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8DA2AZDQV7ZE > Doc # Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays. Analysis of Deterministic Cyclic Gene Regulatory.

By looking further and further with telescopes, we are not "chasing the background radiation out in space", but returning to the point and time of the big bang, when "we" were there. Let's suppose we mentally travel back through space following a direction in the sky. When we reach that particular point back in time when we believe the big bang occurred, where are we? We know what happened at that time: the big bang.

Thus we must be at that singularity point. The point where all of spacetime was concentrated.

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The same is true if we followed another direction in the sky. All directions lead to the same point. Although each direction seems to us straight, we know from relativity that space is curved. It "warps" on itself iii. When we aim to reach that singularity point in time, we reach it in spacetime. The universe is spacetime. We cannot be "outside". Our hypothesis We are not seeing a "halo" of the big bang background radiation , but we are seeing the real thing: the singularity point of the explosion and its observable energy through spacetime, visible as an apparent sphere.

We assume spacetime to be uniformly expanding, as we see it from our spot in the universe. However, we can imagine spacetime as actually curved on itself "inside out" by degrees, between "here, now" and In our ideal representation of the universe, if we keep time as a constant, then the space dimensions must be represented as "warped" or curved iv. Analogy of a flat map of the Earth: Let's imagine a map of the Earth drawn on a flat piece of paper a two dimensional image of a three dimensional sphere.

It can be drawn by maintaining the angles with the meridians v See, for example: maps. If we start by looking at the regions close to the Equator, we see that these are represented with certain relative dimensions that do not warp reality too much, but as we go towards the poles, things appear "warped". For example, northern Canada and Greenland appear bigger than regions closer to the Equator, and appear more distant from each other, until the North Pole appears as a segment as wide as our map when in reality it is a point.

Similarly, our universe appears as a sphere: a three dimensional image of the four dimensional spacetime. Its background after traveling back all the way in spacetime , looks like a sphere, when in reality it was a point.

One way to understand the "warped" spacetime is to think "straight" and compute the mathematics to correct our perception. Some mathematicians can "see" the formulas alive and working in their mind. Most of us instead can better relate to a diagram explaining reality. We can construct a three dimensional model of spacetime, representing the more complicated four dimensional reality. Analogy from elementary geometry: We all know, from elementary geometry, that an ideal line at infinity is cyclical and that parallel line "meet" at infinity. This concept may help visualize our representation of spacetime presented in the following diagram.

Conceptual diagrams of spacetime: As our galaxy travels away from the big bang singularity point at an accelerated speed, our observable universe appears as expanding in every direction, but according to the relativity theory, spacetime is "warped". When choosing a graphic representation for light traveling across a curved space, we could represent light as straight lines and represent spacetime as "warped". However, we do not find this intuitive, as we are accustomed to only three dimensions orthogonal to each other. We find it instead more intuitive to maintain the space dimensions as we are accustomed to, and represent light paths as "curved". A conceptual diagram from our point of view: In the following diagram, we make a further simplification: we represent the moment of the Big Bang and "here, now" in the same point, so that the time coordinate is reduced to a single point where we are.

Their results are published on the 14th September in the journal Science and are based on measurements of an effect called the Shapiro Delay.

It consists of two massive, highly compact neutron stars, each weighing more than our own Sun but only about 20 km across, orbiting each other every 2. It is the only known system of two detectable radio pulsars orbiting each other. Due to the large masses of the system, they provide an ideal opportunity to test aspects of General Relativity:. For these reasons, most physicists who work in general relativity do not believe the pseudo-tensors give a good local definition of energy density, although their integrals are sometimes useful as a measure of total energy.

One other complaint about the pseudo-tensors deserves mention. Einstein argued that all energy has mass, and all mass acts gravitationally. Does "gravitational energy" itself act as a source of gravity? So one can argue that "gravitational energy" does not act as a source of gravity. On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other unlike light waves in Maxwell's linear theory. So one can argue that "gravitational energy" is a source of gravity.

In certain special cases, energy conservation works out with fewer caveats.

The two main examples are static spacetimes and asymptotically flat spacetimes. Light from the Sun appears redshifted to an Earth bound astronomer. In quasi-newtonian terms, we might say that light loses kinetic energy as it climbs out of the gravitational well of the Sun, but gains potential energy. General relativity looks at it differently. In GR, gravity is described not by a potential but by the metric of spacetime.

No problem: the Schwarzschild metric describes spacetime around a massive object, if the object is spherically symmetrical, uncharged, and "alone in the universe". The Schwarzschild metric is both static and asymptotically flat, and energy conservation holds without major pitfalls. For more details, consult MTW, chapter A binary pulsar emits gravitational waves, according to GR, and one expects innocent word! So its orbital period should change.

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Hulse and Taylor were awarded the Nobel prize in Despite this success, Einstein's formula remained controversial for many years, partly because of the subtleties surrounding energy conservation in GR. The need to understand this situation better has kept GR theoreticians busy over the last few years. Einstein's formula now seems well-established, both theoretically and observationally. Each photon gets redder and redder. What happens to this energy? The familiar "expanding balloon speckled with galaxies" belongs to this class of models.

The FRW spacetimes are neither static nor asymptotically flat. Those who harbor no qualms about pseudo-tensors will say that radiant energy becomes gravitational energy. Others will say that the energy is simply lost. It's time to look at mathematical fine points.

## Spacetime curvature measured using a vanishing star - ExtremeTech

There are many to choose from! The definition of asymptotically flat, for example, calls for some care see Stewart ; one worries about "boundary conditions at infinity". In fact, both spatial infinity and "null infinity" clamor for attention—leading to different kinds of total energy. The static case has close connections with Noether's theorem see Goldstein or Arnold.

If the catch-phrase "time translation symmetry implies conservation of energy" rings a bell perhaps from quantum mechanics , then you're on the right track. But two issues call for more discussion. Why does the equivalence between the two forms of energy conservation break down? How do the pseudo-tensors slide around this difficulty? First point: in relativity, energy is not a scalar, but the time component of the energy—momentum 4-vector. If you already know what that means, skip the next paragraph. If you change the coordinate system—say you rotate the axes—then the components change according to a standard formula.

Now in relativity, neither energy nor momentum by themselves have invariant meaning, just like time and space. But if we weld the energy and momentum together, we get a geometric object called a 4-vector that is invariant. That's why we say that the energy is the time component of the energy—momentum 4-vector: it occupies the same slot that time does in the time-space 4-vector.