t" is the time in the standard FRW) I must be at rest in a given surface of homogeneity, can't I move?

#### PeterDonis

Mentor
So if I am an isotropic observer (then "t" is the time in the standard FRW) I must be at rest in a given surface of homogeneity, can't I move?
Of course you "can" move (meaning you can have a worldline that does not stay at the same spatial coordinates for all time in the standard FRW chart). But if you move, you won't be an isotropic observer any more--you won't see the universe as being the same in all directions.

#### PeterDonis

Mentor
Uhm.. if I take a picture of the universe at a frozen time t, then that is a surface of hoogeneity, is it right?
On re-reading this, I realized I should comment on the phrase "take a picture". I was interpreting that to mean "take a particular spacelike slice of constant FRW coordinate time t out of the entire spacetime". Any such spacelike slice is a surface of homogeneity.

However, if you were to literally "take a picture" of the light rays entering your eyes (or your camera's lens) at a particular instant, in a standard FRW spacetime, what would be in the picture would *not* be the same as what is in a spacelike slice. What is in the picture, taken literally, would be what is present on your past light cone at the instant at which the picture is taken. And that will *not* be homogeneous (though it will still be isotropic), because the light coming to you at a given instant was emitted at different times, depending on how far away the objects emitting it are.

#### eoghan

Thank you all for the answers. I've understood the whole thing (I think)

"Isotropic Observer"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving