Metric Tensor: What is it and how does it relate to Riemannian geometry?

[CaraKboom]

I've been trying to get my head around time dilation in order to understand the relationship between space and time. I do not have a maths brain, I tend to understand things better as narrative concepts.

I've read a lot of introductory material, but I'm having trouble making the link between the theory and reality.

For example, my understanding of spacetime is that it is a mathematical model - grid-like in nature - that allows people to plot, measure, and predict space and time interacting. Time is shown as a dimension along with the three spatial dimensions and this makes it easier for people to... do various math things, that I accept are beyond my understanding at this point. So far, so fabulous.

Then things move along to gravity, and I start to lose the plot. Because apparently gravity is caused by objects with mass warping spacetime. But an object - say a planet - is real, but spacetime is just a model that describes space and time in a way that makes it easier to understand. It's abstract. I can see the earth. I can feel gravity. Spacetime is an idea.

So I go back and try to strengthen my understanding of the actual relationship between space and time. I read the light clock thought experiment. There's a lot of threads around this forum grappling with it and the less mathy ones helped me get my head round it.

So the crux of my problem is this.

I can understand that a light beam traveling between two mirrors could be seen to be traveling slower if the mirrors are moving, due to the extra distance traveled by the beam, and that it will depend on where you are in relation to the clock.

But a clock - any clock at all - is just a device that measures time, not time itself. To me time is the way that humans experience events occurring in sequence.

A person standing next to the clock on a horizontally moving spaceship and pouring a cup of tea should not take longer to fill the cup than a person making tea and viewing the moving clock from earth. The light beam may have to move longer between bounces but the tea should not take longer to hit china just because it's traveling horizontally, regardless of where you're seeing it from. A person returning to Earth after a space journey should not be years older than their twin - the beam of light had longer to travel on the spaceship but how does horizontal movement affect how long it takes your cells to decay? Does the clock have magic powers?? What? What am I missing here?

 

[Matterwave]

The point is, the person on the spaceship will, from your point of view, be pouring tea slower than usual. He will, in fact, age slower than usual. This is not due to "light bouncing", this is due to "time slowing". Literally, when you see something that is moving fast relative to you, you perceive its time to be slower. ALL process are slower, including nuclear decay, and cellular decay. The light clock is just one way to realize this, but it's not due to the light clock that time dilates.

 

[ShayanJ]

Yeah, its not a simple concept to grasp, I understand. The problem is those thought experiments like the light clock are for people who were exposed to the fact that why we need time to slow down in SR in the first place.
As you like narrative concepts, let me tell you a story! Imagine Edd and Liz are a happy couple who are both physicists. They decide to measure the speed of light and so they go to get funding for it. Also they want to do it very far from any planet or star, in free space. But when they are struggling to get funding, the hard condition affects their relationship and so they break up after a quarrel. Now things are different. Each wants to do the experiment sooner and better than the other. So they each go to get funding separately and they both succeed but with the difference that Liz gets a little less than Edd and so has to use a spaceship which is a bit slower than Edd's. Now that's annoying for Liz of course but she can do nothing about it. So they both go to deep space and start measuring the speed of light while Edd is moving with speed V and Liz with speed V-w.
Now Edd measures the speed of light and gets c. Then Liz, uses some spying devices to get Edd's result and finds out that he measured c as the speed of light. She thinks to herself that if he measured c, so why should I bother?! I'm moving with speed V-w and so I should find c+a(the sign and magnitude of a depends on the orientation of Liz's speed w.r.t. the beam of light they're using to measure the speed of light.) So let's just accept c+a and get back soon. But then Liz realizes that this is not what a scientist do. She things she shouldn't let her personal affairs affect her scientific integrity. So she stays and measures the speed of light as being c. She's surprised by this result and also a bit sad because she thinks she's wrong somewhere. But again she says to herself, as a scientist, she should report what she found. So she gets back and reports her result. Now its really tough for her because Edd is famous because of his result and she's just someone who got the same thing later and also probably wrong. But Liz is not someone who just goes to a corner and starts crying. She starts to work on her data to see what was wrong with her experiment.
But she finds out that every thing was right. So why she didn't get c+a, but instead c? How can that be? So she thinks, what if actually speed of light isn't the same as other speeds and doesn't actually depend on which frame of reference you're measuring it in? Let's see what happens if we assumed that! Also in her studies, she found realized that equations in Newtonian mechanics have the same form no matter in what frame of reference you are. But Maxwell's equations don't have this property. These two facts make Liz think that things are not as simple as they thought. So she says to herself, I demand that all physical laws have the same form in all frames of reference and speed of light is the same in all frames of reference and try to see what are the implications. After a week of hard work, she finds out that the Galilean transformations between frames don't satisfy these requirements and she finds out the right transformations. The transformations imply that time gets slower for someone who is moving w.r.t. you and also lengths are smaller and these effects are somehow that completely make up that extra a thing that she expected to get. So Liz was moving with speed w w.r.t. Edd and her c+a was converted to c because of the effects she discovered. She also finds out that if she was the one who measured it first and found out c and then Edd spied on her and though it should be c+b, Edd also would measure c and found what she found. So she finds out that its all symmetric between them. So she understands that because the speed of light should remain the same in all frames, it should be that time and space change accordingly to make up for the relative motion. So she publishes her results and gets even more famous than Edd.
Now Liz thinks about that symmetry between herself and Edd and this makes her go back and think about their relationship. She remembers her mistakes and is now ready to accept if Edd comes by and asks to start their relationship anew. Also when Edd finds out how smart is Liz and how she kept her scientific integrity and this led her to success, and this makes Edd think more too. But now Edd thinks if he goes to Liz, she may think its because of her success. Now this is a tough condition because both want to return but Edd thinks this may be understood badly and Liz wants Edd to ask for starting the relationship again. This is now a friend of both who invites both of them to a party and arranges everything so that everything goes smoothly and now Edd and Liz are again a happy couple, this time also famous scientists.
Did you enjoy the story?

