On time dilation and length contraction

[trees and plants]

Hello there.About time dilation, could we provide a derivative of time in relation to one of the coordinates of the manifold we have taking time as a function and get something as a result?Or its integral?And about time dilation we have the formula that gives it between two clocks and an observer measuring the clocks so what gravitational potential is needed for the time to change constantly at a specific rate like at a fixed rate?Could someone also help me about how to write math formulas in posts?Thank you.Now, about length contraction could we somehow make perhaps with calculus of moving manifolds about deforming manifolds a way or some formulas to measure how the objects are deformed from length contraction?Thank you.

 

[jbriggs444]

Hello there.About time dilation, could we provide a derivative of time in relation to one of the coordinates of the manifold we have taking time as a function and get something as a result?Or its integral?And about time dilation we have the formula that gives it between two clocks and an observer measuring the clocks so what gravitational potential is needed for the time to change constantly at a specific rate like at a fixed rate?Could someone also help me about how to write math formulas in posts?Thank you.Now, about length contraction could we somehow make perhaps with calculus of moving manifolds about deforming manifolds a way or some formulas to measure how the objects are deformed from length contraction?Thank you.

I am having great difficulty extracting any useful meaning from anything above except the bit about math formulas.

https://www.physicsforums.com/help/latexhelp/

 

[trees and plants]

I am having great difficulty extracting any useful meaning from anything above except the bit about math formulas.

https://www.physicsforums.com/help/latexhelp/

What I want to say is if time is not only a coordinate but a function, could we take its derivative in relation to one of its spatial coordinates or its integral?And the other about length contraction is if we have a point moving on the object that has the length contraction what parts of the object are deformed and in what way, could we with the help of point moving on the object describe its deformation?Thank you.

 

[PeroK]

What I want to say is if time is not only a coordinate but a function, could we take its derivative in relation to one of its spatial coordinates or its integral?And the other about length contraction is if we have a point moving on the object that has the length contraction what parts of the object are deformed and in what way, could we with the help of point moving on the object describe its deformation?Thank you.

Has this text gone through a translation engine?

 

[trees and plants]

Has this text gone through a translation engine?

No, my friend.It hasn't.

 

[PeroK]

No, my friend.It hasn't.

Well, I'm sorry to say, it doesn't make any sense.

 

[trees and plants]

Well, I'm sorry to say, it doesn't make any sense.

I think I used the correct words.What parts of it do
not make sense?I am sorry.

 

[PeroK]

I think I used the correct words.What parts of it do
not make sense?I am sorry.

I can't understand anything about either of your posts in this thread. I understand literally nothing of what you are trying to ask.

 

[trees and plants]

Could time be a function?Is the length contraction the same to all objects?Does it vary?

 

[martinbn]

Could time be a function?

A function of what? What do you mean by time? A timelike coordinate? Then it is a function, just like any other coordinate. It maps points of the manifold to real numbers.

 

[trees and plants]

A function of what? What do you mean by time? A timelike coordinate? Then it is a function, just like any other coordinate. It maps points of the manifold to real numbers.

A function that contains in its formula the three spatial coordinates, like t(x,y,z)=x²+y²+z².

 

[Dale]

What I want to say is if time is not only a coordinate but a function, could we take its derivative in relation to one of its spatial coordinates or its integral?

Yes. The easy way to derive time dilation is as follows:
$$c^2 d\tau^2=c^2 dt^2 - dx^2 - dy^2 - dz^2$$ $$\frac{d\tau^2}{dt^2}=1-\frac{dx^2}{c^2 dt^2}-\frac{dy^2}{c^2 dt^2}-\frac{dz^2}{c^2 dt^2}$$ $$\frac{1}{\gamma}=\frac{d\tau}{dt}=\sqrt{1-\frac{v^2}{c^2}}$$

 

[trees and plants]

Yes. The easy way to derive time dilation is as follows:
$$c^2 d\tau^2=c^2 dt^2 - dx^2 - dy^2 - dz^2$$ $$\frac{d\tau^2}{dt^2}=1-\frac{dx^2}{c^2 dt^2}-\frac{dy^2}{c^2 dt^2}-\frac{dz^2}{c^2 dt^2}$$ $$\frac{1}{\gamma}=\frac{d\tau}{dt}=\sqrt{1-\frac{v^2}{c^2}}$$

Thank you.To take its first derivative, we integrate the second equation with respect to t and to have the integral of t, we integrate the the second equation three times?What about having time as function and its derivatives as a partial differential equation and then try to solve it?

 

[Dale]

To take its first derivative, we integrate

No. The integral is an anti-derivative

What about having time as function and its derivatives as a partial differential equation and then try to solve it?

It seems already solved. What do you think is unsolved?

 

[martinbn]

A function that contains in its formula the three spatial coordinates, like t(x,y,z)=x²+y²+z².

Then, no.

 

[trees and plants]

No. The integral is an anti-derivative

It seems already solved. What do you think is unsolved?

I mean because the second derivative of t is given by the second equation in your post, then we could integrate the equation I think.

 

[martinbn]

I mean because the second derivative of t is given by the second equation in your post, then we could integrate the equation I think.

