# Calculate mass from time dilation

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aneikei
Based on the exponential growth of time dilation 0.0 - 1.0 if given the radius of an object how do you calculate the mass of the object?

Time dilation is a function of gravity. Which can be thought of as escape velocity from a gravitational field. So if you have to achieve .866 c to escape. Then the time dilation would be 1.99 years for every one year on earth.

using 1/(sqrt(1-(.866^2)/(1^2)))

Now that I have that and if given the radius of the object how do I calculate the mass of the object if keeping the time dilation the same?

The time dilation range is (0.1 - 1.0) = (1.005037815 - infinty)

*this is not homework I'm just trying to scratch an itch

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## Answers and Replies

Mentor
the exponential growth of time dilation 0.0 - 1.0

What are you referring to here?

if given the radius of an object how do you calculate the mass of the object?

The radius alone is not enough information to calculate the mass.

aneikei
Based on the exponential growth of time dilation 0.0 - 1.0 if given the radius of an object how do you calculate the mass of the object?

Time dilation is a function of gravity. Which can be thought of as escape velocity from a gravitational field. So if you have to achieve .866 c to escape. Then the time dilation would be 1.99 years for every one year on earth.

using 1/(sqrt(1-(.866^2)/(1^2)))

Now that I have that and if given the radius of the object how do I calculate the mass of the object if keeping the time dilation the same?

The time dilation range is (0.1 - 1.0) = (1.005037815 - infinty)

Mentor
Time dilation is a function of gravity.

It's a function of gravitational potential.

Which can be thought of as escape velocity from a gravitational field.

Yes, you can think of the time dilation factor as equal to the ordinary SR time dilation for an object moving at escape velocity, relative to an object that is stationary at the same location.

Now that I have that and if given the radius of the object how do I calculate the mass of the object if keeping the time dilation the same?

None of this has anything to do with the radius of the object. It has to do with the distance from the object's center. In relativistic geometric units (where ##G = c = 1##), escape velocity at a distance ##r## from the object's center is ##\sqrt{2M / r}##. So the time dilation factor is

$$\sqrt{1 - v^2} = \sqrt{1 - \frac{2M}{r}}$$

But in fact, just knowing the escape velocity at a distance ##r## from the center is enough to give you the mass, as you can see from the formula for escape velocity that I just gave above.

(Note that, strictly speaking, ##r## is not exactly the distance from the center; it's a radial coordinate. For weak fields, the error in thinking of it as the distance from the center is very small; but for strong fields, it isn't. All of the fields in the solar system are weak so it isn't an issue there, but it is, for example, around neutron stars or black holes.)

aneikei
"None of this has anything to do with the radius of the object. It has to do with the distance from the object's center." isn't that the very definition of the radius? And thank you.

I know the formula to calculate the mass using the escape velocity. However, I'm looking the equation to calculate the mass from the amount of the time dilation from a range of 0.1 - 1.0. This can be derived from this formula 1-(1/((1/(sqrt(1-(v^2)/(1^2)))))) where v is the escape velocity in the percentage of c., for example, this is the formula depicting the amount of time dilation for something moving at .886 c

1/(sqrt(1-(.886^2)/(1^2))) = a dilation of 2.156635402

However, to convert it to an exponential scale of 0.1 - 1.0 we use

1-(1/((1/(sqrt(1-(.886^2)/(1^2)))))) = 0.5363147619

based off this scale and given the radius of an object I'm looking for the object's mass.

Mentor
I'm looking the equation to calculate the mass from the amount of the time dilation

I gave you equations for both in my last post. The equation in terms of time dilation is directly derived from the equation in terms of escape velocity, since the formula for time dilation is directly derived from the formula for escape velocity.

o convert it to an exponential scale of 0.1 - 1.0

Why are you doing this? What does it have to do with anything?

aneikei
Why are you doing this? What does it have to do with anything?

