Time Dilation with more than one observer

[Diaboro]

I have been thinking recently about this, but I haven't really found a clear answer to this online. Almost all examples consider only two things, a stationary observer+Moving object. Now I wanted to know what exactly would happen if there was more than one observer, each one affecting spacetime differently (Through Gravity or Velocity), and each one can observe one another. As an example, let's remove velocity TIme Dilation and only use Gravitational Time dilation for now. Also, let's specify who exactly are the observers and where they are exactly.

 

The first one is Tim. Tim is orbiting around a Supermassive black. His clock will be ticking the slowest

 

The second one is Kim. She is on Earth. Her clock will be ticking on Earth's Scale

 

The third is Jim. Jim is in the center of a big void area (Affected mostly by his own gravity). His clock will be ticking the fastest

 

Now let's assume they can "observe" each other "Simultaneously"

 

Now to Kim, Tim should seem like he's traveling into the future. To Jim, Kim would also seem like she's traveling into the future, but at the same "time", Jim would also see Tim as he's traveling into the future, but because of the gravitational differences between each other, technically speaking, Jim should see Both Tim and Kim as traveling into the future, but he would see Tim as he's traveling into the future more so than Kim.

 

It's a headspinner this one, so I've attached an illustration to explain my point. Let me know what you think.

 

 

[Iggy]

What is the question?

Ignoring velocity (assuming everything is static) the relationship between observers is,

[math]dt' = dt \left( 1= \frac{\Delta \phi}{c^2} \right)[/math] where [math]\Delta \phi[/math] is the change in gravitational potential between them.

If the earth clock runs half as fast as deep space and the black hole clock runs half as fast as earth then the clock in deep space would run 4 times as fast as the black hole clock. (the inverse of .25 is 4)

Edited February 1, 2013 by Iggy

[beefpatty]

All I could think while reading this was "Poor Jim..."

[Diaboro]

What is the question?

 

Ignoring velocity (assuming everything is static) the relationship between observers is,

 

[math]dt = dt \left( 1= \frac{\Delta \phi}{c^2} \right)[/math] where [math]\Delta \phi[/math] is the change in gravitational potential between them.

 

If the earth clock runs half as fast as deep space and the black hole clock runs half as fast as earth then the clock in deep space would run 4 times as fast as the black hole clock. (the inverse of .25 is 4)

 

You answered it . Just wanted to confirm that this is the case. I would assume this is also true for Relative Velocity Time Dilation?

 

E.X:

 

Person A Stationery, V=0M/S, γ=1

Person B Moving, V=0.87 C, γ=2

Person C Moving, V= 0.97 C, γ=4

 

So does this mean that if Person C experiences 1 second of time, person B will experience 2? And at the same "time", will Person A Experience 4 seconds?

 

 

 

All I could think while reading this was "Poor Jim..."

 

Pictures are taken for a presentation that I will be giving to some High School kids soon. (Have to try to be funny or they will sleep). I actually will be mentioning that Jim will be is in the center of the biggest void discovered by mankind (CMB coldspot)

[Iggy]

You answered it . Just wanted to confirm that this is the case. I would assume this is also true for Relative Velocity Time Dilation?

 

E.X:

 

Person A Stationery, V=0M/S, γ=1[/size]

Person B Moving, V=0.87 C, γ=2[/size]

Person C Moving, V= 0.97 C, [/size]γ=4[/size]

 

So does this mean that if Person C experiences 1 second of time, person B will experience 2? And at the same "time", will Person A Experience 4 seconds?

Velocity is much more complicated because velocity is reciprocal. Gravitational time dilation depends on gravitational potential which isn't reciprocal and therefore much more straightforward. It is ok to say "person A has more gravitational potential than person B" in an absolute sense. It then follows to say "person A's clock is faster than person B's clock".

 

But, with velocity you can't really say "person A has more velocity than person B" because velocity is relative. Compared to A, B has velocity and compared to B, A has velocity. It's complicated and nasty but it really can't be avoided.

 

The best you can say is "from A's perspective, B's clock runs slow (half as fast as A's clock)" and "from B's perspective, A's clock runs slow (half as fast as B's clock)".

 

The concept of reciprocal time dilation is a major pitfall for a lot of people. It is impossible to make complete sense out of it without also knowing about length contraction and the relativity of simultaneity.

 

I could recommend wikibooks -- special relativity but it's pretty heavy reading.

[einystieny]

If we (I am talking about the human race right now) could master that technology then we would have major connundrom as you could theoretically conduct experiments at a faster pace then normal without the need to use large amounts of computing power. However it would/could cause major problems becuase anybody with enough money or influence to get into an area that would allow to happen.

Velocity is much more complicated because velocity is reciprocal. Gravitational time dilation depends on gravitational potential which isn't reciprocal and therefore much more straightforward. It is ok to say "person A has more gravitational potential than person B" in an absolute sense. It then follows to say "person A's clock is faster than person B's clock".

But, with velocity you can't really say "person A has more velocity than person B" because velocity is relative. Compared to A, B has velocity and compared to B, A has velocity. It's complicated and nasty but it really can't be avoided.

The best you can say is "from A's perspective, B's clock runs slow (half as fast as A's clock)" and "from B's perspective, A's clock runs slow (half as fast as B's clock)".

The concept of reciprocal time dilation is a major pitfall for a lot of people. It is impossible to make complete sense out of it without also knowing about length contraction and the relativity of simultaneity.

I could recommend wikibooks -- special relativity but it's pretty heavy reading.

I am forced to agree with Igge as the wikibook -- special relativity is a very heavy read but a very informing read. i would reccomend it to anyone who is interested in special relativity.

[elfmotat]

Let's say that Tim is using a rocket ship to hover at a constant distance from the black hole for simplicity (or else you have to take into account time dilation due to velocity). As long as the black hole is significantly far away from Earth, the approximate relation between the times experienced by each person's clock is:

 

[math]t_{Jim}=\frac{t_{Kim}}{\sqrt{1-R_\oplus/r_\oplus}}=\frac{t_{Tim}}{\sqrt{1-R_{BH}/(R_{BH}+r)}}[/math]

 

where [math]R_\oplus[/math] is the Schwarzschild radius of the Earth, [math]r_\oplus[/math] is Earth's radius, [math]R_{BH}[/math] is the Schwarzschild radius of the black hole, and [math]r[/math] is (approximately) the distance from Tim's ship to the black hole's event horizon.

[Diaboro]

Thank you for all help,

 

Will definitely be taking a look at that book.