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18

In a question like this you need to ask what does the volume change relative to. So it's a little bit ambiguous. However, the answer to your question is "yes" in the following restricted sense. Imagine having a "swarm" of test objects, with mass so small that their effect on the spacetime around them is negligible. Assume that they are in freefall, i.e. ...


6

I can offer up something similar to this, which is an isomorphism between something called the Tsirelson bound and the spacetime metric. This is not exactly the emergence of spacetime from quantum mechanics, but it does illustrate how spacetime could be seen as quantum mechanics in diguise. Suppose we have four operators $A_1, A_2, B_1, B_2$  such that: $$ ...


5

You are presumably thinking of the FLRW metric for a universe with greater than critical density i.e. a closed universe. We normally use comoving coordinates to describe this, in which case the time coordinate is not curved and at every point along this time coordinate the three spatial coordinates have the topology of a 3-sphere. That is, if we draw a ...


4

The development of general relativity has led to a lot of misconceptions about the significance of general covariance. It turns out that general covariance is a manifestation of a choice to represent a theory in terms of an underlying differentiable manifold. Basically, if you define a theory in terms of the geometric structures native to a differentiable ...


4

It depends what you mean by their clocks both being at noon when they're at positions A and B, and their speeds differing by 161,000 miles per second. By the way the question was stated, I'll assume you meant their positions were measured in an outside observer's frame, i.e., at noon in the outside observer's frame ship A is at position A and ship B is at ...


4

Quite clearly the answer to this is that no, it does not. In particular, consider two inertial observers moving (in flat spacetime) relative to one another. We know that neither of these two observers is more privileged than the other: the laws of physics are the same for each of them and so on. Yet they will draw different hypersurfaces of simultenaity ...


4

The hypersurface of simultaneity does not represent the present. It is just a plane cut through spacetime. If you changed your own state of motion this would tilt the plane by some angle. So the notion of "now" as the hypersurface of simultaneity would depend on your state of motion, which is of course not meaningful. In fact, the notion of "now" itself is ...


4

Conservation of energy relies on the symmetry of your system under time translation (see Noether Theorem). In a system that is not time translation invariant, eg expanding universe, energy doesn't have to be conserved.


4

You sound as though you may have heard of Gullstrand Painlevé co-ordinates, which are a particular system of co-ordinates for labelling spacetime defined by the Schwarzschild metric around a nonspinning, noncharged black hole. The analogy is often made of a "spacetime river" with this depiction; if you stand still with respect to the co-ordinates you are ...


3

We don't believe this is possible. The justification for this belief is nothing less and nothing more than experimental observation. We have never observed a process where an effect comes before its cause, so we simply reason inductively to establish a postulate that the preferred order of events in physical processes is always the same, for any observer. ...


2

Dark energy is a form energy that appears to have a constant density in space even as the universe expands. The simplest model for this is a cosmological constant in the gravitational field equations. This model agrees with observations. What dark energy really is or whether this model is correct is beside the point here. My answer assumes this hypothesis. ...


2

My understanding (which is based somewhat on Jackson's chapter on SR in Classical Elecrodynamics) is that the invariance of the interval is not enough to derive the Lorentz transformations - you also need the second postulate (that the speed of light is constant in all frames). The invariance of the interval follows from the fact that spherical light waves ...


2

The Thorne time machine, a wormhole with one opening accelerated or Lorentz boosted outwards and then conversely brought back, does not permit time travel prior to the Cauchy horizon. This is the point where the time machine is "turned on." This Cauchy horizon has in regions of spacetime prior to its formation a set of curves winding through the wormhole ...


2

You need to learn to use mathematics to tackle such problems. If you try to do without math using only your intuition then you'll make many hidden assumptions that may not be valid. Physicists who understand some theory well enough can get away with using their intuition, but then that intuition is based on a rigorous mathematical understanding of the theory....


2

"If inertia is a property of the matter form of mass-energy, and it is a property that allows for the transfer of energy, then why doesn't the energy dissipated in a vacuum, as does applied radiant/free energy" The problem with your logic is that is flawed. It is equivalent to "some fruits are apples; oranges are fruits: why do not oranges taste like ...


