Is the Universe Precisely Flat and Infinite?

[BurtJordaan]

I fail to spot any spacetime events indicated in either of your last two diagrams. …To make your depictions more understandable, please indicate the two events that you are referring to and in which inertial frames their coordinates are measured.

OK, the two events that I'm interested in is the origin (0,0) and the orange one (d,t') as per black frame, because they would correspond to e.g. the original colocation event and the turnaround event in the 'twins parable' respectively. To make it easier to grasp, let's put in typical "twins-values", e.g. v/c = 0.6 and t=1 yr. Now what is the spacetime interval between these two 'tangible' events?

The green twin is the only one present at both events and her clock reads 1 yr, so for her the spacetime interval is 1 yr (because in her inertial frame she is at dx=0, dt=1). There are no other related events happening in the black twin's inertial frame (after the origin where they have crossed paths). But we know that the spacetime interval in the black frame between the two relevant events must also be 1 yr, by the invariance of the interval between inertial frames.

So I repeat my statement that the black arrow d is simply the coordinate distance between events (0,0) and (d,t') according to the black frame (with the given values, it is 0.6 lyr).

The black arrow d would be the spacetime interval between some (unrelated) red event that happen simultaneously with the orange event as per the black frame, but it would not be simultaneous as per the green frame. I say again, label the horizontal axes "space" and the objection will disappear!

I agree that the your other diagram:

can be called an "interval-time" diagram.

But I fail to see the utility of it in terms of practical situations, where massive objects and photons crisscross spacetime, e.g. what would the world-lines of the twins look like? And of photons send between them?

[BurtJordaan]

Though I prefer a closed curved-space, radial-time model, I don't rely on critical mass-energy density vs. dark energy as the current model does. In fact, I suspect mass-energy actually accelerates cosmic expansion (and may obviate dark energy) in contrast to conventional models.

Then you must have some other theories than QM and Einstein's GR in mind as the foundation - only the energy of the vacuum has properties that is compatible with accelerated expansion. What we can't fathom is why is vacuum energy so weak. It must be leaking to somewhere...

[bangstrom]

But I fail to see the utility of it in terms of practical situations, where massive objects and photons crisscross spacetime, e.g. what would the world-lines of the twins look like? And of photons send between them?

In the diagrams by Minkowski, Epstein, or Faradave, the so-called “speed of light” is represented as a spacetime dimensional constant rather than as a speed which is valid because this is consistent with observations. There is no suggestion in the diagrams that light has a speed.

The delay we see in light-related events is depicted in the diagrams as the constant ratio of space to time where the constant c is equal to one second of time for every 300,000 km of distance. This is a dimensional constant and not a speed. Einstein’s twins do have a speed relative to their background environment of both space and time so they have identifiable paths through space that can be drawn as world lines.

Light acts instantly between source and sink so the delay we observe between the two is due to the presence of time as a inseparable part of the intervening space rather than a delay caused by waiting for the photon to arrive. Light does not have a world line.

Then you must have some other theories than QM and Einstein's GR in mind as the foundation -

Dark matter and dark energy are not a part of either QM or GR so any model that includes either one or both of these can not be said to be “founded” on either QM or GR.

[BurtJordaan]

In the diagrams by Minkowski, Epstein, or Faradave, the so-called “speed of light” is represented as a spacetime dimensional constant rather than as a speed which is valid because this is consistent with observations. There is no suggestion in the diagrams that light has a speed.

It may be so in Epstein and Faradave diagrams, but traditional Minkowski spacetime diagrams show the speed of light as 1 light-second per sec, or whatever units, provided that are the same on the space and time axes have compatible units, so that the light-cone has a 45 degree slope.

The correct statement is that the speed of light is equal to the constant c.

Then you must have some other theories than QM and Einstein's GR in mind as the foundation -

Dark matter and dark energy are not a part of either QM or GR so any model that includes either one or both of these can not be said to be “founded” on either QM or GR.

Not true. GR includes all forms of energy, whether we can 'see' it or not, as long as it produces spacetime curvature. Einstein's full GR equations (10 potentials) of 1916 included the cosmological constant as one of its solutions. As is still practiced today, Einstein selected the simplest possible solution that would conform to what he thought was a static universe at that time.

QM predicts the cosmological constant as the energy of the vacuum, but they have the its density wrong by a factor of some 10⁶⁰.

[BurtJordaan]

We have now deviated into many side-issues, which are important (and interesting) in their own rights, but the central question (topic) is not well discussed yet.

We have seen many references to popular science articles in the chats, which seems to indicate that the (marginally) closed universe gained some traction. The articles are all based on this new physics letter: Planck evidence for a closed Universe and a possible crisis for cosmology, still unrefereed, but reputable enough to make it into preprint.

