PDA

View Full Version : calculating dilation



tommac
2009-May-17, 06:10 PM
How can I figure out the time dilation at a certain distance from a primordial black hole that has the mass of say the sun.

I would like to know

1) the size of the event horizon
and
2)the point where time would run 10x slower than it does at the distance of the earth from the sun ( from the same BH ).

StupendousMan
2009-May-17, 10:33 PM
How can I figure out the time dilation at a certain distance from a primordial black hole that has the mass of say the sun.

I would like to know

1) the size of the event horizon


The event horizon of a (non-rotating, uncharged)
black hole with mass M is given
by the formula R0 = sqrt( 2*G*M / c^2 ), where
G is the constant of universal gravitation and c is
the speed of light. For an object with M = 1 solar mass,
the radius is a few km.



2)the point where time would run 10x slower than it does at the distance of the earth from the sun ( from the same BH ).

If you're at a distance R from the black hole, then the
time dilation factor is gamma = sqrt ( 1 - R0/R ).

Really, you COULD find all this in just a few minutes
of web surfing.

robross
2009-May-17, 10:35 PM
How can I figure out the time dilation at a certain distance from a primordial black hole that has the mass of say the sun.

I would like to know

1) the size of the event horizon
and
2)the point where time would run 10x slower than it does at the distance of the earth from the sun ( from the same BH ).

I'm working through a book on Special Relativity, so I can answer question #1. The simple formula is

radius of bh = 2.94km X (# of solar masses).

So the radius of a black hole with one solar mass would be 2.94 km.

I can't compute #2 yet until I work through GR, but I'm pretty sure you will not experience such a large time dilation with an object of only one solar mass unless you are *extremely* close to the EV.

Rob

WayneFrancis
2009-May-18, 12:24 AM
So you want the distance from the EH where the escape velocity is 298289729.4m/s

Do you want the distance of this point from the EH relative to the "Earth" location or the "10x slower then earth location"?

Remember the EH of a BH is different depending how far in the gravity well you are.

I think what you want to know is
Given a point in space at 1AU distance from another point in space if you introduce a 1 solar mass black hole at that same point how far would you have to travel towards the BH to get an escape velocity of 298289729.4m/s compared to ~42,100 m/s

the problem with using
Ve = √2GM/r

is that r in that equation is in terms of the pre-bh time space coordinate system.

You can use the formula of
tr/t = √1-rs/r
so 10/1 = √1-(2.94km/r)

My napkin calculation puts you at ~30m from the EH to get a 10/1 time dilation.

Again this "30m" is differenent depending at what reference frame you are observing it.

tommac
2009-May-18, 12:38 AM
I'm working through a book on Special Relativity, so I can answer question #1. The simple formula is

radius of bh = 2.94km X (# of solar masses).

So the radius of a black hole with one solar mass would be 2.94 km.

I can't compute #2 yet until I work through GR, but I'm pretty sure you will not experience such a large time dilation with an object of only one solar mass unless you are *extremely* close to the EV.

Rob

I am looking for a very small black hole where the black hole is equal to 1 solar mass.

tommac
2009-May-18, 12:44 AM
So you want the distance from the EH where the escape velocity is 298289729.4m/s

Hmmm ... not sure. I would like the distance from the EH where the time dilation is 10x slower that what it would be at earths distance. Where are you getting this number: 298289729.4m/s



Do you want the distance of this point from the EH relative to the "Earth" location or the "10x slower then earth location"

Remember the EH of a BH is different depending how far in the gravity well you are.

Yes, I would like to solve for "r" where my clock would run 10x slower than a clock at the earths distance. I would assume ( and this isnt based on much but ... ) i would need to be within a foot or two ( or even closer) of the EH ( which is probably will only be a few inches in diameter ).

tommac
2009-May-18, 12:50 AM
The event horizon of a (non-rotating, uncharged)
black hole with mass M is given
by the formula R0 = sqrt( 2*G*M / c^2 ), where
G is the constant of universal gravitation and c is
the speed of light. For an object with M = 1 solar mass,
the radius is a few km.



If you're at a distance R from the black hole, then the
time dilation factor is gamma = sqrt ( 1 - R0/R ).

Really, you COULD find all this in just a few minutes
of web surfing.


Yes but how does gamma relate to clock slowness ;-) like if I want my clock to be 10x slower ... does that mean i need a gamma of 10?

also is R from the EH or from the singularity? I would assume that it would need to be from the EH as the singularity does not exist in our universe right?

For example ... what would be the time dilation at 1 ft outside of the a EH of a BH of one solar mass?

WayneFrancis
2009-May-18, 01:16 AM
I've edited my post Tommac, re-read it ...I've included the appropriate calculation

tommac
2009-May-18, 01:30 AM
So you want the distance from the EH where the escape velocity is 298289729.4m/s

Do you want the distance of this point from the EH relative to the "Earth" location or the "10x slower then earth location"?

Remember the EH of a BH is different depending how far in the gravity well you are.

I think what you want to know is
Given a point in space at 1AU distance from another point in space if you introduce a 1 solar mass black hole at that same point how far would you have to travel towards the BH to get an escape velocity of 298289729.4m/s compared to ~42,100 m/s

the problem with using
Ve = √2GM/r

is that r in that equation is in terms of the pre-bh time space coordinate system.

You can use the formula of
tr/t = √1-rs/r
so 10/1 = √1-(2.94km/r)

My napkin calculation puts you at ~30m from the EH to get a 10/1 time dilation.

Again this "30m" is differenent depending at what reference frame you are observing it.


Thank you.

So if I have an observer B at 30m away from the EH and observer A at 149,600,000 kilometers from the EH. ** clock will tick 1 hour for ever 10 hours that observers A would right?