2

The geodesic equation is given by, \begin{equation} \frac{\mathrm d^2}{\mathrm d\tau^2}x^{\mu}+\Gamma^{\mu}_{\lambda\sigma}\frac{\mathrm dx^{\lambda}}{\mathrm d\tau}\frac{\mathrm dx^{\sigma}}{\mathrm d\tau}=0 \end{equation} which is a set of 4 equations for $x^{i}$. $\Gamma^{i}_{jk}$ tells us about the curvature of the space time which can be written in ...


2

A 3D cube with pacman topology is translationally invariant and not rotationally invariant. A space like this is a possible (but unlikely) flat space part of a cosmological spacetime


2

How can we look into the past? Light has a fixed velocity of almost 300.000 meters per second. Sunlight takes about 8 minutes to reach us. So we see the sun always 8 minutes ago. As the other answer says, stars are much further away and it takes light that much longer to reach us. How do we know how far away the stars are? There are various methods that ...


2

Stars are very far away. So light takes a while to get from stars to you. The light arriving now shows you what the stars looked like when the light left. It is like getting a letter from a far away friend. The letter took a few days to arrive. It has news from a few days ago.


2

Within the Schwarzschild metric, the volume does change. It is the rectangle formed by the radial dimension and time which is invariant: The dilating effect of the Schwarzschild metric $$ \mathrm ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 ~\mathrm dt^2 + \frac{1}{1 - \frac{2GM}{c^2 r} }~\mathrm dr^2 + r^2 (\mathrm d\theta^2 + \sin^2 \theta~\mathrm d\...


2

I can answer some of it, and in such a way that it has invariant general relativistic meaning. However, not a general answer. You do have to, and can, treat curvature and some measures of volume invariantly. There are two questions. 1)Does negative/positive curvatures have more volume, that some (in some sense) equivalent spacetime with no curvature? And 2)...


1

The thing about the relativity principle is just that: all non-accelerating frames are equal, in the sense that no inertial frame is more "real" or "accurate" than an other inertial frame. If two frames are moving in a Minkowskian manifold with a constant velocity relative to each other, it doesn't matter which one you choose. No matter what, all the ...


1

Yes, curved spacetime does change the volume of space. When space is curved by mass it is stretched more in some dimensions than others. Picture a balloon being stretched or squeezed--the volume changes.


1

Given a few plausible assumptions about the universe its spacetime geometry is described by a solution to the Einstein equations called the FLRW metric. If we know the densities of various types of matter/energy present, e.g. photons/matter/dark energy/anything else, then we can calculate how the expansion of the universe varies with time. Generally ...


1

This is not what I, and I would posit most physicists, understand as a physical treatment of what general covariance is in physics. General covariance is that the equations look the same in any coordinate frame - any meaning that the transformations can be any function. The only limitation is that the functions be differentiable, maybe n or infinite times (...


1

It is a little different in General Relativity. Let's start with Special Relativity and all the 3 forces of the Standard Model in physics. Then we will talk about gravity and the universe. In The Standard Model spacetime is Minkowski, meaning flat in all 4 dimensions. If it is that way clearly any direction and position is equivalent. That's called ...


1

Infinity is a mathematical concept, as well as the concept of variables describing dimensions. Physics is about observations, either in the laboratory or of the cosmos, which are fitted with mathematical models. It started with the geocentric system, became the heliocentric system and then the realization that the galaxy is composed out of sun like stars, ...


1

Actually, the explanation as to why rotation of a mass affects the metric in principle is simple. Rotation means there is angular momentum, and angular momentum contributes to the energy-momentum-stress tensor in general relativity. If this was a nonrelativistic rotation we would say that the rotation carries kinetic energy. The rotation contributes as a ...


1

As the saying goes, a picture is worth a thousand words. Below is a time-spatial axis diagram of the time and x-axis in two frames. One is at rest relative to the blue, and the other is Lorentz boosted to some velocity. What is the plane of simultaneity is dependent upon the frame that you are on


1

The FLRW energy equation for the motion of test masses in the universe is $$ \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G\rho}{3}. $$ the scale factor for space is $a$ and its time derivative is $\dot a$. I derived this from Newtonian dynamics. The density of mass $\rho$ for the case of a quantum vacuum energy level is constant. I now replace this with ...



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