It for does to me.I'm working on a paper and need it converted this way

Thank you for the help

Science Advisor
For clarification, it sounds to me like you are asking about a test object sitting on the surface of a massive object. A clock sitting on a planet, for example. And you wish to observe the passage of time as measured by that clock, compared to a clock “outside” the gravitational influence of that object, and work backwards to determine the strength of gravitational pull at the surface, and use that to determine the mass of the object. This would give you a way to determine mass when orbital dynamics are not observable.

Is that right?

aneikei
For clarification, it sounds to me like you are asking about a test object sitting on the surface of a massive object. A clock sitting on a planet, for example. And you wish to observe the passage of time as measured by that clock, compared to a clock “outside” the gravitational influence of that object, and work backwards to determine the strength of gravitational pull at the surface, and use that to determine the mass of the object. This would give you a way to determine mass when orbital dynamics are not observable.

Is that right?
Sort of.
I'm looking for the equation to calculate the mass of an object if given its radius and the amount of the time dilation it causes using a range of 0.1 - 1.0. Where 1 would be infinite time dilation.

To derive this scale we use the formula 1-(1/((1/(sqrt(1-(v^2)/(1^2)))))) where v is the escape velocity in the percentage of c.

for example, if the escape velocity is .886 c the object has a time dilation of 0.5363147619

1-(1/((1/(sqrt(1-(.886^2)/(1^2)))))) = 0.5363147619

based off this scale and given the radius of an object I'm looking for the object's mass.

Science Advisor
Gold Member
I'm looking for the equation to calculate the mass of an object if given its radius and the amount of the time dilation it causes using a range of 0.1 - 1.0. Where 1 would be infinite time dilation..
Peter has given you enough info in #4 above to do what you have asked - check again.

But this is a very round-about way for obtaining the mass of an astronomical object. Normally one would simply observe the period and eccentricity of any object orbiting the prime mass from some distance ('static' relative to the object) and calculate the mass of the prime object from Kepler's laws of planetary motion.

jartsa
I'm looking for the equation to calculate the mass of an object if given its radius and the amount of the time dilation it causes using a range of 0.1 - 1.0. Where 1 would be infinite time dilation.

Why not solve this formula for M:

##x = \sqrt{1 - \frac{2M}{r}} ##

I got this:

##M = \frac{r}{2} - \frac{rx^2}{2}##

The x is the "time dilation factor". I guess your "amount of time dilation" is the inverse of x. No, that was not quite what you wanted. "1 would be infinite time dilation"

Hmmm... speed 1 corresponds to infinite gamma, speed zero corresponds to gamma of one. I see, it's supposed to work that way. Right? (Gamma is the inverse of the kinetic time dilation factor)

Science Advisor
Gold Member
Hmmm... speed 1 corresponds to infinite gamma, speed zero corresponds to gamma of one. I see, it's supposed to work that way. Right? (Gamma is the inverse of the kinetic time dilation factor)

I think the Op's interest is not velocity time dilation, but rather converting the (static) gravitational time dilation at a distance r from a large prime mass, to the actual mass of the prime. It is easily coming out of
$$d\tau/dt = \sqrt{1-2M/r}$$
provided that ##d\tau/dt## is known, but how is that obtained? I suppose it can be organized to that a distant observer beams a constant (agreed upon) wavelength to the observer at r, who then observes the gravitational blueshift and so determines the time dilation factor ##d\tau/dt## at r, in the range 0 to 1.

Mentor
1-(1/((1/(sqrt(1-(.886^2)/(1^2))))))
I cannot follow this. Please use LaTeX! And simplify your expression. ##1^2=1## and division by 1 does nothing and can be dropped.

aneikei
Why not solve this formula for M:

##x = \sqrt{1 - \frac{2M}{r}} ##

I got this:

##M = \frac{r}{2} - \frac{rx^2}{2}##

The x is the "time dilation factor". I guess your "amount of time dilation" is the inverse of x. No, that was not quite what you wanted. "1 would be infinite time dilation"

Hmmm... speed 1 corresponds to infinite gamma, speed zero corresponds to gamma of one. I see, it's supposed to work that way. Right? (Gamma is the inverse of the kinetic time dilation factor)

Yes, that's exactly right. If gamma is 1 then it means I'm standing on a black hole which would have an infinite time dilation which is also 1. So if given that and if I know that the radius of the black hole I can calculate its mass.