 

[vela]

A person standing next to the clock on a horizontally moving spaceship and pouring a cup of tea should not take longer to fill the cup than a person making tea and viewing the moving clock from earth. The light beam may have to move longer between bounces but the tea should not take longer to hit china just because it's traveling horizontally, regardless of where you're seeing it from. A person returning to Earth after a space journey should not be years older than their twin - the beam of light had longer to travel on the spaceship but how does horizontal movement affect how long it takes your cells to decay? Does the clock have magic powers?? What? What am I missing here?

For both people (let's call them Alice and Bob), time will proceed as normal from their own perspective. Neither will notice the tea taking longer to fill his or her own cup. But if Alice watches Bob pouring tea, she will say it takes longer (as measured by her) for Bob to fill his cup. Similarly, he will say the same thing about Alice because she is moving relative to him. From his perspective, she takes more time (as measured by him) to fill up her cup.

Why does this happen? Because our universe works in such a way that the speed of light is the same for both Alice and Bob. Space and time adjust in just the right way so that this paradoxical fact holds true. In other words, Alice and Bob don't experience space and time in the same way.

 

[THE HARLEQUIN]

the fact is as you say the time a cup actually should take to fill anywhere shouldn't vary ... and actually it doesn't vary ... it depends on the frame of reference you are talking about ..
suppose you are filling up the cup and another person is doing the same except you are traveling in a vehicle ... then what you will see is that the other person is pouring tea slower than you are and the other person will observe just the opposite ...
the fact that yes a clock is just a machine and it just does 1 tick per second as its made to do ... suppose you are in a spaceship and your twin is on Earth then if you synchronize those two clocks ( suppose you send electromagnetic pulse every second ) then the person standing on Earth will count that your time is passing slower as the more distance it needs to travel ... and if u are traveling with v velocity the person on Earth travels with -v velocity from your frame of reference ... so if she sends a pulse you will also get it a bit later than she wanted you to get ... so its pretty funny because both of you will calculate wrong and think that the other person is younger than you ...
now the fact that the Earth has a gyroscopic motion unlike rockets and a rocket doesn't always move with a constant speed and u won't be able to keep contact with others for long in a spaceship ... say if u whirl around the galaxy for a year according to your calendar and clock when you land on Earth you will see a bit more time has passed than u counted ... the other people won't also feel the same that u have passed a bit more time than they have ...
i've tried my best ...

 

[Matterwave]

I'd like to clarify that what you "see" and what you "measure" are not the same in these scenarios. What you "see" will also include Doppler effects due to the person moving towards or away from you (as well as abberation, etc.) and as such, a person moving towards you at high speed will actually "look" sped up instead of slowed down. But once you take this Doppler effect into account, you will find the usual time dilation.

 

[ghwellsjr]

the fact is as you say the time a cup actually should take to fill anywhere shouldn't vary ... and actually it doesn't vary ... it depends on the frame of reference you are talking about ..
suppose you are filling up the cup and another person is doing the same except you are traveling in a vehicle ... then what you will see is that the other person is pouring tea slower than you are and the other person will observe just the opposite ...

No, the other person will observe the same that you are observing. If you see the other person pouring tea slower than you are then that other person will see you pouring tea slower than they are.

the fact that yes a clock is just a machine and it just does 1 tick per second as its made to do ... suppose you are in a spaceship and your twin is on Earth then if you synchronize those two clocks ( suppose you send electromagnetic pulse every second ) then the person standing on Earth will count that your time is passing slower as the more distance it needs to travel ... and if u are traveling with v velocity the person on Earth travels with -v velocity from your frame of reference ... so if she sends a pulse you will also get it a bit later than she wanted you to get ... so its pretty funny because both of you will calculate wrong and think that the other person is younger than you ...

Why do you say both of you will calculate wrong? It's true that according to each of your frames of reference, the other person is younger than each of you. There's nothing wrong with that.

now the fact that the Earth has a gyroscopic motion unlike rockets and a rocket doesn't always move with a constant speed and u won't be able to keep contact with others for long in a spaceship ... say if u whirl around the galaxy for a year according to your calendar and clock when you land on Earth you will see a bit more time has passed than u counted ... the other people won't also feel the same that u have passed a bit more time than they have ...
i've tried my best ...

I'm not sure what you're saying here. I hope you are saying that you will see a bit (actually a lot) more time has passed on Earth than for you and the other people will agree that you have passed a bit less time than they have.

 

[THE HARLEQUIN]

No, the other person will observe the same that you are observing. If you see the other person pouring tea slower than you are then that other person will see you pouring tea slower than they are.

... yes i meant exactly what you meant ... both will count the other persons time slower .. i was trying to say that she ll say " u r slow " and the other person will say , "no , u are slow ."... and yes of course both will calculate wrong in the sense that if the spaceship weren't accelerating in any point (which is not possible) and if the other twin didnt know that she was on a spacecraft , then after her arrival they will look the same ( not that one will be an aged lady and the other will be pretty young ) ... and this doesn't mean that the whole calculation we do for time dilation is wrong ... time dilation is more real than anything ... its just i meant it in the naked eye it seems wrong to a person without the knowledge of frame of reference

 

[A.T.]

But a clock - any clock at all - is just a device that measures time, not time itself.

Physics doesn't make a distinction between "measured time" and "time itself", because it's only concerned with stuff that you can measure.

the beam of light had longer to travel on the spaceship but how does horizontal movement affect how long it takes your cells to decay?

