How would the solar system appear if you approached at near c?

[Chris Miller]

Approaching Earth from 100 light years (by Earth's measurement) at the fastest theoretically (not practically) possible velocity where from your relative near-c frame the distance is foreshortened to 1 Planck length and a century elapses in Earth's time frame to 1 Planck time in yours, how would SR, from your ~c frame, generally describe the Earth's solar orbit? It seems, because everything is flattened to almost 2D in your direction of travel, the Earth would be oscillating back and forth across its orbit's 186 million mile diameter perpendicular to your approach at a very high frequency, which, I know can't be right, since this far exceeds c.

I use the word "present" vs. "see" because I have no clue how you'd observe any of this, but am more interested in how SR would expect Earth's orbit around the sun to present if you, somehow, could.

 

[Ibix]

An orbit is like the pendulum of a clock. How do clocks moving with respect to you look?

There is no "theoretical maximum speed". You can get arbitrarily close to c as measured in some frame, but you can always accelerate more.

 

[PeroK]

Approaching Earth from 100 light years (by Earth's measurement) at the fastest theoretically (not practically) possible velocity where from your relative near-c frame the distance is foreshortened to 1 Planck length and a century elapses in Earth's time frame to 1 Planck time in yours, how would SR, from your ~c frame, generally describe the Earth's solar orbit? It seems, because everything is flattened to almost 2D in your direction of travel, the Earth would be oscillating back and forth across its orbit's 186 million mile diameter perpendicular to your approach at a very high frequency, which, I know can't be right, since this far exceeds c.

I use the word "present" vs. "see" because I have no clue how you'd observe any of this, but am more interested in how SR would expect Earth's orbit around the sun to present if you, somehow, could.

We're back to "time dilation" in a given direction again, I see! Somehow, you need to get the idea out of your head that time dilation is directional.

Time dilation is time dilation. If the Earth clock is moving slow, it's moving slow and the Earth isn't going anywhere fast with respect to the sun. Not in any direction.

 

[Chris Miller]

We're back to "time dilation" in a given direction again, I see! Somehow, you need to get the idea out of your head that time dilation is directional.

Time dilation is time dilation. If the Earth clock is moving slow, it's moving slow and the Earth isn't going anywhere fast with respect to the sun. Not in any direction.

I'm getting contradictory responses to this question, which is why I'm trying to clarify. Most answer that 1 century passes on Earth in the traveler's Planck time interval. It's not in my head that dime dilation is directional, only length foreshortening.

 

[Chris Miller]

An orbit is like the pendulum of a clock. How do clocks moving with respect to you look?

There is no "theoretical maximum speed". You can get arbitrarily close to c as measured in some frame, but you can always accelerate more.

If I am approaching the pendulum, head on, then like a pendulum, unless my velocity brings SR into effect. The theoretical maximum speed is not c, but anything less, of course.

 

[PeroK]

I'm getting contradictory responses to this question, which is why I'm trying to clarify. Most answer that 1 century passes on Earth in the traveler's Planck time interval. It's not in my head that dime dilation is directional, only length foreshortening.

In the reference frame of a particle or spaceship traveling close to $c$
relative to the solar system, then the solar system is foreshortened in the direction of motion and its clocks (including the orbit of the planets, which is a fairly good clock of sorts) run slow.

If you imagine approaching from "above" the plane of the solar system, then it would be normal in size and shape (*), but very thin and traveling very fast towards you. The planets would hardly move as it passes you by.

(*) as I think you understand these are the measurements in your reference frame, not what you would see, which would be significantly affected by the finiteness of the speed of light.

That's just basic SR.

 

[Ibix]

If I am approaching the pendulum, head on, then like a pendulum, unless my velocity brings SR into effect.

I have no idea what you are trying to say here. All I was pointing out was that you can use the Earth's orbital motion as a clock. Do clocks moving with respect to you tick fast or slow?

The theoretical maximum speed is not c, but anything less, of course.

There is no theoretical maximum speed, as I just said. You can always accelerate.

 

[Dale]

Approaching Earth from 100 light years (by Earth's measurement) at the fastest theoretically (not practically) possible velocity where from your relative near-c frame the distance is foreshortened to 1 Planck length and a century elapses in Earth's time frame to 1 Planck time in yours,

Why do you insist on using annoying numbers in your examples? Why not 0.6 c?

If you are not going to do the calculations yourself then just be a little courteous of other people and don't make your question unnecessarily difficult to answer.

