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# Geometry of Spacetime

Author Topic:   Geometry of Spacetime
NoNukes
Inactive Member

 Message 6 of 41 (701440) 06-19-2013 2:08 AM Reply to: Message 4 by Coyote06-19-2013 12:02 AM

 Why do you + x, y, and z but –t? Why not +t?

You mean -i*t rather than -t since the coordinates are squared in the given equation.

The glib answer would be that the universe just does not work that way. This is not a calculus question per se, but a special relativity question.

http://www.iep.utm.edu/proper-t/#H1

quote:
STR predicts that motion of a system through space is directly compensated by a decrease in real internal processes, or proper time rates. Thus, a clock will run fastest when it is stationary. If we move it about in space, its rate of internal processes will decrease, and it will run slower than an identical type of stationary clock. The relationship is precisely specified by the most profound equation of STR, usually called the metric equation (or line metric equation).

(1) [sorry I'm too lazy to latex the equation]

This applies to the trajectory of any physical system. The quantities involved are:

Dt is the amount of proper time elapsed between two points on the trajectory.Dt is the amount of real time elapsed between two points on the trajectory.
Dr is the amount of motion through space between two points on the trajectory.
c is the speed of light, and depends on the units we choose for space and time.

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 4 by Coyote, posted 06-19-2013 12:02 AM Coyote has taken no action

NoNukes
Inactive Member

 Message 9 of 41 (701473) 06-19-2013 6:44 PM Reply to: Message 7 by New Cat's Eye06-19-2013 11:09 AM

 The time coordinate is at a right angle, but the time direction of your path is only at a right angle to your distance when you're at rest.

What does it mean to be 'at rest'?

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 7 by New Cat's Eye, posted 06-19-2013 11:09 AM New Cat's Eye has replied

 Replies to this message: Message 10 by Taq, posted 06-19-2013 9:11 PM NoNukes has replied Message 13 by New Cat's Eye, posted 06-20-2013 10:47 AM NoNukes has replied

NoNukes
Inactive Member

 Message 12 of 41 (701507) 06-20-2013 10:19 AM Reply to: Message 10 by Taq06-19-2013 9:11 PM

 It means relative velocity. All frames of reference are equally valid, so at rest means not moving in relation to your indicated frame of reference.

Thanks. I was really checking to see what Catholic Scientist meant by 'at rest'?

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 10 by Taq, posted 06-19-2013 9:11 PM Taq has taken no action

NoNukes
Inactive Member

 Message 14 of 41 (701517) 06-20-2013 11:51 AM Reply to: Message 13 by New Cat's Eye06-20-2013 10:47 AM

 The impossible state of not-moving. Since everything is moving, there's always going to be some contribution to your path from the time direction. That is, it would never be at a true right angle.

Taq has anticipated where my question is going. As I understand your answer, you seem to believe that there is some rest state that is impossible to achieve. Instead the situation is that some objects are at rest with respect to some frame, possibly an inertial frame. However there is no special inertial frame which defines motion at rest.

Edited by NoNukes, : No reason given.

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 13 by New Cat's Eye, posted 06-20-2013 10:47 AM New Cat's Eye has replied

 Replies to this message: Message 15 by New Cat's Eye, posted 06-20-2013 12:05 PM NoNukes has seen this message

NoNukes
Inactive Member

 Message 19 of 41 (701559) 06-21-2013 1:22 AM Reply to: Message 18 by nwr06-20-2013 7:34 PM

 If we stretch out the x-axis, say rescale it so that what was one unit of length becomes 2 units, then angles change -- assuming that we don't also stretch out the y-axis. So, in some sense, the magnitude of angles is an artifact of how we measure them.

This is true. However, there is a natural set of units that eliminates this problem. If the time and distance units are chosen so that the speed of light equals one, then we have eliminated the issue you describe above.

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 18 by nwr, posted 06-20-2013 7:34 PM nwr has seen this message

NoNukes
Inactive Member

 Message 21 of 41 (701687) 06-24-2013 3:19 AM Reply to: Message 20 by Iblis06-23-2013 10:06 PM

 Look, in my original dealie, distance is 3, the square is 9; time is 4, the square is 16; 9-16=-7, the root of minus 7 is a broken calculator.

Assuming units where c=1, you can express the proper time as follows:

Where distances are changing at a constant rate with respect to time, then you can use this expression:

This is the form that I have seen used to show how the twin paradox works. By the way, just because an expression generates an error on your calculator does not mean that the expression is improper.

Edited by NoNukes, : No reason given.

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 20 by Iblis, posted 06-23-2013 10:06 PM Iblis has replied

 Replies to this message: Message 22 by Iblis, posted 06-25-2013 10:01 PM NoNukes has replied

NoNukes
Inactive Member

 Message 23 of 41 (701769) 06-25-2013 10:13 PM Reply to: Message 22 by Iblis06-25-2013 10:01 PM

 This will give us i's in the right place, I suspect. Why is this different from what we were seeing before, as from Son Goku for example?