Two ways to understand this:

In general, If a light clock and some other (say biological) process (at relative rest to each other) would not be both affected in the same way, you would get actual paradoxes: What if the light clock releases food rations to the human? For observes at rest to the two, the human would be well fed. But for those who observe the light clock to slow down extremely, the human would starve. Assuming that all processes are slowed down in the same manner, is the simplest way out of this dilemma.

In this specific case, you can also argue that the biological processes are based on chemical reactions and thus electromagnetic forces, which are affected in the same way electromagnetic waves (light) are.

 

[harrylin]

You already received some great answers, so here's a little addition:

I've been trying to get my head around time dilation in order to understand the relationship between space and time. I do not have a maths brain, I tend to understand things better as narrative concepts.

I've read a lot of introductory material, but I'm having trouble making the link between the theory and reality.

For example, my understanding of spacetime is that it is a mathematical model - grid-like in nature - that allows people to plot, measure, and predict space and time interacting. Time is shown as a dimension along with the three spatial dimensions and this makes it easier for people to... do various math things, that I accept are beyond my understanding at this point. So far, so fabulous.

Then things move along to gravity, and I start to lose the plot. Because apparently gravity is caused by objects with mass warping spacetime. But an object - say a planet - is real, but spacetime is just a model that describes space and time in a way that makes it easier to understand. It's abstract. I can see the earth. I can feel gravity. Spacetime is an idea. [..]

Yes indeed; however, it's a little more than an idea: "space" and "time" in a "spacetime plot" account for actual measurements with rulers and clocks. The "warping of spacetime" expresses that the presence of a massive object such as the Earth affects our measurements of height and time. According to general relativity, objects near a heavy mass are deformed (they are shrunken in the direction of the gravitational field) and natural processes are slowed down compared to objects and processes far away from heavy mass.
Verifying the deformation is extremely difficult and I think that it has not yet been done, but verifying the slowdown of natural processes is done as a routine with atomic clocks.

 

[ghwellsjr]

... yes i meant exactly what you meant ... both will count the other persons time slower .. i was trying to say that she ll say " u r slow " and the other person will say , "no , u are slow ."... and yes of course both will calculate wrong in the sense that if the spaceship weren't accelerating in any point (which is not possible) and if the other twin didnt know that she was on a spacecraft , then after her arrival they will look the same ( not that one will be an aged lady and the other will be pretty young ) ... and this doesn't mean that the whole calculation we do for time dilation is wrong ... time dilation is more real than anything ... its just i meant it in the naked eye it seems wrong to a person without the knowledge of frame of reference

I'm still not sure what you are saying. Are you saying that it is possible for one twin to accelerate away, take a long trip, turn around and come back to the other twin and they won't see an obvious difference in their ages? Are you denying that one will be an aged lady and the other will be pretty young? Are you saying that the naked eye is not enough to see a difference in their ages but rather they need to understand Special Relativity and have knowledge of frames of reference in order to establish Time Dilation?

 

[THE HARLEQUIN]

'm still not sure what you are saying. Are you saying that it is possible for one twin to accelerate away, take a long trip, turn around and come back to the other twin and they won't see an obvious difference in their ages? Are you denying that one will be an aged lady and the other will be pretty young? Are you saying that the naked eye is not enough to see a difference in their ages but rather they need to understand Special Relativity and have knowledge of frames of reference in order to establish Time Dilation?

yes i am saying that there won't be any visible difference between them if she always travels with a constant velocity ...but if that's not possible then the space traveller might look younger .
actually traveling in space with constant velocity is not possible in real life and a point on Earth is nowhere near an ideal frame of reference for a rocket ( there are all sorts of motion going on ) ... if u observe very carefully , in real life every second the rocket might be accelerating with a variable acceleration if u take the point from which it left the Earth as the frame of reference ...

 

[harrylin]

yes i am saying that there won't be any visible difference between them if she always travels with a constant velocity ...but if that's not possible then the space traveller might look younger .
actually traveling in space with constant velocity is not possible in real life and a point on Earth is nowhere near an ideal frame of reference for a rocket ( there are all sorts of motion going on ) ... if u observe very carefully , in real life every second the rocket might be accelerating with a variable acceleration if u take the point from which it left the Earth as the frame of reference ...

What matters is speed with respect to inertial reference systems, and not with respect to the Earth which indeed is not exactly an inertial reference system (I hope that that is what you meant!). In such examples the effects of the Earth's speed and gravitation are commonly neglected compared to the effect of a very high speed of the traveller's rocket.

 

[Ibix]

But a clock - any clock at all - is just a device that measures time, not time itself.

...and this clock is doing just that - measuring the passage of time. What's special about the light-clock is that you only need high school maths to show that it must be ticking at different rates according to different observers. Therefore time must be passing at different rates for different observers.

 

[ghwellsjr]

I'm still not sure what you are saying. Are you saying that it is possible for one twin to accelerate away, take a long trip, turn around and come back to the other twin and they won't see an obvious difference in their ages? Are you denying that one will be an aged lady and the other will be pretty young? Are you saying that the naked eye is not enough to see a difference in their ages but rather they need to understand Special Relativity and have knowledge of frames of reference in order to establish Time Dilation?

yes i am saying that there won't be any visible difference between them if she always travels with a constant velocity ...but if that's not possible then the space traveller might look younger .
actually traveling in space with constant velocity is not possible in real life and a point on Earth is nowhere near an ideal frame of reference for a rocket ( there are all sorts of motion going on ) ... if u observe very carefully , in real life every second the rocket might be accelerating with a variable acceleration if u take the point from which it left the Earth as the frame of reference ...

But I didn't ask you about the traveling twin going at a constant velocity, I asked if she accelerates away, that's not constant velocity. And then I said she turns around, that's not constant velocity. And please, let's not get stuck on the fact that we don't have the technology to actually do this experiment.

