# Can We Pair Composition of Velocities with an Expanding Universe?

• Tibriel
In summary, the velocity addition formula of SR does not apply to the expanding universe as there is no unique definition of relative velocity at a distance. While the distance between two objects can increase faster than the speed of light, their velocity relative to the local background radiation cannot exceed c. This is due to the fact that velocity is measured locally and cannot be defined as change in global distance over time.
Tibriel
Hi All,

I get that one object can't travel faster than light with respect to another object because of SR and length and time contraction. What I don't get is how this meshes up with expanding universe.

If the space between 2 objects is expanding then wouldn't two objects be accelerating away from each other at a pace which would eventually exceed the speed of light? Yet my understanding of SR says this isn't possible. Does this mean there is a limit to how fast the universe can expand with respect to any observer?

Oh, and I realize the above questions are yes/no questions so if you could also explain that would be helpful.

Thanks!

Velocity addition formula of SR only applies locally in GR, which is required to discuss expanding universe. In GR, there is no unique definition of relative velocity at a distance, so no strong statement can be made.

Instead, different velocities are used for different purposes. Recession velocity (which is based on growth of proper distance between galaxies) can greatly exceed c. On the other hand, if one parallel transports the 4-velocity of one galaxy along its light path to another galaxy, and compare the 4-vectors, you always get a sub-luminal relative speed.

Upshot: your intuition that expanding universe creates a problem for the SR statement is correct.

The distance between two objects in expanding space can increase faster than c, but velocity (relative to the local background radiation) cannot exceed c. That's because distance is a global quantity, and velocity is measured locally. Unfortunately, you can't define velocity as change in global distance over time. It is the change in infinitesmal distance over infinitesmal time, measured between two objects at the same place and time. Kinda makes it hard to actually use for anything.

## 1. What is the composition of velocities in an expanding universe?

The composition of velocities in an expanding universe refers to the concept that the expansion of the universe affects the velocities of objects within it. This is due to the fact that as the space between objects increases, their relative velocities also increase.

## 2. How does the expansion of the universe affect the composition of velocities?

The expansion of the universe causes the composition of velocities to increase. This means that objects moving away from each other will have higher velocities than if the universe were not expanding. This effect is known as the Hubble flow.

## 3. Can we observe the composition of velocities in an expanding universe?

Yes, we can observe the composition of velocities in an expanding universe through various methods such as measuring the redshift of distant galaxies, which indicates their relative velocities, or through studying the cosmic microwave background radiation.

## 4. What role does dark energy play in the composition of velocities in an expanding universe?

Dark energy, which is the mysterious force driving the acceleration of the universe's expansion, has a significant impact on the composition of velocities in an expanding universe. It is believed to be responsible for the increasing rate of expansion, which in turn affects the velocities of objects within the universe.

## 5. How does the composition of velocities in an expanding universe relate to the theory of general relativity?

The composition of velocities in an expanding universe is a key aspect of the theory of general relativity. This theory, proposed by Albert Einstein, describes the relationship between matter, energy, and the curvature of spacetime. The expansion of the universe is accounted for in this theory and plays a crucial role in understanding the composition of velocities and the overall dynamics of the universe.

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