## Calculate Mass from Time Dilation and Radius

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aneikei
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### Calculate Mass from Time Dilation and Radius

Hello, I'm new to this site and I'm been searching the internet all over for this answer. I'm looking for an equation, that if given the time dilation at the surface and the radius of an object, it will allow me to calculate the mass of the object.

However, the current way of calculating time dilation (dτ/dt) does not work for my model and is not what I am looking for.

The time dilation range I'm using is 0.0 - 1.0. Where 1 would be infinite time dilation.

To derive this new time dilation scale I 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.

To calculate the time dilation without the range 0.0 - 1.0 it's
1/(sqrt(1-(.886^2)/(1^2))) = 2.15

This does not work for me. I need the output of mass from the unknown equation using the input time dilation range of 0.0 - 1.0
Last edited by aneikei on Mon Jan 14, 2019 3:58 am UTC, edited 1 time in total.

gmalivuk
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### Re: Calculate Mass from Time Dilation and Radius

1^2 = 1, and 1/(1/x) = x, so as you've written it,
1-(1/((1/(sqrt(1-(v^2)/(1^2)))))) = 1-sqrt(1-(v^2))
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aneikei
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Joined: Sun Jan 13, 2019 11:36 pm UTC

### Re: Calculate Mass from Time Dilation and Radius

Thank for the response. The equation is the Lorentz factor (gamma) used to determine the amount of time dilation.

1/(sqrt(1-(v^2)/(c^2)))

Where c is the speed of light

However when c = 1 you can chart it out as gamma having a range of 0.1 - 1.0

However, I want the Y values to correspond with 0.1 - 1.0 also so the equation becomes

1-(1/((1/(sqrt(1-(v^2)/(1^2))))))

Eebster the Great
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### Re: Calculate Mass from Time Dilation and Radius

So you are just doing 1 - 1/γ? I don't understand your objective. Anyway, the range is 0-1, not 0.1-1.

You cannot calculate the mass of an object just by its radius and the gravitational field on the surface unless you know some additional things about it, like that its density is distributed with spherical symmetry (for GR, we also need it to be nonrotating and neutrally charged). But if you do know these things, you get t0/tf = sqrt(1-2Gm/rc²), where t0 is proper (dilated) time for someone on the surface and tf is coordinate (undilated) time for a distant observer. Solving for m gives m = rc²/(2G) (1 - t0²/tf²).

Now I think your desired time dilation scale is ť = 1 - t0²/tf², so if that's true, your desired formula is m = ťrc²/(2G).

(r is the radial distance in Schwarzschild coordinates of the dilated observer from the center of the uniform, nonrotating, neutral spherical mass; c is the speed of light in a vacuum; and G is the universal gravitational constant.)

gmalivuk
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### Re: Calculate Mass from Time Dilation and Radius

The analogue of gamma for the Schwarzschild metric is 1/sqrt(1-2GM/(rc^2)). The "time dilation"
aneikei wrote:
The time dilation range I'm using is 0.0 - 1.0. Where 1 would be infinite time dilation.

Wanting it to go from 0 to 1 isn't unreasonable, as that could just be the reciprocal of the usual figure. (In other words, if you wanted to know how many ticks a clock on the surface makes for each tick at a distance, whereas the number that goes from 1 to infinity is the number of ticks a distant clock makes for each tick of a clock on the surface.)

What's strange is why you want to flip this around and use 1 for "infinite" time dilation. That doesn't seem to accomplish anything but removing the intuitive meaning the original 0 - 1 scale had.

But in any case, your "time dilation" number is d = 1 - 1/γ (as Eebster points out above). The analogue to 1/gamma for the Schwarzschild metric is sqrt(1 - 2GM/(rc^2)) instead of sqrt(1-v^2/c^2). Conveniently time dilation is the same outside an unmoving, uncharged, spherically symmetric matter distribution as it is in empty space if you're traveling at the escape velocity of that matter distribution. This is a surprising but convenient coincidence.

So you know the value of d = 1 - sqrt(1 - 2GM/(rc^2)), and you know what r is, and you want to calculate M?

sqrt(1 - 2GM/(rc^2)) = 1 - d
1 - 2GM/(rc^2) = (1 - d)^2
2GM/(rc^2) = 1 - (1 - d)^2
2GM = rc^2 (1 - (1 - d)^2)
M = rc^2 / (2G) * (1 - (1 - d)^2)

So, if clocks on the surface (at radius 10km) tick 10% slower than clocks in distant space (so d = 0.1), you have
M = 10000 c^2 / (2G) * (1-0.9^2) = 1.28e30 kg. (I picked 10km because even a "modest" time dilation factor of 0.9 seemed neutron-star-ish, and indeed the Sun has a mass of about 2e30 kg, so that's definitely the right order of magnitude for everything.)
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