Redshift Ques: Applying Doppler Formula Can Lead to Error

[jostpuur]

The formula for redshift according to the special relativity is

<br /> f = \sqrt{\frac{c-v}{c+v}} f_0.<br />

Here f\to 0 when v\to c. So for arbitrarily strong redshift effect we can always solve the corresponding velocity v&lt;c of the emitter moving away. Why then does Carrol explain this:

"As an example of how you can go wrong, naive application of the Doppler formula to the redshift of galaxies implies that some of them are receding faster than light, in apparent contradiction with relativity."

 

[yuiop]

I guess he is talking about the application of the non-relativistic Doppler equation

f=f_o\frac{c}{(c+v)}

where the velocity equates to v=c\left(\frac{f_o}{f}-1\right)

Modern cosmology considers the vacuum to be expanding and considers the recession of distant galaxies to be a result of being carried along by the expanding vacuum. Ignoring any local "peculiar" proper motion, distant galaxies are basically stationary with respect to the local vacuum medium and so the relativistic equation that includes time dilation does not apply. In other words a distant galaxy being carried along by the expansion of the universe does not experience time dilation due to its motion relative to us the observers because it is (mostly) stationary with respect to the local medium.

 

[jostpuur]

"stationary with respect to the local vacuum medium" sounds very suspicious thing... something that shouldn't exist in relativity.

It's not my intention to criticize cosmology now, but I must say that I haven't fully understood why cosmologists believe that the redshift of distant galaxies isn't mere naive redshift according to the special relativity. What is the reason for this belief?

 

[MeJennifer]

Expansion of space is a coordinate effect. Spacetime on the other hand is static and does not change.

 

[rahuldandekar]

"stationary with respect to the local vacuum medium" sounds very suspicious thing... something that shouldn't exist in relativity.

It's not my intention to criticize cosmology now, but I must say that I haven't fully understood why cosmologists believe that the redshift of distant galaxies isn't mere naive redshift according to the special relativity. What is the reason for this belief?

A variety of reasons. Firstly, for the galaxies to be actually "moving" away from us at such high velocities, initially, they'd have to be given a very big "push". What is the source of this push (or explosion)? On the other hand, the interpretation as FRW metric follows naturally from a homogenous and isotropic matter distribution.

Also, there is an interpretational problem: I see galaxies moving away from ME at high speeds, causing Doppler Shift. What would happen if I looked from some other galaxy's point of view?

If galaxies indeed are moving away because of an explosion, why do I see about the same distribution in every direction? Why is it that there are as many galaxies "ahead" of me as "behind" me in the expansion?

Hope I was not very silly in raising these objections. ;)

 

[yuiop]

Expansion of space is a coordinate effect. Spacetime on the other hand is static and does not change.

Hi Jennifer,

I am troubled by your statement that spacetime is static. After all isn't the spacetime around a mssive body supposed to be "warped" or curved? Wheeler once said something like "Matter tells space how to curve and space tells matter how to move". I am not sure if acually said or meant spacetime rather than space but I am sure in GR time and space are inextricably linked and you cannot warp space without having some effect on time. In fact, I personally think curvature or warping of space is defined by the relationship of time to space. In short, spacetime is not static but a maleable quantity, no?

Second of all you often mention "coordinate effects" in the context that coordinate effects are somehow not real or just just artifacts because they made by a distant observer while local "proper" measurements are somehow more real physical effects. I would argue that coordinate measurements are tangible real physical effects with this example I will call the "gravitational twins paradox".

Two twins are born on the top of an extremely high tower that is standing on a non rotating neutron star. On PF we can not resist the urge to separate twins at birth and perform experiments on them in the time honoured way :P. So one of the twins is sent to the base of the tower. We are not totally cruel so we allow them to send birthday messages to each other each year. The one at the base sees the birthday messages from the one at the top arrive every 3 months while the one at the top sees birthday messages from the one at the base arrive every 4 years. We see that the clock at the base is running four times slower than the one at the top and that this is a coordinate measurement. The proper time according to each of the twins is running at one second per second or one year per year and the coordinate time is just an illusion or is it? When the twin at the top gets to 80 years old he is not feeling well and we allow him his dying wish to see his sibling before he dies. He is transported down to his brother at the surface and finds his brother is only 20 years old.