My take out of reading it is that they did the analysis over many datasets, in which there are some 'tensions', meaning that for some parameters, the published error bars do not overlap with each other. Then they change the modeling of the flat LCDM cosmology by allowing a small positive curvature. This seemed to ease the tensions between some parameters, while making others worse.

The results are hence inconclusive and one will have to wait for more scrutiny of the very complex analysis methods used. The Planck space mission ended in 2013 and since then there have been updated "public data releases" at least every 2 years - the current data is 2018 and is still being analyzed worldwide.

The one thing that I'm very glad of is that it seems to point away from a negatively curved spatial model, which sports horribly complex math (IMO), going into the complex number math regime.

[bangstrom]

My replies are more relevant to the "Photons" thread so that is where I will put them.

[Faradave]

I'm interested in the origin (0,0) and the orange (d,t') as per black frame, … the original colocation event and the turnaround event in the 'twins parable' respectively. … what is the spacetime interval between these?

Thanks for your patience with such thought provoking questions!

"Twins" is an OK term, but I'll refer to black clock and green clock. Humans can't reset ages the way clocks reset times (e.g. synchronizing at the original colocation event). Doing so, with the green clock passing at constant speed, we can neglect start up and stop forces f. However we can NOT neglect the single turnaround force, which is a problem.

I'm happy with the one-way diagram. I'd be just as happy with another made for a return trip but the turnaround force makes combining them more tricky than I thought. I'll try to fix it.

For simplicity, turnaround is treated as a perfectly elastic reflection (ricochet) at a single event. However, force is lightlike. Inasmuch as force is conventionally mediated by massless "force carriers", force points in the same direction. That's why force applies horizontally (like light rays) in interval-time coordinates. Being perpendicular to proper time, this explains why light is non-aging. It also means that force imparts a non-aging component to the worldline of its object.

On a cosmic scale, a curved-space, radial-time model provides a non-aging, tangent path to the future for light quanta. Thus, incident light harbors an age gap compared to subluminal travelers. Similarly, lightlike forces impose a tangent interval component, to an object's worldline, seen from the object's rest frame. (High spatial curvature, seen only at very high speed or early cosmic age, aids illustration.)

Critically, this is how time dilation stops being relative and instead applies unilaterally to the object being accelerated. It's why the traveling clock (or twin) returns having actually aged less. The entire age gap occurs by force application. The rest of the trip out and back (with straight world lines) is just relative clock slowing that each might mutually see of the other.

I say again, label the horizontal axes "space" and the objection will disappear!

As I associate interval-time coordinates with a curved-space, radial time model, technically space can never be truly horizontal. Local flatness makes it almost impossible to distinguish space from horizontal intervals. In the same way, light was considered instantaneous for millennia until sensitive measures became available. At near lightspeed, spatial curvature dramatically increases in the direction of motion (becoming vertical at limit c).

I fail to see the utility of it in terms of practical situations, where massive objects and photons crisscross spacetime, e.g. what would the world-lines of the twins look like? And of photons send between them?

I admit, I developed interval-time coordinates primarily to provide an undistorted (i.e. Euclidean) view of lightlike phenomena such as light transmission, force and fields. I hadn't given much thought to non-inertial frames.

Since we imagine instantaneous turnaround, the traveling clock (green) is literally "forced" to the future abruptly (as if massless). This shows up as a discontinuity in the diagram. Thus, the turnaround is not a single event!

Here an instantaneous turnaround force splits the turnaround "event" in two, each with different coordinate times. This asymmetrically "forces" the moving clock to the a future time without aging.

This should not be too much of a surprise. In depicting light reflection previously, the incident event and the reflection event were discontinuous. Instant turnaround acts the same way.

A conventional rocket engine would generate a small continuous force over a long, curved trajectory instead a discontinuity. But it would accomplish the same unilateral future displacement (i.e. age gap).

[BurtJordaan]

Humans can't reset ages the way clocks reset times (e.g. synchronizing at the original colocation event). Doing so, with the green clock passing at constant speed, we can neglect start up and stop forces f. However we can NOT neglect the single turnaround force, which is a problem.

Yup. Another problem that you face is that forces cause accelerations, but acceleration per se does not cause changes in perfect clock times. So the 'jump' on your diagram seems seriously contrived. And as I tried to explain before, the horizontal axis does not show the relevant spacetime interval.