That is a square not second derivative.

 

[trees and plants]

We have the differential of t but to have t only we integrate I think.Sorry for talking about it too much I think.I am sorry.But about the length contraction there is the formula so I think I get the answer about it.

 

[trees and plants]

That is a square not second derivative.

You are right, it is.It is the differential squared.Another question about time dilation is every external observer measures time dilation as the same?If there are two,three or n external observers I mean.

 

[Dale]

I mean because the second derivative of t is given by the second equation in your post, then we could integrate the equation I think.

No, that is not the second derivative. It is the first derivative squared. A second derivative would be written $$\ddot y(x)=\frac{d^2 y}{dx^2}$$ as opposed to what was written here $$(\dot y(x))^2=\frac{dy^2}{dx^2}$$

 

[Dale]

Another question about time dilation is every external observer measures time dilation as the same?If there are two,three or n external observers I mean.

Time dilation is a relationship between coordinate time and proper time, as shown above. Proper time is invariant. All observers that use the same coordinates will get the same time dilation.

 

[trees and plants]

Time dilation is a relationship between coordinate time and proper time, as shown above. Proper time is invariant. All observers that use the same coordinates will get the same time dilation.

So, if they use different coordinates they get different time dilations.You mean if they use coordinate systems like spherical, cylindrical,,curvilinear etc?

 

[jbriggs444]

So, if they use different coordinates they get different time dilations.You mean if they use coordinate systems like spherical, cylindrical,,curvilinear etc?

Plain old cartesian coordinates with three perpendicular axes for space and a standard of rest so that the time axis can be oriented.

If you change the standard of rest, you get time dilation.

 

[Dale]

So, if they use different coordinates they get different time dilations.You mean if they use coordinate systems like spherical, cylindrical,,curvilinear etc?

Well, changing the spatial coordinates will indeed change the formulas, but to get a substantive difference in time dilation you have to change the time coordinate. E.g. the relativity of simultaneity.

 

[PeroK]

So, if they use different coordinates they get different time dilations.You mean if they use coordinate systems like spherical, cylindrical,,curvilinear etc?

That would be a neat trick: you switch to polar coordinates and your watch runs slow!

 

[vanhees71]

Nothing at all must change by just changing any spacetime coordinates. If it does, you made a mistake. The most "safe" way to do it right is to use the manifestly covariant formalism of tensor algebra and tensor calculus.

 

[Dale]

Nothing at all must change by just changing any spacetime coordinates

Coordinate-based quantities can and do change when changing coordinates

 

[vanhees71]

Of course, components of vectors and tensor change when you change the basis, and often you change the basis when changing the coordinates, but physical quantities must not depend on any of these arbitrary choices. Remembering this from time to time when involved in complicated transformation issues helps!

 

[trees and plants]

Could time be a generalisation of a spatial coordinate?Spatial coordinates show the position but time coordinate changes and moves only as a fourth dimension to the future naturally.

 

[trees and plants]

Another question is could we have change of mass in the measurements while length contraction happens?I think it stays the same according to the math.But could it only in the measurements show this?

 

[jbriggs444]

Another question is could we have change of mass in the measurements while length contraction happens?I think it stays the same according to the math.But could it only in the measurements show this?

By mass we conventionally mean the invariant mass. The term "invariant" means that it does not change depending on one's choice of reference frame. Length contraction, on the other hand, depends on reference frame.

So no, length contraction has no effect on mass. Nor do measurements referenced against any particular reference frame show any effect on mass.

 

[Dale]

Could time be a generalisation of a spatial coordinate?Spatial coordinates show the position but time coordinate changes and moves only as a fourth dimension to the future naturally.

See $dt$ in the first equation in post 12.

 

[jbriggs444]

Could time be a generalisation of a spatial coordinate?Spatial coordinates show the position but time coordinate changes and moves only as a fourth dimension to the future naturally.

If we draw a "timelike" curve or follow the path of a massive particle and if we parameterize that curve (say with parameter $\tau$) and if we use a reasonably ordinary coordinate system then the time coordinate t will always increase along with the $\tau$ parameter, yes.

Technically, it is possible to reverse the direction of the curve so that the time coordinate always decreases with increases in the $\tau$ parameter, but that changes nothing physical.

 

[Marcus Parker-Rhodes]

As a total amateur, my guess is that Universe Function is asking for a way of calculating the "Speed Of Time" (can I say that?) in locations where the average gravity is different from where we are standing. For example, something that has been been puzzling me is that the edges of our galaxy appear to move faster than they should. The centre is deep in a gravity well where passes more slowly than at the edges, which could explain why the galaxy appears a more rigid structure than it really is.

 

[jbriggs444]

As a total amateur, my guess is that Universe Function is asking for a way of calculating the "Speed Of Time" (can I say that?) in locations where the average gravity is different from where we are standing.

Gravitational time dilation scales with gravitational potential (how deep you are in a gravitational well) rather than with gravitational acceleration (how hard gravity is "pulling" on you where you are).

In the case of galaxies, clocks in the core run only a fractional percentage point slow when compared to clocks on the rim.