So can we agree that coordinate time is a real tangible physical effect and not just an an illusion or artifact of distant measurements?

 

[jostpuur]

Also, there is an interpretational problem: I see galaxies moving away from ME at high speeds, causing Doppler Shift. What would happen if I looked from some other galaxy's point of view?

If galaxies indeed are moving away because of an explosion, why do I see about the same distribution in every direction? Why is it that there are as many galaxies "ahead" of me as "behind" me in the expansion?

Actually I haven't checked mathematics behind this thing, but I don't think that is a problem. Doing a boost and translation into some other galaxy's frame should keep the big picture the same... or at least my intuition says so. Maybe I'll calculate something with this at some point.

 

[MeJennifer]

I am troubled by your statement that spacetime is static.

Let me expand a litttle bit on what I wrote because the word "static" was confusing in this context.

Static is a well defined term when talking about spacetimes. Spacetimes are either stationary or non-stationary. A static spacetime is a subset of stationary spacetimes. The Schwarzschild metric is an example of a static spacetime. The Kerr metric is an example of a stationary spacetime. The metric of a spacetime with two black holes is an example of a non-stationary spacetime.

But that is not what I meant by static in this context, what I meant is that the metric of a given spacetime is fixed, it does not fluctuate, expand or contract, it is static. Spacetime contains all of space and time.

Wheeler once said something like "Matter tells space how to curve and space tells matter how to move". I am not sure if acually said or meant spacetime rather than space but I am sure in GR time and space are inextricably linked and you cannot warp space without having some effect on time. In fact, I personally think curvature or warping of space is defined by the relationship of time to space. In short, spacetime is not static but a maleable quantity, no?

Spacetime is not a maleable quantity. Einstein's equation of GR in the most simple terms is

Warping of spacetime = Mass-Energy distribution over time.

It is not an expression in space and in time but an expression over all of time and space together. Nothing can change in a given spacetime.

 

[yuiop]

Let me expand a litttle bit on what I wrote because the word "static" was confusing in this context.
...

Thanks Jennifer, for taking the time to expand on your statement :) ..but I am still a little confused:(

Would the spacetime around a pair of mutually orbiting neutron stars be an example of non static spacetime?

 

[MeJennifer]

Thanks Jennifer, for taking the time to expand on your statement :) ..but I am still a little confused:(

Would the spacetime around a pair of mutually orbiting neutron stars be an example of non static spacetime?

Yes, such a spacetime is not only non static but also non stationary.

With regards to the idea of the "expansion of space" and coordinate effects one can for instance say with equal validity in GR that instead of space expanding we have clocks that are speeding up, or any mixture of space expanding and clocks speeding up, it all depends on the coordinate chart you use.

 

[stillwonder]

gravitational redshift is explained by differing clock speeds because of differing gravity
i was wondering about redshift in a *uniform gravitational field*.
Since the field is uniform, all clocks are of same speed.
but a photon will still redshift traveling parallel to the field. Why?

 

[yuiop]

gravitational redshift is explained by differing clock speeds because of differing gravity
i was wondering about redshift in a *uniform gravitational field*.
Since the field is uniform, all clocks are of same speed.
but a photon will still redshift traveling parallel to the field. Why?

Hi, and welcome to PF :)

In the gravitational field of spherical mass the acceleration of gravity is proportional to GM/(R^2) and the gravitational potential (the integral of the acceleration) is proportional to -GM/R. It can be seen that the gravitational time dilation factor of 1/sqrt(1-2GM/R) is more a function of potential than acceleration. In a uniform gravitational field the acceleration would be proportional to GM (i.e. independent of height) and the potential of such a field would be +GMR. So even in a uniform gravitational field the gravitational potential is not independent of height and it is reasonable to assume that the gravitational time dilation factor would also not be independent of height in such a field. So a photon moving upwards would still red shift. Note that the gradient defined by the gravitational potential in a uniform gravitational field still allows a notion of upwards and falling bodies still generally move from a high potential to a lower potential towards the massive body.