 

[Chris Miller]

The "hundred years" bit is flexible and depends on your choice of synchronisation convention. And you are moving into GR territory with talk of the universe (is it really too much trouble for you to write the whole word?) expanding. So the answer is "it depends". The universe around you would be cold and dark by the time one second passed for you, however.

Only that your use of those terms makes your specification of the scenario incomplete. "When", in reference to spatially separated events (like it being 2001 on Earth "when" you pass the star), always needs to come with a specification of a frame: in this case, what you probably meant was something like "the event at which I pass the star is simultaneous, in the Earth's rest frame, to the event of the clock striking noon, Greenwich Mean Time, in London on January 1, 2001". But you need to specify all that to make "when" meaningful. Similarly, you need to specify distances in some particular frame--for example, "at the event when I pass it, the star is 100 light years from Earth in the Earth's rest frame".

Given the specifications as I have fixed them above, the answer to your question is that the event of you reaching Earth will be at noon GMT in London on January 1, 2101, by Earth clocks, plus some very small amount of time which I haven't done the exact math to calculate.

(Peter's response best clarifies my question maybe.) The above to excerpted responses would seem to suggest Earth's clocks running faster, not slower?

 

[Chris Miller]

Why do you insist on using annoying numbers in your examples? Why not 0.6 c?

If you are not going to do the calculations yourself then just be a little courteous of other people and don't make your question unnecessarily difficult to answer.

Thanks, Dale, and sorry. I don't expect anyone to perform any calculations. I'm trying to clarify generalities by using extreme parameters.

 

[PeroK]

(Peter's response best clarifies my question maybe.) The above to excerpted responses would seem to suggest Earth's clocks running faster, not slower?

That looks like an entirely different question. What a muddle!

 

[Chris Miller]

I have no idea what you are trying to say here. All I was pointing out was that you can use the Earth's orbital motion as a clock. Do clocks moving with respect to you tick fast or slow?

Sorry, misunderstood: Slow.

There is no theoretical maximum speed, as I just said. You can always accelerate.

Yes, the maximum theoretical speed approaches c. Thanks for honing my semantics. (I suppose one might say that the fastest theoretical v foreshortens the radius of the universe in the direction of travel to 1 Planck length?)

 

[Chris Miller]

That looks like an entirely different question. What a muddle!

Hm... clearly I don't speak your language yet. To me it's basically the same.

 

[PeroK]

Hm... clearly I don't speak your language yet. To me it's basically the same.

Here's what you can't do:

1) A spaceship is approaching the solar system at nearly $c$ - gamma factor of 50, say. In the Earth's reference frame, the ship's time is dilated (running 50 times slow). From passing Alpha Centauri, the ship takes 4.5 years (Earth time) to reach us. The ship time has advanced only 1 month (approx) in this time. This is all in the Earth's reference frame.

False conclusion: Earth time is faster than ship time.

2) In the ship's frame, the distance from Alpha Centauri is only 1 light month. The journey takes only 1 month for the ship and, in that time, 4.5 years pass on Earth. The ship measures the Earth orbit the sun 4.5 times in 1 month. (False conclusion)

The correct statement 2) is:

2) In the ship's frame, the distance from Alpha Centauri is only 1 light month and that's how long (approx) the journey takes. The Earth clocks are running slow, so only about 12 hours pass on Earth during the journey (as measured by the ship).

False conclusion: Earth time is slower than ship time.

By changing your question from one frame to the next (ask one question in Earth frame and one question in ship frame) you appear to get different answers about whose clock is "really" running slow. Then, you set the two answers up against each other and the waters just get muddier.

Your questions should be posed such as:

In the reference frame of a ship, it passes Alpha Centauri at $t=0$ and reaches Earth 1 month later ...

 

[Ibix]

Sorry, misunderstood: Slow.

Correct.

Yes, the maximum theoretical speed approaches c. Thanks for honing my semantics. (I suppose one might say that the fastest theoretical v foreshortens the radius of the universe in the direction of travel to 1 Planck length?)

I'm not correcting your semantics. I'm telling you that there's no maximum speed in relativity - three times now. Planck's length isn't anything special. It's not the "shortest possible length", if that's what you're thinking.

 

[Dragon27]

It seems, because everything is flattened to almost 2D in your direction of travel, the Earth would be oscillating back and forth across its orbit's 186 million mile diameter perpendicular to your approach at a very high frequency

The Earth would stand still. Like, almost not move at all. Why do you think it wouldn't? Because 100 years should pass while you approach the Earth from 100 light years at almost light speed? Well, it should, in the Earth's frame of reference, not in you near-c frame of reference. In that frame of reference the Earth has already done it's 100 hundred revolutions around the Sun, and now is waiting for you to come by (which would happen in a Planck length over speed of light seconds) practically motionless. You should never forget about the relativity of simultaneity.

