Clarifying Robertson-Walker Metric Math Objects

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In summary, the Robertson Walker metric, which is described in the link provided, is being used to derive general relativistic mathematical objects. The scale factor, R(t), is causing confusion as to whether it should be treated as a constant or as a function of x0 when differentiating terms. The correct approach is to treat R(t) as a function of x0, otherwise there would be no curvature.
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Here is the Robertson Walker metric:

ds2= (cdt)2 - R2(t)[dr2/(1- kr2) + r2(dθ2 + sin2(θ)dΦ2)]

This metric is seen and discussed in this link: http://burro.cwru.edu/Academics/Astr328/Notes/Metrics/metrics.html

Now I am in the process of deriving the general relativistic mathematical objects for this metric such as the Christoffel symbols, Ricci tensor, etc... However, one thing is bothering me.

As you can see both in the link and at the top of this post, they did not omit the c term using the c=1 convention in the first term of the metric. However, that scale factor R(t) only has t in it and not ct.

This bothers me because I am on the fence about whether I should treat R(t) as a constant when deriving my Christoffel symbols or if I should treat it as a function of x0 and differentiate accordingly when deriving my Christoffel symbols. Note that x0 = ct , x1= r , x2=θ , x3 = Φ

It is possible that they may be assuming that c=1 inside of the R(t) function and that is why they omit the c there, or it could just simply be that R(t) is not a function of x0 and I should just treat it as a constant when differentiating terms of my metric tensors.

Which option is the correct choice?

For those who need clarification on what I am asking, here is a numerical example:

The metric tensor element g11 = -R2(t)/(1- kr2)

While deriving the Christoffel symbols, one of the derivatives I will have to take is:
∂g11 /∂x0

If I treat the term -R2(t) as a function of x0, then the above derivative would evaluate to be:

-2R(t)R'(t)/(1- kr2) where R'(t) is simply the derivative of R(t) with respect to t.

However, if I treat the term -R2(t) as a constant, then the derivative is 0.

Which case is the correct case?
 
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  • #2
I think your difficulty is in ##x^0=ct##. Just drop the ##c## there. ##R(t)## is a function of ##x^0##. If not you get no curvature as you say.
 
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FAQ: Clarifying Robertson-Walker Metric Math Objects

1. What is the Robertson-Walker metric?

The Robertson-Walker metric is a mathematical equation used in cosmology to describe the geometry of the universe. It is based on the general theory of relativity and is used to model the expansion of the universe over time.

2. How is the Robertson-Walker metric used in cosmology?

The Robertson-Walker metric is used to describe the large-scale structure of the universe, including the distribution of matter and energy. It is also used to make predictions about the evolution of the universe and the behavior of objects within it.

3. What are the components of the Robertson-Walker metric?

The Robertson-Walker metric has four components: the scale factor, the spatial curvature, the time coordinate, and the spatial coordinates. These components describe the size, shape, and expansion of the universe at a given point in time.

4. What is the significance of the scale factor in the Robertson-Walker metric?

The scale factor is a key component of the Robertson-Walker metric as it represents the expansion of the universe over time. It is used to calculate distances between objects in the universe and how these distances change as the universe expands.

5. How is the Robertson-Walker metric related to the Big Bang theory?

The Robertson-Walker metric is closely tied to the Big Bang theory, which is the prevailing theory for the origin and evolution of the universe. The metric is used to model the expansion of the universe from a single point of origin, as described by the Big Bang theory.

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