You can develop the equations with either negative time contributions and positive space contributions or vice versa.

The "d"s in the first equation indicate that dt, dx, dy, dz are differential values. The integral is a line integral over the trajectory of a particle.

In the equation with the deltas, deltas mean 'change'. The second equation can be be used where the change in time, and x, y, z coordinates in a case where those quantities each vary linearly with time from a starting point to the ending point.

Your questions suggest that you've got quite a bit of studying including learning a tiny bit of math before you are going to understand special relativity. Try reading the wikipedia article I linked as a starting point.

Edited by NoNukes, : No reason given.

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 22 by Iblis, posted 06-25-2013 10:01 PM Iblis has taken no action

NoNukes
Inactive Member

 Message 27 of 41 (701958) 06-28-2013 10:47 AM Reply to: Message 24 by Iblis06-28-2013 12:14 AM

Re: Anybody?
 why we are acting like this...are in some sense interchangeable, or descriptive of the same situation, or, whatever it is they are that allows them both to be here.

I have some patience, but to date you don't seem willing to make much of an effort on your own.

Let's discuss what the equations are mean. The equation for proper time gives the time that an observer traveling a trajectory would measure. Observers not following that same trajectory could measure different times. This is explained in a fairly detailed manner with a couple of example calculations at the Wikipedia article on proper time located here.

We might also discuss the "proper length" between events rather than proper time. That formulation leads to those equations with the negative signs.

http://en.wikipedia.org/wiki/Proper_length

[quote]In special relativity, the proper length between two spacelike-separated events is the distance between the two events, as measured in an inertial frame of reference in which the events are simultaneous. So if the two events occur at opposite ends of an object, the proper length of the object is the length of the object as measured by an observer which is at rest relative to the object.[\quote]

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 24 by Iblis, posted 06-28-2013 12:14 AM Iblis has replied

 Replies to this message: Message 29 by Iblis, posted 07-12-2013 11:24 PM NoNukes has replied

NoNukes
Inactive Member

 Message 31 of 41 (702979) 07-13-2013 11:16 AM Reply to: Message 29 by Iblis07-12-2013 11:24 PM

Re: Anybody?
 If the events were simultaneous, what would the t represent?

Did you even bother to look at the example calculations in those wikipedia articles I recommended?

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 29 by Iblis, posted 07-12-2013 11:24 PM Iblis has replied

 Replies to this message: Message 32 by Iblis, posted 07-13-2013 9:58 PM NoNukes has replied

NoNukes
Inactive Member

 Message 33 of 41 (703025) 07-13-2013 11:01 PM Reply to: Message 32 by Iblis07-13-2013 9:58 PM

 I am noticing that there are other equations in the article, and some multiplication by c-squared and so on.

Yes, but when distance is measured in light-years and time is measured in years then c squared = 1.

 But I'm giving you the benefit of the doubt

Don't be silly. There are no proper length examples. However the discussion in the article answers the question you asked me. If you believe I've provided you with an improper equation, I'll be happy to acknowledge and correct that.

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 32 by Iblis, posted 07-13-2013 9:58 PM Iblis has replied

 Replies to this message: Message 34 by Iblis, posted 07-13-2013 11:52 PM NoNukes has replied

NoNukes
Inactive Member

 Message 35 of 41 (703032) 07-14-2013 12:00 AM Reply to: Message 34 by Iblis07-13-2013 11:52 PM

Re: Did you even bother to look
 If the c² is just 1=discardable, then my proper length appears to be the same as my proper time

With c² = 1, the equation appears to be exactly the same as what I posted. Thanks we can do away with that accusation.

By discussion I definitely don't mean what's posted on the talk page. Instead I mean the discussion that talks about the meaning of proper length and proper time.

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 34 by Iblis, posted 07-13-2013 11:52 PM Iblis has replied

 Replies to this message: Message 36 by Iblis, posted 07-14-2013 12:13 AM NoNukes has replied

NoNukes
Inactive Member

 Message 37 of 41 (703038) 07-14-2013 1:06 AM Reply to: Message 36 by Iblis07-14-2013 12:13 AM

Re: Did you even bother to look
 So why is it being called invariant

Imagine an object moving relative to the earth. The observed length of the object varies with the velocity relative to earth, but the proper length is independent of the relative velocity. The proper length then is a property of the object, while in general, the observed length is not.

I yield. This is not fun anymore, and I usually enjoy talking about relativity.

Edited by NoNukes, : No reason given.

Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)

I would say here something that was heard from an ecclesiastic of the most eminent degree; ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how the heaven goes.’ Galileo Galilei 1615.

If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass

 This message is a reply to: Message 36 by Iblis, posted 07-14-2013 12:13 AM Iblis has taken no action

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