Let me ask again, suppose both twins are 20 years old when one twin takes a trip away from Earth at 99% of the speed of light where she spends 5 years according to her clock traveling away at 99%c and another 5 years getting back. She will be and look 30 years old when she gets back and her twin will be and look 91 years old, according to Special Relativity. Do you agree?

 

[THE HARLEQUIN]

But I didn't ask you about the traveling twin going at a constant velocity, I asked if she accelerates away, that's not constant velocity. And then I said she turns around, that's not constant velocity. And please, let's not get stuck on the fact that we don't have the technology to actually do this experiment.

Let me ask again, suppose both twins are 20 years old when one twin takes a trip away from Earth at 99% of the speed of light where she spends 5 years according to her clock traveling away at 99%c and another 5 years getting back. She will be and look 30 years old when she gets back and her twin will be and look 91 years old, according to Special Relativity. Do you agree?

general relativity says nothing about an accelerating frame of reference ... note that this post's about general relativity ... and yes they ll look same with acceleration being neglegible ...and u can't prove what u r saying

 

[ghwellsjr]

general relativity says nothing about an accelerating frame of reference ... note that this post's about general relativity ... and yes they ll look same with acceleration being neglegible ...and u can't prove what u r saying

We're not talking about general relativity and we're not talking about an accelerating frame of reference. This thread is not just about general relativity. The OP asked about Time Dilation and "A person returning to Earth after a space journey". That's what you discussed in your first post and what we have been discussing ever since.

Now you are claiming that for the scenario I described, the twins will look the same. That's wrong.

Maybe a spacetime diagram will help:

Are you saying that my spacetime diagram is correct but that the twins will look the same age when they reunite? In other words, are you saying that the red twin's clock will register 30 years since she was born but she will look like she is 91 years old, just like the blue twin?

 

[Nugatory]

yes i am saying that there won't be any visible difference between them if she always travels with a constant velocity

"Constant velocity" isn't a complete specification of her journey and hence of the amount she ages on that journey. To know how much she ages, you have to specify the point in spacetime where the journey starts, the point where it ends, and the path between those two points.

Furthermore, to compare the age of the traveller with her stay-at-home twin both twins must be at the same place at the same time (although not necessarily at rest relative to one another, which makes it easier to construct acceleration-free thought experiments) at the beginning and end of the comparison period. This is why the problem is often posed using idealized twins - we can reasonably assume they were at the same place at the same time at the moment of birth.

 

[THE HARLEQUIN]

We're not talking about general relativity and we're not talking about an accelerating frame of reference. This thread is not just about general relativity. The OP asked about Time Dilation and "A person returning to Earth after a space journey". That's what you discussed in your first post and what we have been discussing ever since.

Now you are claiming that for the scenario I described, the twins will look the same. That's wrong.

Maybe a spacetime diagram will help:

View attachment 77392​

Are you saying that my spacetime diagram is correct but that the twins will look the same age when they reunite? In other words, are you saying that the red twin's clock will register 30 years since she was born but she will look like she is 91 years old, just like the blue twin?

u r talking about minkowski diagram ? yes ... special relativity kinda tries to solve this... i agree with u ... but also raise further many more questions ...
what if both of them are traveling at exactly opposite directions with high speed and meets on Earth after some time what will we see ?

 

[ghwellsjr]

u r talking about minkowski diagram ? yes ... special relativity kinda tries to solve this...

What do you mean, "kinda tries to solve this"? Special Relativity exactly solves this, no equivocation.

i agree with u ... but also raise further many more questions ...
what if both of them are traveling at exactly opposite directions with high speed and meets on Earth after some time what will we see ?

Obviously, they will end up at the same age. What age do you think that will be if they both take the path of the red twin but in opposite directions?

That was one question that was totally unrelated to anything discussed before. What are your many more questions?

 

[THE HARLEQUIN]

the fact is if u take his initial path as the frame of reference ... then u get to see that the Earth is going on a straight line with .99c ... i think the graph is also reversible for the Earth twin ... and t shouldn't also be a straight line for the homebound one cause in space time his point is accelerating millions of time and each acceleration causes his position to change in space time .
i know what u r saying ... i understand that ... special relativity has in fact solved this from all frames of reference ... but i believe there s a massive hole lain beneath those methods .as they all seem pretty misguiding and full of limitations ... and many more question you want to hear ?? i will post them in a thread ...

 

[ghwellsjr]

the fact is if u take his initial path as the frame of reference ... then u get to see that the Earth is going on a straight line with .99c ...

Ok, I'll do that:

i think the graph is also reversible for the Earth twin ... and t shouldn't also be a straight line for the homebound one cause in space time his point is accelerating millions of time and each acceleration causes his position to change in space time .

I have no idea what you are trying to say here. Isn't the Earth twin the same as the homebound one?

i know what u r saying ... i understand that ... special relativity has in fact solved this from all frames of reference ... but i believe there s a massive hole lain beneath those methods .as they all seem pretty misguiding and full of limitations ...

You need to learn more about Special Relativity so you can get rid of those massive holes and your misguided ideas that are full of limitations.

and many more question you want to hear ?? i will post them in a thread ...

Yes, I want to hear your questions. If I can't answer them, someone else can. That's the purpose of this forum.

 

[Dale]

But a clock - any clock at all - is just a device that measures time, not time itself. To me time is the way that humans experience events occurring in sequence.

In relativity there are two distinct concepts of time. One is called "proper time", this is the time that is measured by a clock. The other is called "coordinate time", this is the time as determined by a coordinate system (often an inertial reference frame).

I would probably say that "the way that humans experience events occurring in sequence" is a clock (a rather inaccurate and imprecise clock, but a clock nonetheless). So what you are calling "time" seems to correspond to the concept of "proper time" in relativity.