However, a gamma of .50 would give me a time dilation of 0.1339745962 so there isn't a 1-to-1 ratio between gamma and the time dilation factor. And for my purposes, I need the time dilation factor to find my mass. I need to calculate it from the amount of time dilation and the radius of the object.

The equation I used to convert gamma to the time dilation factor

##x = 1 - \frac{1}{\sqrt{1 - .50^2/c^2}} = 0.1339745962##

and yes gamma is the inverse of the kinetic time dilation factor

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aneikei
I cannot follow this. Please use LaTeX! And simplify your expression. ##1^2=1## and division by 1 does nothing and can be dropped.
Sorry about that, the time dilation here is 0.13

##x = 1 - \frac{1}{\sqrt{1 - .50^2/c^2}} = 0.1339745962##

So if given for x and the radius of the object. I'm looking for the mass of the object

Mentor
Thanks, that is much more legible!

Mentor
Sorry about that, the time dilation here is 0.13

##x = 1 - \frac{1}{\sqrt{1 - .50^2/c^2}} = 0.1339745962##

So if given for x and the radius of the object. I'm looking for the mass of the object
OK, so you should never refer to this quantity as time dilation. Time dilation is not defined this way. Time dilation is defined as ##dt/d\tau##. For velocity time dilation ##dt/d\tau=\gamma=(1-v^2/c^2)^{-1/2}##. For gravitational time dilation ##dt/d\tau= (1-\frac{2GM}{rc^2})^{-1/2}##

I have that and if given the radius of the object how do I calculate the mass of the object if keeping the time dilation the same?
Just solve that last equation for ##M## in terms of ##r## and ##dt/d\tau##

aneikei
OK, so you should never refer to this quantity as time dilation. Time dilation is not defined this way. Time dilation is defined as ##dt/d\tau##. For velocity time dilation ##dt/d\tau=\gamma=(1-v^2/c^2)^{-1/2}##. For gravitational time dilation ##dt/d\tau= (1-\frac{2GM}{rc^2})^{-1/2}##

Just solve that last equation for ##M## in terms of ##r## and ##dt/d\tau##
Is this right?
## M = \frac{rc^2}{2G}##

If that is the case there no term to plug in for my ##x = 1 - \frac{1}{\sqrt{1 - .50^2/c^2}} = 0.1339745962##

Shouldn't it be something like this ## M = \frac{c^2\cdot r \cdot (1-1/x^2)}{2\cdot G} ##

Mentor
Is this right?
## M = \frac{rc^2}{2G}##
No. You dropped the time dilation term, ##dt/d\tau##

aneikei
No. You dropped the time dilation term, ##dt/d\tau##

Then I think there is a small bit of confusion as to what I'm looking for.

If I'm standing on a neutron star and all I'm given is its radius and the amount that it dilates time at its surface. Which is 0.0910729244

Using the time dilation range 0.1 - 1.0 What is its mass?

If it had a time dilation = 1 it would be an infinite time dilation or in other words, a black hole.

So for Sagittarius A the radius would be 22,000,000,000 meters

How do I find its mass using its radius and its time dilation factor of 1?

Mentor
If I'm standing on a neutron star and all I'm given is its radius and the amount that it dilates time at its surface. Which is 0.0910729244

Using the time dilation range 0.1 - 1.0 What is its mass?

I do not understand why you insist on going through this elaborate procedure to change how you define "time dilation", when the ordinary time dilation formula, that has been posted for you a number of times in this thread (first by me in post #4), already tells you how to get the mass of the object from the ordinary time dilation and the distance you are from its center. Just invert the formula I gave you in post #4. Done.

Thread closed.

Dale