Your green arrows are simply space-propertime wordlines, identical to Epstein's. Cut out the jump and you have a diagram perfectly consistent with SR. The green clock will obviously not reach the black clock at 2T, but rather at 2T(1-v²/c²)^0.5, as we know is the correct answer. In this type of diagram, a single event is not necessarily a single point, but can be vertically spaced, depending on their respective histories.

I like your term: the turn-around force and acceleration "locks in" the relative aging at that point, although it is perhaps a little misleading. The explanation that stood the test of time is that the force puts the green clock in a different inertial frame and that the two combined green inertial frames simply followed a spacetime path with less elapsed propertime than the 'unforced' black frame.

I would like to see how you would show the same scenario on this interval-time diagram of yours :;=)

[Faradave]

forces cause accelerations, but acceleration per se does not cause changes in perfect clock times. So, the 'jump' on your diagram seems seriously contrived.

Hmm… I'‘m trying to think of an acceleration that doesn't alter clock times. Force and acceleration are pretty tightly linked, so attribution can be ambiguous. Recall that conventional massless "force carriers" are ageless, so they always have stopped clocks associated with total age gaps.

I can't cut out the jump. That's the asymmetric aging part. It's how much less green clock will age than 2t in coordinate time. But it's only discontinuous in the idealized, instantaneous turnaround scenario.

as I tried to explain before, the horizontal axis does not show the relevant spacetime interval.

Points on a circle may be considered normal to a radius at any point (e.g. horizontal to a vertical radius) and the same is true for tangent lines. With local flatness, circumference and tangent may seem to coincide. Nevertheless, light takes a tangent path in interval-time coordinates (horizontal in the diagrams).

Your green arrows are simply space-propertime wordlines, identical to Epstein's. …

Identical to what Epstein was trying to accomplish (Euclidean simplicity). But to call his coordinates "space-propertime" is still a problem. θ correlates to speed (slope is technically inverse speed in spacetime). We seem to agree: t' = tcosθ, where t' is the amount a moving clock is observed to have aged in the black t rest frame. The d's are implied light paths over which those observations are made. Thus d = d' (invariance) and each is horizontal in a clock's rest frame.

In either rest frame d is the horizontal path delivering light and information to the moving clock.

Cut out the jump and you have a diagram perfectly consistent with SR.

Here's the problem. t' goes to zero at 90°. Speed (v = c sinθ) goes to limit c at 90°. In spacetime c = 1 is at 45°. It can't be both. Interval-time puts c horizontal. And light is not a special case. All worldlines have twice the angle in interval-time than they do in spacetime. I think Epstein must give up calling his horizontal coordinate space. Interval time shows superluminal trips going backward in time, which is prevented by unidirectional time. At 45° in spacetime light still appears to be aging (by t’ = tcos45°) and we lose a structural cause for the limit.

The green clock will obviously not reach the black clock at 2t, but rather at 2t(1-v²/c²)0.5

Yup! 2t is the aging of the black non-traveling clock. Green clock has aged less by the amount of the age gap. (∆age on diagram). As indicated on the black coordinate, green has aged 2t – ∆age = 2t –[(2t – t’) – t’], which is just 2t’ = 2tcosθ.


By the identity: cos² + sin² = 1, we have cos = √(1-sin²). Since v = csinθ, cosθ = √(1-v²/c²).
You indicate the green clock aged 2t√(1-v²/c²), which is 2tcosθ. We agree!

a single event is not necessarily a single point, but can be vertically spaced, depending on their respective histories.

That spacing is my indicated age gap (∆age).

The explanation that stood the test of time is that the force puts the green clock in a different inertial frame and that the two combined green inertial frames simply followed a spacetime path with less elapsed propertime than the 'unforced' black frame.

I have no problem with that.

I would like to see how you would show the same scenario on this interval-time diagram of yours

When space and time collapse we're looking at light's rest frame (that is interval rest: ∆d/∆t = 0.) It's difficult to say what a light quantum makes of the meanderings of massive particles (if such a quantum even exists as an object). But there is a lot more that diagram can tell us about light, fields & force.

[bangstrom]

The short answer to this question is, we do not know. The bad part is that if space is precisely flat, we can never experimentally confirm it, simply because we cannot observe anything with infinite accuracy.

Even if curved, the universe should appear flat and accuracy is not an issue. Gravity is curved spacetime and there is no such thing as a straight line or flat surface in a gravitational field. That makes it impossible for the universe to be flat since there is no place in the universe where gravity does not exist. A surface that looks perfectly flat in one gravitational reference frame will appear curved to another.

The observation that we live in a perfectly flat universe is nothing more than the observation that we are observing the universe from a reference frame where our global gravitational field, far from any local gravity sources, is uniform in all directions so the universe appears flat meaning that we are at the precise gravitational center of the universe.