A person standing next to the clock on a horizontally moving spaceship and pouring a cup of tea should not take longer to fill the cup than a person making tea and viewing the moving clock from earth. The light beam may have to move longer between bounces but the tea should not take longer to hit china just because it's traveling horizontally, regardless of where you're seeing it from.

Proper time is an invariant quantity, meaning that it is the same in all coordinate systems. So all coordinate systems will agree that "the way humans experience events occurring in sequence" will be the same whether you are pouring your tea on Earth or on an earthlike planet moving at near c relative to earth.

What changes from frame to frame is coordinate time. In an inertial coordinate system where one guy is at rest, the other guy's tea-pouring amount of proper time corresponds to a much longer amount of coordinate time.

 

[georgir]

Then things move along to gravity, and I start to lose the plot. Because apparently gravity is caused by objects with mass warping spacetime.

Two comments here:
One, you do not need to care for gravity to understand special relativity. There is no warped space in it. There is time dilation and length contraction, but that is not a warping of spacetime.
Two, "caused" is a bit misleading in that statement. It could just as well have said "gravity is a warping of spacetime". But again, see the last point. You can leave dealing with gravity for later.

I can understand that a light beam traveling between two mirrors could be seen to be traveling slower if the mirrors are moving, due to the extra distance traveled by the beam, and that it will depend on where you are in relation to the clock.

Not quite "depend on where you are in relation to the clock", rather "how the clock moves relative to what you think is rest.".
In other words, you could say that a moving light clock would not function correctly and would show a wrong (slower) time.

But a clock - any clock at all - is just a device that measures time, not time itself. To me time is the way that humans experience events occurring in sequence.

Indeed, a clock is just a clock. Us seeing it go slower when it is moved is no grounds for us declaring time itself is doing anything weird to cause it... it could just be considered broken because of its motion.

... until the principle of relativity kicks in, and we realize that from someone else's point of view, that clock is not actually moving. Our rest clock is moving for them instead. At first we may try to convince them that their clock is the wrong one, but weirdly enough, they are trying to convince us that ours is, and we are getting nowhere... the situation is in fact completely symmetrical. This is the key you need to grasp. Both clocks are equally correct, each is running slower than the other according to different observers.

Does the clock have magic powers??

Well, it is not just clocks. Any process and any measurement will give you the same result, including counting number of cups of tea drank or age of a person.
The magic is not in the clocks, it is in "spacetime" - or if you insist on regarding it as just a mathematical model, the magic is in reality itself, which our mathematical model... well, models.

 

[ghwellsjr]

I can understand that a light beam traveling between two mirrors could be seen to be traveling slower if the mirrors are moving, due to the extra distance traveled by the beam, and that it will depend on where you are in relation to the clock.

Not quite "depend on where you are in relation to the clock", rather "how the clock moves relative to what you think is rest.".

It's important to remember what Matterwave said in post #4:

I'd like to clarify that what you "see" and what you "measure" are not the same in these scenarios. What you "see" will also include Doppler effects due to the person moving towards or away from you (as well as abberation, etc.) and as such, a person moving towards you at high speed will actually "look" sped up instead of slowed down. But once you take this Doppler effect into account, you will find the usual time dilation.

 

[Jimster41]

I have struggled with this so much also. But for me when it clicks it's worth it.

I just came across a Minkowski diagram like the one below in a book. I hadn't seen it before and I found it a helpful thing. My understanding is that the accelerated or "dilated" coordinate frame (x',ct') is just a squished version of the original frame (x,ct). The diamond shapes in it are squished (or dilated) squares in the original. And this is what actually happens in reality (not just in the math or graphs). The geometry of space and time is actually getting distorted.

The trick is that the squishing always occurs in such a way as to keep the blue line (the velocity of a photon traveling the given frame) exactly half-way in between the x and y axes of that frame (as it is in both below). This is saying that the squished geometry is consistent with the un-squished geometry in such a way that if you are inside the squished frame (you got yourself accelerated and dilated), the diamonds still look to you, like squares...

Definitely rocks the imagination...

I think it's also worth pointing out that the theory says you can't actually experience (be an observer) in one moment inside the different geometries shown in the diagram. You are either in the accelerated (dilated) space-time frame or you are in the original un-dilated frame, or you are in some other frame that obeys the same rules. All the geometry in one observer's space-time frame is the same - and though it's not the same as the geometry in the other frames that are accelerated, it is consistent in that to observers in all frames, squares - still look like squares.

Also, as I understand it, we can't tell the difference between the dilation or distortion caused by acceleration in space, and the dilation caused by gravity around mass. My understanding is different from those above who are saying that dilation is not warped space-time. I think GR (General Relativity) says that "gravity" is the general term for space-time dilation (or curvature). So GR says it is exactly that.

To hold onto the feeling this all gives my gut, (like really pleasant vertigo) I like to imagine my body being immersed in a space-time distortion as I stand on the earth... my feet, and all my particles are subject to a continuum of internally consistent but infinitesimally different geometries of space-and-time, because of the gravitational distortion around Earth's mass.

If this sounds like it contradicts the statement above that you can't simultaneously experience two different frames of reference - it kind of does. And it begs the question as to what experience spread across curving space-time means, and also importantly, whether or not space-time geometry is continuous or whether it comes in little chunks. And if it comes in chunks what is happening to two of my particles in different chunks?!

My understanding is that these are very much open questions, being vigorously pursued.

I also love to think about the idea that Photons, because they travel at the speed of light, do not age - so when I stare at a candle, I am looking at little eddy where time, is stopped.

Having said all this (partly because practicing thinking about it helps) the last thing I would want to do is give you an incorrect imago, so I hope someone corrects me here if I am wrong about any of this. I'm really in the same boat as you.