 

[Chris Miller]

In that frame of reference the Earth has already done it's 100 hundred revolutions around the Sun, and now is waiting for you to come by (which would happen in a Planck length over speed of light seconds) practically motionless. You should never forget about the relativity of simultaneity.

This is helpful. Thanks. I should probably learn about and understand "the relativity of simultaneity" before I never forget about it. I'm having trouble with the "already" in your explanation.

 

[Mister T]

I'm getting contradictory responses to this question, which is why I'm trying to clarify. Most answer that 1 century passes on Earth in the traveler's Planck time interval.

If a century passes on an Earth clock, that's the elapse of a proper time (measured with a single clock). If $10^{-44}\ \mathrm{s}$ passes on the traveler's clock, that's also the elapse of proper time (measured with a single clock).

There's no meaningful way to say the century elapses here while the $10^{-44}\ \mathrm{s}$ elapses there. It therefore appears to me that your question doesn't make sense.

For Earth and the traveler to each measure an elapsed amount of proper time between the same two events, they would each have to be present at each of those two events. The twin paradox is famous for measuring a difference in this way, but if your traveler and Earth are in inertial motion relative to each other, they can never both be present at two separate events.

I tried to straighten out your thinking on this in another thread. Until you realize that proper time is measured with one clock, and that at least two clocks (or their equivalent) are needed to measure dilated time, you'll never resolve the issue you're attempting to get at with your questions.

I don't expect anyone to perform any calculations.

The only way to fulfill that expectation is to leave numbers out of your discussions.

I'm trying to clarify generalities by using extreme parameters.

They're so extreme that they do the opposite of clarify. With a speed negligibly close to the speed of light all you have to do is mention the value of $\gamma$. For example, $\gamma=100$. Then for every year of proper time that elapses between two events in either of the frames, 100 years of dilated time is measured between those same two events in the other.

 

[Dragon27]

This is helpful. Thanks. I should probably learn about and understand "the relativity of simultaneity" before I never forget about it. I'm having trouble with the "already" in your explanation.

One need to understand exactly what the events look like in different frames of reference, of course. Here's what I meant by "already". We've got two frames of reference: the stationary one and the near-c one (these are just names). In the stationary frame of reference, at $t=0$, there's a spaceship at the origin (which has near-c velocity in the direction of the Earth's orbit), and the Earth 100 light years away from the ship. The Earth has 0 revolution's around the Sun (years) on its clock (for convenience's sake). Now, in the near-c frame of reference, the ship is at the same origin (at $t'=0$), the Earth's orbit is (because of the length's contraction) 1 Planck's length away from the ship, and the Earth has 100 revolutions (and years) on its clock. When did they happen? In our near-c frame of reference, they happened in the past, $t'<0$, before the event $A$ (the ship is 100 light years away in the stationary frame or 1 Planck's length away in the near-c frame from the Earth).

 

[A.T.]

Related:
http://www.spacetimetravel.org/

 

[Dale]

Thanks, Dale, and sorry. I don't expect anyone to perform any calculations. I'm trying to clarify generalities by using extreme parameters.

If you are interested in generalities and not specific numbers then ask the question in terms of generalities instead of specific numbers. All you have to say is "relativistic speed". Everything you add beyond that is a distraction.

Your first post could/should have been something like: "I am traveling inertially at relativistic speed on a path parallel to the plane of the ecliptic. What is the Earth's orbit like in my frame?" That's all.

 

[Battlemage!]

Approaching Earth from 100 light years (by Earth's measurement) at the fastest theoretically (not practically) possible velocity where from your relative near-c frame the distance is foreshortened to 1 Planck length and a century elapses in Earth's time frame to 1 Planck time in yours, how would SR, from your ~c frame, generally describe the Earth's solar orbit? It seems, because everything is flattened to almost 2D in your direction of travel, the Earth would be oscillating back and forth across its orbit's 186 million mile diameter perpendicular to your approach at a very high frequency, which, I know can't be right, since this far exceeds c.

I use the word "present" vs. "see" because I have no clue how you'd observe any of this, but am more interested in how SR would expect Earth's orbit around the sun to present if you, somehow, could.

If the bold is your main question, I think someone pointed out that an orbit can be considered a clock, and if you're at rest in the spaceship then it appears to me that the Earth's orbit would constitute a "moving clock" according to your reference frame. Yes, no? Maybe so?

