Undergrad Calculating Time Dilation & Galaxy Rotation Curve

[sha1000]

Hello,

What I understood from multiple answers on different threads is that the effect of the time dilation is too small to explain the galaxy rotation curve. I was advised to do some calculations in order to see it myself. And this is what I would like to do but I need some help.

- What is the best simple equation which represents the mass distribution as a function of R, so I can calculate the orbital velocity.
- What is the mass that I need to consider in order to calculate the time dilation for any star in the galaxy? It can't be only the mass of the central black hole but all the mass contained in the volume defined by R (distance between the star and the center of the galaxy); is it right?

 

[PeroK]

Hello,

What I understood from multiple answers on different threads is that the effect of the time dilation is too small to explain the galaxy rotation curve. I was advised to do some calculations in order to see it myself. And this is what I would like to do but I need some help.

- What is the best simple equation which represents the mass distribution as a function of R, so I can calculate the orbital velocity.
- What is the mass that I need to consider in order to calculate the time dilation for any star in the galaxy? It can't be only the mass of the central black hole but all the mass contained in the volume defined by R (distance between the star and the center of the galaxy); is it right?

Calculating the expected galaxy dynamics takes you into the realm of a computer simulation. Instead, you can work with the observed quantities:

1) We know that the galaxy is about 100,000 light years across.

2) We know that the galaxy's rotation speed is about $200km/s$ in the neighbourhood of the Sun. That's less than 1% of the speed of light.

3) You can calculate the time dilation associated with a speed of $200km/s$ and find it's negligible.

That's all you need. The relative motion of stars in our galaxy is not significantly relativistic.

 

[sha1000]

Calculating the expected galaxy dynamics takes you into the realm of a computer simulation. Instead, you can work with the observed quantities:

1) We know that the galaxy is about 100,000 light years across.

2) We know that the galaxy's rotation speed is about $200km/s$ in the neighbourhood of the Sun. That's less than 1% of the speed of light.

3) You can calculate the time dilation associated with a speed of $200km/s$ and find it's negligible.

That's all you need. The relative motion of stars in our galaxy is not significantly relativistic.

Hi,

Thank you for your reply.

Since you mentioned the speed of the sun 200km/s, I suppose that you are referring to the time dilation related to the speed. What I would like to calculate is the orbital velocity of the stars and the gravitational time dilation as a function of R using the equation: T' = T*sqrt(1 - 2GM/rc^2). I need the mass distribution equation in order to get it right.

Once I have a acceptable approximation of the mass distribution it's easy to calculate the orbital velocity (using the mass which is contained in the volume defined by R). But I have a doubt that I must use the same mass for the gravitational time dilation.

 

[PeroK]

Hi,

Thank you for your reply.

Since you mentioned the speed of the sun 200km/s, I suppose that you are referring to the time dilation related to the speed. What I would like to calculate is the orbital velocity of the stars and the gravitational time dilation as a function of R using the equation: T' = T*sqrt(1 - 2GM/rc^2). I need the mass distribution equation in order to get it right.

Once I have a acceptable approximation of the mass distribution it's easy to calculate the orbital velocity (using the mass which is contained in the volume defined by R). But I have a doubt that I must use the same mass for the gravitational time dilation.

The galaxy is an extended non-rigid, roughly elliptical body. The simplifying assumptions you suggest are many orders of magnitude more significant that time dilation can ever be.

It's a bit like factoring in time dilation to terrestrial projectile motion, but still ignoring air resistance!

I suggest you look for a paper on galaxy dynamics. I suspect one can't be hard to find. It's not going to use high-school mathematics and physics.

 

[Ibix]

Gravitational time dilation is of the same order of magnitude as kinematic time dilation for objects in orbits. That follows from the fact that $v^2=GM/r$ for a circular orbit and $\gamma=1/\sqrt{1-v^2/c^2}\approx 1+v^2/2c^2$.

If you insist on doing full calculations, the very first search result for "galaxy rotation curves" is the relevant Wikipedia page, which includes graphs (and supporting citations) for M33. That would be a good place to start.

 

[sha1000]

Gravitational time dilation is of the same order of magnitude as kinematic time dilation for objects in orbits. That follows from the fact that $v^2=GM/r$ for a circular orbit and $\gamma=1/\sqrt{1-v^2/c^2}\approx 1+v^2/2c^2$.

If you insist on doing full calculations, the very first search result for "galaxy rotation curves" is the relevant Wikipedia page, which includes graphs (and supporting citations) for M33. That would be a good place to start.

Thanks.

Can I conclude that we use the same M (mass) in both equations ($v^2=GM/r$ and T' = T*sqrt(1 - 2GM/rc^2)) which leads to the same order of magnitude for both kinematic and gravitational time dilation? This was actually my question since the beginning :).

 

[PeroK]

Thanks.

Can I conclude that we use the same M (mass) in both equations ($v^2=GM/r$ and T' = T*sqrt(1 - 2GM/rc^2)) which leads to the same order of magnitude for both kinematic and gravitational time dilation? This was actually my question since the beginning :).

Essentially, yes, although $v^2 = GM/r$ is for a spherically symmetric distribution of mass. And the galaxy is not even approximately spherical.

 

[Dale]

Essentially, yes, although $v^2 = GM/r$ is for a spherically symmetric distribution of mass. And the galaxy is not even approximately spherical.

Engineers famously approximate even cows as spherical.