 

[ghwellsjr]

I have struggled with this so much also. But for me when it clicks it's worth it.

I just came across a Minkowski diagram like the one below in a book. I hadn't seen it before and I found it a helpful thing. My understanding is that the accelerated or "dilated" coordinate frame (x',ct') is just a squished version of the original frame (x,ct).

It's not an accelerated frame, it's a frame at a velocity with respect to the original frame. It's also not a "dilated" frame. You should just call it the "primed" frame.

The diamond shapes in it are squished (or dilated) squares in the original. And this is what actually happens in reality (not just in the math or graphs). The geometry of space and time is actually getting distorted.

No, nothing is actually happening in reality, it is just in the math or graphs. The geometry of space and time are not actually getting distorted. It is merely a change in coordinates.

A good analogy is a thermometer with two scales on it, one for Fahrenheit and one for Celsius. The two scales merely let you see what the temperature is in those two systems of units without having to do a calculation. If someone said to you that something was actually happening in reality to the temperature when you go from the Fahrenheit to the Celsius, I'm sure you would think they were nuts.

In the same way, you can look at a position on the Minkowski diagram and see what the coordinates are in the unprimed frame and the primed frame, that is, you could if the diagram included grid lines for the unprimed frame.

I have included such grid lines in this diagram and marked a position on it with a dot:

The coordinates of the dot in the unprimed frame using the squared grid lines are x=3 and t=4. The coordinates of the dot in the primed frame, which is moving at 51%c with respect to the unprimed frame, are approximately x'=1.2 and t'=2.8. Putting two sets of coordinates and grid lines on the one diagram saves you the bother of calculating the primed coordinates using the Lorentz Transformation but if we did, we would see that the primed coordinates are x'=1.15 and t'=2.89.

The trick is that the squishing always occurs in such a way as to keep the blue line (the velocity of a photon traveling the given frame) exactly half-way in between the x and y axes of that frame (as it is in both below). This is saying that the squished geometry is consistent with the un-squished geometry in such a way that if you are inside the squished frame (you got yourself accelerated and dilated), the diamonds still look to you, like squares...

Definitely rocks the imagination...

I think it's also worth pointing out that the theory says you can't actually experience (be an observer) in one moment inside the different geometries shown in the diagram. You are either in the accelerated (dilated) space-time frame or you are in the original un-dilated frame, or you are in some other frame that obeys the same rules. All the geometry in one observer's space-time frame is the same - and though it's not the same as the geometry in the other frames that are accelerated, it is consistent in that to observers in all frames, squares - still look like squares.

You should not think of an observer as being inside just one of the frames any more than you should think of a temperature as being Fahrenheit and not Celsius. All temperatures are both. In the same way all observers are in both the unprimed and primed frames (and all other frames) all at the same time. We might arbitrarily choose to define a scenario according one frame which we usually call the unprimed frame and then transform it to a primed frame. And if you start out at rest with respect to one frame and then accelerate to a speed so that you are at rest in a second unprimed frame, you are still in both frames, it's just that in the first one you start out at rest and end up in motion while in the second frame you start out in motion and end up at rest.

Furthermore, you cannot see the shape of the grid lines (squares or diamonds). You can establish where they are by using radar techniques but it takes time to do this both before and after the coverage of diagram so by the time you get all the information you need, you will be long gone from the area that the diagram covers.

Also, as I understand it, we can't tell the difference between the dilation or distortion caused by acceleration in space, and the dilation caused by gravity around mass. My understanding is different from those above who are saying that dilation is not warped space-time. I think GR (General Relativity) says that "gravity" is the general term for space-time dilation (or curvature). So GR says it is exactly that.

A Minkowsky diagram assumes flat spacetime with no gravity. There is no distortion of spacetime caused by acceleration. I think you have been mislead by the "squished" grid lines on a Minkowsky diagram to think that it is the same or similar to the curved spacetime due to gravity.

To hold onto the feeling this all gives my gut,...

I can't help you with the feelings you get in your gut so I won't comment on the rest of your post.

 

[Jimster41]

I know this isn't a classroom and it's not realistic to expect the volunteer mentors and moderators to step in at every point and educate all us poor souls. And I can also imagine that CaraKBoom has long since tuned out, or gone elsewhere for clarity.

So, I appreciate the corrections ghwellsjr. I'll admit I'm confused, and I'd like to try to keep learning...

I guess I do use the phrase "accelerated frame" to mean a frame at some velocity w/respect to a rest frame, because in order for that frame to have gotten that way (as you later say)... it must have been accelerated. I am not one to throw equations around because I find even the simplest ones confusing at times. I do feel moderately comfortable with one like this though, and I'll add it because it gives me comfort that I am not misunderstanding the relationship of acceleration and velocity. However I probably should just say the primed frame is at some velocity with respect to the un-primed frame.

[itex]a\quad =\quad \frac { { v }_{ 2 }\quad -\quad { v }_{ 1 } }{ { t }_{ 2 }\quad -\quad { t }_{ 1 } } [/itex]

I have done zero calculations of Lorentz transformations but I know enough about them to believe you are correct that the Lorentz transforms map one coordinate system to another. But why the need for a mapping in the case of a change in velocity? Why has the coordinate system changed, why does it need to change? I thought that what Einstein figured out was that it had to change in order to preserve the speed of light. This was the outcome of his "what if you are driving at the speed of light and you turn on your headlights" thought experiment. The baffling result he had to explain, is that you get the exact same speed (this has been experimentally tested lots and lots). Since speed is distance in space divided by time, and supposedly you've added a whole bunch of speed to the ruler (you have on the seat beside you), how come distance/time for light doesn't change?