 

[Chris Miller]

When did they happen? In our near-c frame of reference, they happened in the past, $t'<0$, before the event $A$ (the ship is 100 light years away in the stationary frame or 1 Planck's length away in the near-c frame from the Earth).

Again, very helpful, thanks so much. Been trying to get my head around all weekend. Googled "relativity of simultaneity" which seemed, in the example (Alpha Centauri attacking us) paradox to do with direction. Is there any time in my near-c frame prior to Earth's 100 revolutions around the sun? Seems not possible. If I extrapolate by continuing at this gamma factor of >10⁵⁰ velocity for a second/hour/year, will the universe have "already" (at my t'<0) expanded into flat, dark, cold and be waiting for me? In other words, is an event's occurrence at my t'<0 tantamount to its never having happened in my my frame of reference?

I'm also having trouble resolving the seeming asymmetry of Earth's and my perspectives. E.g., to me Earth appears very close, but to earthlings, although my clock is running similarly slow (virtually stopped), I'm 100 light years away. Why no reciprocal foreshortening?

 

[Dragon27]

Is there any time in my near-c frame prior to Earth's 100 revolutions around the sun? Seems not possible. If I extrapolate by continuing at this gamma factor of >10⁵⁰ velocity for a second/hour/year, will the universe have "already" (at my t'<0) expanded into flat, dark, cold and be waiting for me? In other words, is an event's occurrence at my t'<0 tantamount to its never having happened in my my frame of reference?

It's really hard to understand what you're trying to say. I think it would be better if you took it slow and easy, one basic concept and aspect at a time.
What do you mean by "event at $t'<0$ never happened"? It happened at the time $t'<0$.
An inertial frame of reference exists, so to speak, "forever". There were events in the past (relative to the moment $t'=0$ in it), there will be events in the future. If something happens in the future relative to some event in one frame of reference, it may have happened in the past relative to the same event in some other frame of reference, but only if this "something" happens far enough from the event - far enough to become unreachable.

E.g., to me Earth appears very close, but to earthlings, although my clock is running similarly slow (virtually stopped), I'm 100 light years away. Why no reciprocal foreshortening?

Your ship is foreshortened.

 

[Nugatory]

I'm also having trouble resolving the seeming asymmetry of Earth's and my perspectives. E.g., to me Earth appears very close, but to earthlings, although my clock is running similarly slow (virtually stopped), I'm 100 light years away. Why no reciprocal foreshortening?

All of time dilation and length contraction is inextricably connected to relativity of simultaneity.

The length of something is the distance between where its endpoints are at the same time; thus different observers with different notions of "at the same time" are talking about different things when they talk about "the" length of something. In your example of the ship approaching from alpha centauri, the Earth observer is considering the position of the Earth at the same time that the ship passes alpha centauri while defining "at the same time" using a frame in which the Earth and alpha centauri are at rest. The ship observer is considering the position of the Earth at the same time that the ship passes alpha centauri while defining "at the same time" using a frame in which the Earth and alpha centauri are approaching at high speed. Those are two different points, so naturally they find different distances between them and alpha centauri.

 

[Mister T]

If I extrapolate by continuing at this gamma factor of >10⁵⁰ velocity for a second/hour/year, will the universe have "already" (at my t'<0) expanded into flat, dark, cold and be waiting for me? In other words, is an event's occurrence at my t'<0 tantamount to its never having happened in my my frame of reference?

The "second/hour/year" is an amount of proper time that elapses in the near-c frame. The solar system dying out is an amount of proper time that elapses in the Earth frame. As I told you before, it makes no sense to ask what happens during the amount of proper time that elapses in one inertial frame while a different amount of proper time elapses in another inertial frame when the two frames have a large relative speed. Changing "solar system" to "universe" doesn't alter that.

A remedy is to make this a twin paradox scenario. Spend a year somehow traveling at a large enough speed to make $\gamma \approx 10^{50}$ and somehow manage to return to Earth in the process. When you do that you will find that $10^{50}$ years have elapsed on Earth, but you will be a year older.

(I'm assuming there's enough flat space to accommodate the trip. And that Earth will still be here and will have spent that entire time in inertial motion.)

I'm also having trouble resolving the seeming asymmetry of Earth's and my perspectives. E.g., to me Earth appears very close, but to earthlings, although my clock is running similarly slow (virtually stopped), I'm 100 light years away. Why no reciprocal foreshortening?

You need something located 100 light years away from Earth, as measured in Earth's rest frame. Say it's a planet at rest relative to Earth.

You need something located 100 light years away from you, as measured in your rest frame. Say it's a pod at rest relative to you.