 

[MeJennifer]

So a photon moving upwards would still red shift.

Where is upwards in a uniform gravitational field?

 

[yuiop]

Yes, such a spacetime is not only non static but also non stationary.

With regards to the idea of the "expansion of space" and coordinate effects one can for instance say with equal validity in GR that instead of space expanding we have clocks that are speeding up, or any mixture of space expanding and clocks speeding up, it all depends on the coordinate chart you use.

hmmm.. so in the context of clocks speeding up (no mixing), presumably the speed of light does not speed up correspondingly or we would not notice the apparent expansion. In that context, the length of meter sticks does not change but it eventually be noticed that it takes longer for light to travel the length of a meter stick so light would appear to be gradually slowing down over time. Presumably it would be many thousands of years before we could actually be able to notice a measurable difference. Are you also suggesting (in the pure clock speeding up interpretation) that all clocks universally speed up at the same rate everywhere at the same rate and that eventually the the size of the solar system or local galaxy would also appear to be increasing gradually over time?

 

[yuiop]

Where is upwards in a uniform gravitational field?

In the very next sentence I said "Note that the Gradient defined by the gravitational potential in a uniform gravitational field still allows a notion of upwards ..."


Perhaps we have different interpretations of what a uniform gravitational field is? My understanding is that the non zero magnitude of the acceleration is independent of the horizontal and vertical position of the test particle relative to a hypothetical infinite plane gravitational body, but all the acceleration vectors are parallel and point to the gravitational body.

 

[MeJennifer]

Are you also suggesting (in the pure clock speeding up interpretation) that all clocks universally speed up at the same rate everywhere at the same rate and that eventually the the size of the solar system or local galaxy would also appear to be increasing gradually over time?

Let me make it clear I am not suggesting anything here. I am simply saying that in GR one arrange charts in many ways. The view of "expanding space" or "speeding up clocks" is simply equivalent under GR. It would be impossible to distinct between the two.

 

[yuiop]

Let me make it clear I am not suggesting anything here. I am simply saying that in GR one arrange charts in many ways. The view of "expanding space" or "speeding up clocks" is simply equivalent under GR. It would be impossible to distinct between the two.

I was just wondering how that interpretation works and what we can learn from it. I did not mean to suggest it was your personal view. Sorry, if I implied that.

 

[stillwonder]

Hi, and welcome to PF :)

In the gravitational field of spherical mass the acceleration of gravity is proportional to GM/(R^2) and the gravitational potential (the integral of the acceleration) is proportional to -GM/R. It can be seen that the gravitational time dilation factor of 1/sqrt(1-2GM/R) is more a function of potential than acceleration. In a uniform gravitational field the acceleration would be proportional to GM (i.e. independent of height) and the potential of such a field would be +GMR. So even in a uniform gravitational field the gravitational potential is not independent of height and it is reasonable to assume that the gravitational time dilation factor would also not be independent of height in such a field. So a photon moving upwards would still red shift. Note that the gradient defined by the gravitational potential in a uniform gravitational field still allows a notion of upwards and falling bodies still generally move from a high potential to a lower potential towards the massive body.

thanks
the gravitational potential dependency would make it even worse, since every point in uniform g.field has infinite g.potential going by definition.
on the other hand, if the equation for time dilation is a clever retrofitting only, then it will be independent of height and there will only be redshift but no time dilation.
also, since all g.fields are non-uniform, given enough time (and money) one can find out if one is being accelerated or attracted.

 

[yuiop]

thanks
the gravitational potential dependency would make it even worse, since every point in uniform g.field has infinite g.potential going by definition...

I can see why you would think that, due to my over simplication when I stated the acceleration would be proportional to GM in a uniform field. I have checked the calculations and a better expression for the acceleration of a uniform field orthogonal to flat surface of a thin gravitational disk of infinite radius would be proportional to 2\pi G p d where d is the thickness of the disk and p is the density of the disk. Expressed like that the effective gravitational mass of the disk is not infinite. I struggle a bit with calculus so maybe one of the better mathematicians here could check that. The integration method is fairly involved when you take the thickness of the disk into account, but it is fairly satisfying to get the same result as the simpler Gaussian surface method.