Bottom line I have to stand by my understanding that space-time geometry is actually getting distorted by mass and relative difference in velocity (acceleration) and that the space time occupied by an observer at some velocity is a different geometric landscape than that occupied by an observer at rest, though the two geometric landscapes are conformally the same - meaning measurements taken in either show the same results. This is just so weird, and yet it's true.

There are too many differences between the physical questions that GR and Psychrometrics are tackling to fully de-tangle your analogy, but I think the Fahrenheit vs. Celsius example is a good one to illustrate just how counter-intuitive Relativity (and real space-time) really is. In the cases of F and C you are right, one phenomenon at one place and one time, with two different rulers, just different labels for the same phenomenon. But GR is about place and time, a whole different level of confusing, at least in my experience.

Here's a random quote from the chapter on "Acceleration and the Warping of Space and Time" from the really great book "The Elegant Universe" by Physicist Brian Greene. (p65, First Vintage Books softcover edition). If CaraKBoom is still around, I would highly recommend this book by the way. He's a spectacular teacher - especially for 20 bucks.

"... And so Einstein realized that the familiar geometrical spatial relationships codified by the Greeks, relationships that pertain to 'flat' space figures like a circle on a flat table [my comment - or squares], do not hold from the perspective of an accelerated observer... In fact, accelerated motion not only results in a warping of space, it also results in an analogous warping of time..."

I'm not sure what's going on with the "radar techniques". I have some feeling you might be teasing me. But if not I'd be interested to hear more what you mean.

 

[ghwellsjr]

I know this isn't a classroom and it's not realistic to expect the volunteer mentors and moderators to step in at every point and educate all us poor souls. And I can also imagine that CaraKBoom has long since tuned out, or gone elsewhere for clarity.

So, I appreciate the corrections ghwellsjr. I'll admit I'm confused, and I'd like to try to keep learning...

I guess I do use the phrase "accelerated frame" to mean a frame at some velocity w/respect to a rest frame, because in order for that frame to have gotten that way (as you later say)... it must have been accelerated.

No, I didn't say that a frame got accelerated. I said that an observer in one frame started out at rest and then accelerated to a constant velocity. But he's still in the same frame, that is, we describe his state of motion according to one frame. First he's at rest and then he accelerates to a constant speed. But the frame hasn't accelerated, it's the same before and after his acceleration. Then I said we could go to a new frame and describe his state of motion from that one where he started out at some constant speed and then accelerated to a stop. Both frames are inertial and we use the Lorentz Transformation to get from one inertial frame to the other. We can also do the same thing for an inertial observer who is at rest in one frame and we can then see what it looks like in another frame moving at any arbitrary speed with respect to his rest frame. There's no acceleration involved.

I am not one to throw equations around because I find even the simplest ones confusing at times. I do feel moderately comfortable with one like this though, and I'll add it because it gives me comfort that I am not misunderstanding the relationship of acceleration and velocity. However I probably should just say the primed frame is at some velocity with respect to the un-primed frame.

[itex]a\quad =\quad \frac { { v }_{ 2 }\quad -\quad { v }_{ 1 } }{ { t }_{ 2 }\quad -\quad { t }_{ 1 } } [/itex]

I have done zero calculations of Lorentz transformations but I know enough about them to believe you are correct that the Lorentz transforms map one coordinate system to another. But why the need for a mapping in the case of a change in velocity? Why has the coordinate system changed, why does it need to change? I thought that what Einstein figured out was that it had to change in order to preserve the speed of light. This was the outcome of his "what if you are driving at the speed of light and you turn on your headlights" thought experiment. The baffling result he had to explain, is that you get the exact same speed (this has been experimentally tested lots and lots). Since speed is distance in space divided by time, and supposedly you've added a whole bunch of speed to the ruler (you have on the seat beside you), how come distance/time for light doesn't change?

Bottom line I have to stand by my understanding that space-time geometry is actually getting distorted by mass and relative difference in velocity (acceleration) and that the space time occupied by an observer at some velocity is a different geometric landscape than that occupied by an observer at rest, though the two geometric landscapes are conformally the same - meaning measurements taken in either show the same results. This is just so weird, and yet it's true.

There are too many differences between the physical questions that GR and Psychrometrics are tackling to fully de-tangle your analogy, but I think the Fahrenheit vs. Celsius example is a good one to illustrate just how counter-intuitive Relativity (and real space-time) really is. In the cases of F and C you are right, one phenomenon at one place and one time, with two different rulers, just different labels for the same phenomenon. But GR is about place and time, a whole different level of confusing, at least in my experience.

Here's a random quote from the chapter on "Acceleration and the Warping of Space and Time" from the really great book "The Elegant Universe" by Physicist Brian Greene. (p65, First Vintage Books softcover edition). If CaraKBoom is still around, I would highly recommend this book by the way. He's a spectacular teacher - especially for 20 bucks.

"... And so Einstein realized that the familiar geometrical spatial relationships codified by the Greeks, relationships that pertain to 'flat' space figures like a circle on a flat table [my comment - or squares], do not hold from the perspective of an accelerated observer... In fact, accelerated motion not only results in a warping of space, it also results in an analogous warping of time..."

I'm not sure what's going on with the "radar techniques". I have some feeling you might be teasing me. But if not I'd be interested to hear more what you mean.

You need to keep GR separate from SR. There is a warping of spacetime in GR due to mass but there is no such thing in SR because it cannot handle effects of gravity. In SR we exclude scenarios that involve gravitational effects from masses. GR is much more complicated to understand than SR. SR is very simple and I would recommend that you master it before you try GR. That's where I am.

I am not teasing you with regard to radar techniques. You can do a search for "radar" with my name and you will find lots of examples where I explain it in great detail. It can also be used for establishing accelerated frames but it doesn't make sense to do it with only one accelerating observer and no other objects for him to bounce his radar signals off of.