In your rest frame the Earth-planet distance is very small. In Earth's rest frame the you-pod distance is very small.

That's your reciprocal "foreshortening". Although it's more commonly called length contraction.

(I'm assuming there's 100 light years of flat space.)

 

[Dale]

If I extrapolate by continuing at this gamma factor of >1050 velocity for a second/hour/year, will the universe have "already" (at my t'<0) expanded into flat, dark, cold and be waiting for me?

At relativistic speeds the universe is not spatially homogenous or isotropic, so it doesn't have a single uniform temperature. It is hot in front of you and cold behind you. Similarly with density and many other properties.

 

[Chris Miller]

Your ship is foreshortened.

But not the distance between Earth and my ship?

I know I'm failing to grasp some essential concept here, and just keep re-expressing this non-understanding. I get something's happening in my t's past (t<0), with t=0 marking some specific moment (after Earth's century has passed) in my frame. But can't find/imagine/describe a point in my ~c frame that precedes Earth's 100 solar revolutions.

 

[Dale]

But not the distance between Earth and my ship?

The distance between Earth and your ship is not constant. The length contraction formula assumes a constant distance. It simply doesn't apply to the distance between the ship and the earth.

The formula you would use is the Lorentz transform.

 

[Mister T]

But can't find/imagine/describe a point in my ~c frame that precedes Earth's 100 solar revolutions.

Suppose the first of those revolutions occurs during the year 1901 and the 100^th during the year 2000. What do you find/imagine/describe happened during the year, say, 1865?

Among other things, Earth revolved about the sun one time.

 

[Battlemage!]

The distance between Earth and your ship is not constant. The length contraction formula assumes a constant distance. It simply doesn't apply to the distance between the ship and the earth.

The formula you would use is the Lorentz transform.

Quick question regarding this: since the Lorentz transform is Δx = γ(Δx'+vΔt') and Length contraction is L₀/γ = L, is it technically speaking correct that for a transformation where Δt'=0 that you have the length contraction formula? [i.e. Δx = γ(Δx'+vΔt') at Δt' = 0 is Δx = γΔx', which is Δx'/γ = Δx]

 

[Dale]

is it technically speaking correct that for a transformation where Δt'=0 that you have the length contraction formula? [i.e. Δx = γ(Δx'+vΔt') at Δt' = 0 is Δx = γΔx', which is Δx'/γ = Δx]

Not quite. If you use that formula you find that $\Delta t \ne 0$ so the quantity does not represent a length in the unprimed frame. That is because you are transforming the difference between two events instead of the distance between two worldlines

 

[Dragon27]

But not the distance between Earth and my ship?

In the stationary frame, where ship was initially 100 lights away from Earth? No, by definition of this frame. If we place an asteroid, or whatever, at this origin point, then we will have an "asteroid-Earth" object, that is 100 light years long and is at rest in the stationary frame. This object is 1 Planck length long and is moving towards the ship in the near-c frame. Likewise, the ship has normal rest length in the near-c frame and almost zero length in the stationary frame. Seems reciprocal enough.

I know I'm failing to grasp some essential concept here, and just keep re-expressing this non-understanding.

You can just keep hitting the wall with your head, hoping that understanding will somehow just fall onto your head, or you can try and work through a textbook. Special relativity is not too complicated a theory, but it has its subtleties, and if you want to get it, you just have to put in some serious, systematic effort. You have to learn to use and understand space-time diagrams and Lorentz transformation formulas - these are the essential concepts, that you're missing here. All the special effects (time dilation, length contraction, relative simultaneity, etc..) are derived from them.

But can't find/imagine/describe a point in my ~c frame that precedes Earth's 100 solar revolutions.

An inertial near-c frame wouldn't be inertial if it didn't exist forever, so to speak. There should be enough past in it to contain all the Earth's revolutions that have ever happened.

 

[Battlemage!]

Not quite. If you use that formula you find that $\Delta t \ne 0$ so the quantity does not represent a length in the unprimed frame. That is because you are transforming the difference between two events instead of the distance between two worldlines

And length can only be measured if the ends are looked at simultaneously (in your frame), correct? Otherwise you are merely extrapolating?

 

[jbriggs444]

And length can only be measured if the ends are looked at simultaneously (in your frame), correct? Otherwise you are merely extrapolating?

That's not "extrapolating". When you are measuring the distance between the two ends at different times, you are not measuring length at all.

At the risk of repeating an example, if you compute the length of a moving car by subtracting the position of the rear bumper at 2:00 pm from the position of the front bumper at 3:00 pm then your result will not represent the length of the car.