This is no different from the observation that expansion is equal in all directions which places us at the precise center of universal expansion. The appearance of being at the center of the universe is a consequence of our universe having a curved 4D geometry as described by Riemann where any point within a 4D hypersphere appears to be at the exact center for all observers no matter what their location may be. So the appearance of a perfectly flat universe has no meaning except that our universe has a curved Riemann geometry.

[BurtJordaan]

forces cause accelerations, but acceleration per se does not cause changes in perfect clock times. So, the 'jump' on your diagram seems seriously contrived.

Hmm… I'‘m trying to think of an acceleration that doesn't alter clock times.

See Time in a Centrifuge

Your green arrows are simply space-propertime wordlines, identical to Epstein's. …

Identical to what Epstein was trying to accomplish (Euclidean simplicity). But to call his coordinates "space-propertime" is still a problem.

The x-axis is the coordinate distance (space) of the reference frame and the cT-axis indicates both the proper time of the reference frame clock (1 unit) and of the relatively moving clock (0.8 unit), as projected on the cT axis - hence it is a propertime axis.

I'm actually surprised that you did not embrace the Epstein diagram, because it supports your radial time type cosmology! Complete with uniform scaled time. And your radial time is in fact the propertime of comoving observers, who see the CMB as isotropic. The only problem is that Epstein does not really work for curved spacetime, like in our expanding universe - Epstein is a SR 'empty space' construction (no energy).

The d's are implied light paths over which those observations are made. Thus d = d' (invariance) and each is horizontal in a clock's rest frame.

Light paths in Epstein is perpendicular to the worldlines of respective clocks (as in your diagram) and do not represent spacetime intervals. They represent the distance as observed in either frame and since the relative speed is common to both (just of opposite sign) and the time t is the same for both, it is obvious that the d's must be the same. As I pointed out before, the spacetime interval between the two relevant events equals t in your diagram (like in Epstein).

Cut out the jump and you have a diagram perfectly consistent with SR.

Here's the problem. t' goes to zero at 90°. Speed (v = c sinθ) goes to limit c at 90°. In spacetime c = 1 is at 45°. It can't be both. Interval-time puts c horizontal. And light is not a special case.

I hope you now understand that Epstein is not a Minkowski spacetime diagram and that light paths are indeed horizontal. As far as the "jump" is concerned, the turn-around causes problems for standard Epstein as well, because the change of inertial frame of red causes an effective shift in the origin of the diagram due to the relativity of simultaneity. This makes it less intuitive for more than two inertial frames.

That said, space-propertime diagrams in general can handle accelerations pretty well. I have a thread on this in the Physics Form (http://www.sciencechatforum.com/viewtopic.php?f=2&t=32847.

[BurtJordaan]

The observation that we live in a perfectly flat universe is nothing more than the observation that we are observing the universe from a reference frame where our global gravitational field, far from any local gravity sources, is uniform in all directions so the universe appears flat meaning that we are at the precise gravitational center of the universe.

The same arguments hold for flat and for negatively curved space. Riemann geometry works for all three possibilities.

[BurtJordaan]

As far as the "jump" is concerned, the turn-around causes problems for standard Epstein as well, because the change of inertial frame of red causes an effective shift in the origin of the diagram due to the relativity of simultaneity. This makes it less intuitive for more than two inertial frames.

That said, space-propertime diagrams in general can handle accelerations pretty well. I have a thread on this in the Physics Form (http://www.sciencechatforum.com/viewtopic.php?f=2&t=32847.

Reading that old thread again made me realize that it is in fact the no acceleration flyby's and/or instant turnarounds that cause some confusion in 'twin paradoxes'. If we would specify a proper acceleration for the start and turnaround of your green clock, the issues of gap and origin shift become irrelevant.



One can still make the total duration long enough so that the acceleration periods are negligible and the visualization the same. Once you know that green started with acceleration, its propertime can only be t' (your red dot) at the turnaround and not possibly t. No jump or shifting of origin is required and you can just connect the two greens and end up with the propertime of green as 2t'.

The correct view is that acceleration does not change the clock, but it changes the spacetime structure that the clock experience. The problem with the "no acceleration flyby" at the origin is that you don't know to which clock you should ascribe the changed spacetime view. This causes debate as to what the green clock should read just before the turnaround.

Einstein's original version of the moving clock was like that (acceleration stated beforehand). I can't remember exactly what he wrote, but paraphrased it was something like: "if clocks are synchronized and you quickly move one clock some distance away and stop it there, that clock will record less elapsed time than the one not moved". But again, it is movement that defines the difference in elapsed time, not the acceleration or jerk.