I How can the Universe know when you change directions?

[jnhrtmn]

The best way to think about time in physics is from Einstein: “Time is what a clock measures”.

Clocks are always some form of oscillator. Math is using time as a sequence marker to describe the oscillation or periodic effect. Send a pendulum into space and the clock stops measuring what it was measuring. Remember and anticipate are in the brain. How can I see time as more than numbers on a page?

 

[Ibix]

How can I see time as more than numbers on a page?

The point of Einstein's "time is what clocks measure" is to suggest that you stop philosophising and follow the evidence. Clocks are apparently measuring something because we get repeatable, predictable and consistent measurement from them.

So go back and look at the last diagram I drew. Notice how the spatial axis of the original frame isn't horizontal? That means that a ruler at rest in that frame is not measuring distance in what this frame calls space, but some mixture of space and time. Likewise, the time axis is not vertical, so clocks at rest in that frame do not measure just what this frame calls time, but some mix of space and time. The point is that it's a mistake to think of "time" as a thing in relativity. There is only spacetime. You may split it into space and time, and it's often convenient to do so, but it's an arbitrary distinction that you choose to make.

That flexibility means that there are actually several distinct concepts in relativity that more or less correspond to the Newtonian notion of time. "Proper time" (proper in the Latin sense where "property" comes from) is what clocks measure. There is a concept in spacetime closely analogous to "distance" in space, and proper time is a measure of that analogue of distance that the clock has travelled. As with ordinary distance, the distance along different paths can be different - this is the resolution of the so-called Twin Paradox, where two clocks take paths of different "length" through spacetime and have different readings at the end.

But proper time is personal - two clocks may start at the same place synchronised, separate and return, and be showing different times (not due to mechanical malfunction) when they meet again because they took routes of different "length". It is possible to define a "global" time, which we call coordinate time by having a flock of clocks at rest with respect to each other and synchronised. But it turns out that the synchronisation process is arbitrary, so that global time comes at the cost of being something you can choose differently.

Fundamentally, spacetime is a 4d manifold. Its equivalent of Pythagoras' Theorem is $\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$, and that one term having an opposite sign is why one "direction" is different from all the others. You can't freely reverse that direction and you can't freely exchange it with the others, and that is why time is different from space.

 

[Sagittarius A-Star]

Send a pendulum into space and the clock stops measuring what it was measuring.

A "pendulum clock" is not a clock. The system of "pendulum clock" and a gravitational field is a clock, according to your own definition:

Clocks are always some form of oscillator.

 

[Nugatory]

Send a pendulum into space and the clock stops measuring what it was measuring

And when we send a wooden ruler into a sufficiently hot fire the ruler stops measuring what it was measuring.

Whether we’re measuring time or distance, we need measuring devices that won’t fail while we’re measuring. That tells us something about how we should design our measuring devices, but has nothing to do with the definitions of time as what a clock measures or distance as what a ruler measures.

 

[cianfa72]

But proper time is personal - two clocks may start at the same place synchronized, separate and return, and be showing different times (not due to mechanical malfunction) when they meet again because they took routes of different "length". It is possible to define a "global" time, which we call coordinate time by having a flock of clocks at rest with respect to each other and synchronized. But it turns out that the synchronization process is arbitrary, so that global time comes at the cost of being something you can choose differently.

Ok, therefore the flocks of clocks entirely filling the space you were talking about, are actually "at rest w.r.t. each other" as measured for instance by constant round-trip travel time of bouncing light/radar pulses exchanged between them. Otherwise which could be a different physical criterion to define "at rest w.r.t. each other" ?

 

[Dale]

Clocks are always some form of oscillator. Math is using time as a sequence marker to describe the oscillation or periodic effect. ... How can I see time as more than numbers on a page?

Is the oscillator a number on a page? The math describes the oscillator, but the oscillator is not numbers on a page.

 

[Dale]

The point of Einstein's "time is what clocks measure" is to suggest that you stop philosophising and follow the evidence. Clocks are apparently measuring something because we get repeatable, predictable and consistent measurement from them.

Personally, I think that it is even more than that. The thing that clocks measure is a useful quantity. It deserves a name. That name is "time". If we didn't call it "time" then we would have to call it something else, and as a practical matter that would be the word that we use for talking about when things happen. Time.

 

[cianfa72]

The math describes the oscillator, but the oscillator is not numbers on a page.

You mean that the math provides "a map" that maps what clocks read into a real number.

 

[Nugatory]

Ok, therefore the flocks of clocks entirely filling the space you were talking about, are actually "at rest w.r.t. each other" as measured for instance by constant round-trip travel time of bouncing light/radar pulses exchanged between them. Otherwise which could be a different physical criterion to define "at rest w.r.t. each other" ?

This is drifting into a thread hijack, but be aware that @Ibix is just providing an abridged description of Taylor and Wheeler's ("Spacetime Physics") model of an inertial frame with associated simultaneity convention.

To avoid further drift, you might want to review that section of Taylor and Wheeler and then if you have followup questions, start a new thread.

 

[jnhrtmn]

Back to Doppler, it seems overcomplicated. Using 0.6c added to c gives 1.6c, (contracted length) / (speed multiplier), 0.8/1.6=0.5. The other is 0.8/0.4=2 This works throughout. Is it because1.6c can't be real, but even sqrt 1.6 is still 1.265. Using 1 instead of contracted length is the classical version. I don't understand the use of source and observer in the math if the relative velocity between the frames is the only factor.

 

[Sagittarius A-Star]

Back to Doppler, it seems overcomplicated. Using 0.6c added to c gives 1.6c, (contracted length) / (speed multiplier), 0.8/1.6=0.5. The other is 0.8/0.4=2 This works throughout.

Yes.
$\sqrt{1+v/c \over 1-v/c} = \sqrt{(1+v/c)(1-v/c) \over (1-v/c)(1-v/c)} = {1 \over \gamma (1-v/c)} [= \gamma (1+v/c)]$

Usually, the $\gamma$ factor in the relativistic Doppler formula is explained via time-dilation, but because of $c=\lambda f$ also an argumentation via length contraction is possible.

Is it because1.6c can't be real

The closing speed between 2 objects can be greater than $c$. Only the speeds of each object relativ to the choosen inertial reference frame cannot be greater than $c$.

I don't understand the use of source and observer in the math if the relative velocity between the frames is the only factor.

See
https://en.wikipedia.org/wiki/Relativistic_Doppler_effect

 

[Dale]

it seems overcomplicated.

Sometime things are complicated. Nature has no obligation to be convenient or simple.

The theoretically predicted and experimentally validated formula is derived here: https://www.mathpages.com/home/kmath587/kmath587.htm

 

[Ibix]

I don't understand the use of source and observer in the math if the relative velocity between the frames is the only factor.

You need to be careful here. If you and I stand stationary near each other, we share a rest frame. I throw a ball to you. The ball has a different rest frame with a single velocity with respect to our frame. But to you, light from the ball is blue shifted while it is red shifted to me. So the $v$ in the Doppler shift formula cannot be a frame velocity since we couldn't get different Doppler shifts if it were. And we could get opposite Doppler shifts by reversing our $x$ axes so the velocity switched from $+v$ to $-v$.

The $v$ in the Doppler shift formula is the closing rate (or separation rate, depending on your sign convention) of the source with respect to a specific observer, not a frame velocity. It will typically be equal to plus-or-minus the velocity of the source's rest frame in the observer's frame, but it is a distinct concept.