Finally, I do not have a high opinion of Brian Greene. His ideas create more confusion than enlightenment and your quote of his is a good example. Accelerated motion does not result in a warping of either space or time and it's a shame he tries to put those words in Einstein's mouth.

 

[Jimster41]

No, I didn't say that a frame got accelerated. I said that an observer in one frame started out at rest and then accelerated to a constant velocity. But he's still in the same frame, that is, we describe his state of motion according to one frame. First he's at rest and then he accelerates to a constant speed. But the frame hasn't accelerated, it's the same before and after his acceleration. Then I said we could go to a new frame and describe his state of motion from that one where he started out at some constant speed and then accelerated to a stop. Both frames are inertial and we use the Lorentz Transformation to get from one inertial frame to the other. We can also do the same thing for an inertial observer who is at rest in one frame and we can then see what it looks like in another frame moving at any arbitrary speed with respect to his rest frame. There's no acceleration involved.

I can understand that a change from rest to some velocity in one frame of reference, can look just like a change from some velocity to rest from another frame of reference. But they can't both be inertial frames can they.

Do I understand your example correctly?
In one frame of reference an observer accelerates from rest to some velocity.
From another frame of reference some observer sees that the first "observer" going from some velocity to rest - at which point they are in the same inertial frame.

Were they in the same inertial frame before observer 1 accelerated from rest?

When I look at the Wiki on "Inertial Frames of Reference" I see.

"All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration. Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity)"

I find the sentence "an accelerometer moving with any of them would detect zero acceleration" confusing and ambiguous though. Two observers in motion can have zero reading accelerometers with them. It is only when you compare their velocity (direction and speed) that you find out whether one is accelerated w/respect to the other?

But maybe this is where I'm missing it.

My perhaps incorrect understanding is that no two things traveling in different directions or in the same direction at different speeds, are in the same "inertial frame" because they are not in a state of "constant rectilinear motion" with respect to each other. Though they can be in the same frame of reference (just not the same inertial frame), and they can be considered to be in their own inertial frames.

Anyway, I'd be the first to say I doubt my understanding of this on a regular basis.

I will google your radar stuff.

 

[Dale]

The point is that you don't need to use an accelerated frame of reference simply because you have an accelerating object. You can analyze accelerating objects from inertial frames.

There seems to be a pedagogical obsession with attaching a frame to every single object. It is unnecessary. Students should be taught to use a reference frame that makes the analysis easy.

 

[Jimster41]

Thanks DaleSpam. The clarification is helpful for me. And I think I have an improved understanding of this distinction. Is it correct to say that no two objects moving with respect to each other can be used to define the same inertial frame of reference. But you can define an inertial frame of reference and have it contain stuff moving at constant velocity, moving with accelerating velocity, or at rest, with respect to that frame?

When you say, Pedagogical obsession and unnecessary, am I correct in interpreting that to mean that for practical purposes the problem of Quantizing Gravity (or space-time geometry) doesn't have to be solved in order to use SR and GR (or Newton for that matter) to calculate the behavior of objects in spacetime? I just read something similar and I can understand why such a thing is worth keeping in mind.

Thanks, and thanks ghwellsjr. I feel I am learning something here.

 

[Dale]

Is it correct to say that no two objects moving with respect to each other can be used to define the same inertial frame of reference.

Not exactly. Suppose that I have a long straight level road with a posted speed limit of 100 kph. Now, I can use that road to define frame F where the road is moving at v=0 (at rest), but v=0 is just arbitrary so I can also use that road to define frame F' where the road is moving a v=-100 kph. I can make that definition for F' using only the road even if there are no cars traveling on the road or even if there are cars on the road and none of them is at the posted speed limit.

If I used the phrase "the road's frame" then that would be understood to mean F. However, that does not mean that F is the only frame that I can define using the road, just that is the specific frame in which the road's v=0.

But you can define an inertial frame of reference and have it contain stuff moving at constant velocity, moving with accelerating velocity, or at rest, with respect to that frame?

Yes.

When you say, Pedagogical obsession and unnecessary, am I correct in interpreting that to mean that for practical purposes the problem of Quantizing Gravity (or space-time geometry) doesn't have to be solved in order to use SR and GR (or Newton for that matter) to calculate the behavior of objects in spacetime?

No, it has nothing to do with quantum gravity. "Pedagogical" means "method of teaching". What I mean is that for some reason the usual approach of teaching SR is to do "thought experiments" where you have Alice, Bob, a Train, and an Embankment and you go about "attaching" reference frames to each of them and then laboriously going through the same scenario from each reference frame. It is silly and it conveys several wrong messages:

1) that reference frames are little bubbles that surround a person and which things can enter or exit
2) that a person MUST use the reference frame where they are at rest
3) that an understanding of a problem requires its solution in every reference frame
4) that all reference frames (inertial or non-inertial) are equivalent

There are times when it is important to be able to transform from frame to frame, but mostly you want to pick the frame where working the problem will be the easiest and simply use that frame. The only time that you need to transform from frame to frame are when your initial conditions are given in multiple frames or when the easy calculation frame is not the frame that you want the answer in.

 

[ghwellsjr]

There are times when it is important to be able to transform from frame to frame, but mostly you want to pick the frame where working the problem will be the easiest and simply use that frame. The only time that you need to transform from frame to frame are when your initial conditions are given in multiple frames or when the easy calculation frame is not the frame that you want the answer in.

I would say there's another time when you need to transform between frames and that is when someone proposes a "paradox" by incorrectly claiming what happens in different frames and you need to resolve the "paradox" by correctly transforming between those frames.

 

[Dale]

That is like saying that you need another beer in the morning to resolve a hangover.

That type of "paradox" is the "hangover" caused by the pedagogical overindulgence in reference frames. It is better to avoid it altogether, IMO.