WayneFrancis
2009-May-18, 01:39 AM
...
For example ... what would be the time dilation at 1 ft outside of the a EH of a BH of one solar mass?


given formula above 1' ~.3048m

tr/1 = √1-rs/r

I get 98.217459x

At 1m from the EH your time dilation is only ~54.23x
at 50m it is ~7.733x

Below is the time dilation and distance in m from the EH


54.23098745 1.00
38.35361782 2.00
31.32091953 3.00
27.12931993 4.00
24.26932220 5.00
22.15851981 6.00
20.51828453 7.00
19.19635382 8.00
18.10156531 9.00
17.17556404 10.00
12.16552506 20.00
9.94987437 30.00
8.63133825 40.00
7.73304597 50.00
5.51361950 100.00
3.96232255 200.00
3.28633535 300.00
2.88963666 400.00
2.62297541 500.00
2.42899156 600.00
2.28035085 700.00
2.16217483 800.00
2.06559112 900.00
1.98494332 1,000.00
1.57162336 2,000.00
1.40712473 3,000.00
1.31719399 4,000.00
1.26015872 5,000.00
1.22065556 6,000.00
1.19163753 7,000.00
1.16940156 8,000.00
1.15181017 9,000.00
1.13754121 10,000.00
1.01459351 100,000.00
1.00146892 1,000,000.00
1.00014699 10,000,000.00
1.00001470 100,000,000.00
1.00000147 1,000,000,000.00
1.00000015 10,000,000,000.00
1.00000010 15,000,000,000.00
1.00000001 150,000,000,000.00

WayneFrancis
2009-May-18, 02:10 AM
Thank you.

So if I have an observer B at 30m away from the EH and observer A at 149,600,000 kilometers from the EH. ** clock will tick 1 hour for ever 10 hours that observers A would right?

yea but you can see that anyone from about 100km out onward would still see a ~10/1 ratio between the clocks

The curvature of space time is a very steep curve for a 1 solar mass black hole.

tommac
2009-May-18, 02:22 AM
yea but you can see that anyone from about 100km out onward would still see a ~10/1 ratio between the clocks

The curvature of space time is a very steep curve for a 1 solar mass black hole.

This is very helpful. Thank you very much.

Now one small follow up.

Say that we had observers A + B at 100 km out. B leaves A and travels to 2 meters outside the EH stays there for 1 year then returns back to A.

You would expect ** clock to be off by roughly 900 years right? in other words As clock would have passed 900 years during ** 1 year.

Do you agree with that?

WayneFrancis
2009-May-18, 02:35 AM
This is very helpful. Thank you very much.

Now one small follow up.

Say that we had observers A + B at 100 km out. B leaves A and travels to 2 meters outside the EH stays there for 1 year then returns back to A.

You would expect ** clock to be off by roughly 900 years right? in other words As clock would have passed 900 years during ** 1 year.

Do you agree with that?

Forgetting the time to get from 100km to 2m and back I would expect 37 years 292 days 21 hours 56 min and 3 seconds to have passed.

How are you arriving at 900 years?

tommac
2009-May-18, 02:38 AM
Forgetting the time to get from 100km to 2m and back I would expect 37 years 292 days 21 hours 56 min and 3 seconds to have passed.

How are you arriving at 900 years?
Yes you are right I was using 1 meter not 2 meters.

WayneFrancis
2009-May-18, 02:54 AM
Yes you are right I was using 1 meter not 2 meters.

no...at 1 meter you are still only talking about ~53years.
To get to 900 years you have to get to about 3cm and while 1m 97cm doesn't seem much when you compare it to the original 100km in terms of the black hole's EH it is a HUGE amount

tusenfem
2009-May-18, 07:30 AM
tommac have you EVER REALLY read a book on relativity?
These are such basic questions, that I think you don't even WANT to learn anything from the answers you get.

tommac
2009-May-19, 11:49 AM
tommac have you EVER REALLY read a book on relativity?
These are such basic questions, that I think you don't even WANT to learn anything from the answers you get.

Yes I have ... and I wanted to post this question here to show adsar that I am not the only one that believes that there is a point where 104:1 time dilation is possible.

grant hutchison
2009-May-19, 12:33 PM
Worth pointing out that the values computed here are in Schwarzschild coordinates. The distances as measured by stationary observers positioned at these Schwarzschild coordinates would be greater. Such observers find that they have a lot of room above the event horizon, even when they're very close to it in Schwarzschild measures.

Grant Hutchison

Ken G
2009-May-19, 02:26 PM
Worth pointing out that the values computed here are in Schwarzschild coordinates. Right, it's purely a coordinate thing. The "time dilation" number doesn't explicitly mean anything physically real, unless the rocket turns around and comes back away from the EH-- and then it's the interaction of the gravity and the acceleration (or more generally, the curved path in spacetime) that "causes" the time discrepancy. As soon as you lose the SR concept of a "global inertial coordinate system", every numerical result for a time that isn't a proper time (i.e., the time on a clock that was present at two events whose interval is being measured) is just purely a mathematical choice.

grant hutchison
2009-May-19, 03:08 PM
Right, it's purely a coordinate thing.And free-falling observers have different coordinates again. Although it's neat that a free-faller who "starts" from rest at infinity will measure radial distances that match the Schwarzschild radial coordinates. :)

Grant Hutchison

Ken G
2009-May-19, 07:13 PM
Indeed, I would argue the only global (i.e., integrated) coordinates any user "has" is the proper time on their own clock. Perhaps one can imagine they also hold a ruler, and if there is a reference object of some kind, like a giant stationary string heading into the EH, they could mark off ticks the size of their ruler as they go by the string, but the choice of that string versus some other still seems like an arbitrary choice of coordinate. I only see the clock as endemic to the observer, nothing else.