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10

To make progress we need to be clear what we mean by the laws of physics and observer. A law of physics is just some set of equations that we use to predict what happens. So if for example we're trying to describe how charges interact with light our set of equations, i.e. our law of physics, would be Maxwell's equations. But to write down Maxwell's ...


6

In special relativity, the metric on spacetime is $$\mathrm{d}s^2 = \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2 - \mathrm{d}t^2$$ (or with inverted signs). If you now formally transform $t \mapsto \mathrm{i}t$, this becomes the familiar Euclidean metric on $\mathbb{R}^4$ $$\mathrm{d}s^2 = \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2 + \mathrm{d}t^2$$ ...


4

I think you should understand the passage a bit more abstractly: Take the space $\mathbb{R}$. It's obviously one-dimensional. Now, consider the space $\mathbb{R}\times\mathbb{R} = \mathbb{R}^2$, the vector space over $\mathbb{R}$ with two dimensions. You have thus created a two-dimensional object from a one-dimensional one. Let us now construct the book: ...


4

Imaginary time has no physical meaning. It is an assumption physicists make, namely, that the math will endure the analytic continuation of the time variable onto the complex plane, which makes some calculations easier, but it is not absolutely necessary (e.g. you can do instantons without imaginary time).


3

You're asking two separate questions. To take your second question first, the existance of seven extra spacelike dimensions is a requirement for the consistency of string theory and we have no experimental evidence that extra dimensions exist or that string theory is a good description of reality. So it's impossible to make any definitive comment about why ...


3

Why time is considered to be a dimension? Because, to the extent of the empirical evidence, relatively moving inertial observers are related by the Lorentz transformation. But, the Lorentz transformation mixes time and space coordinates in a particular way. If time were not a dimension, if time were just a universal parameter, this mixing would not be ...


3

You shouldn't use the "subjective/objective" distinction for a place where "relative/absolute" is much more appropriate, because they mean different things. For something to be subjective, it must be dependent on the knowledge or state of mind of an observer. As an example, suppose we define "depth" as "length along the direction an observer is facing". ...


3

I will expand my comment above into an answer, but I will not comment further on it to avoid the usual very long discussions of your posts. In my opinion, you are trying to argue on a logical level, but it is not clear if you have enough knowledge of logical theories to do so on a mathematical/physical level. Without entering too much into details, a ...


3

I've said it before and I will say it again: There are no frames travelling at the speed of light As David Z says in the very link you give, it is meaningless to ask what you would perceive travelling at the speed of light. You cannot. And even though there are particles that can, there are no frames associated with them. Have a look at the Lorentz boost. ...


3

At some time, you need physical postulates. For instance, suppose these two possible transformations applying to the infinitesimal space-time components $dt$ and $dz$, between a frame $R$, and an other frame $R'$ moving at a velocity $v_z=v$ relatively to $R$ : $$\begin{pmatrix} dt'\\dz'\end{pmatrix} = \begin{pmatrix} \cos \lambda & \sin ...


2

The planck length is not necessarily an absolute limit to how small thing can be sub divided. The planck length is theoretical and it is empirically defined by dimensional analysis. At this length scale our knowledge of physics makes no sense. The planck length $\ell_P$ is defined as: $\ell_\text{P} =\sqrt\frac{\hbar G}{c^3} \approx 1.616\;199 (97) \times ...


2

The "Mexican Hat Potential" (although now more politically correctly called the "Champagne Bottle Potential" after the punt at the base) is the potential energy curve for the vacuum expectation value (VEV) of the Higgs field. Think of the blue dot as being "the vacuum", and the radial direction as turning up the strength of a background field that permeates ...


2

It means that time is no longer an absolute concept, yes. The time a specific observer experiences in a specific frame of reference, i.e. his proper time depends on the path (worldline) he takes through spacetime. In other words, it depends on his state of motion, the way he accelerates. This is the reason for the famous twin paradoxon: the resolution is ...


2

Ok, before we fill up the comment section with this, I will write this as an answer: Proper time $\tau$ along a path $\gamma$ is $$ \tau := \int_\gamma \sqrt{\mathrm{d}x^\mu\mathrm{d}x_\mu}$$ and a clock moving along $\gamma$ will have $\tau$ as its elapsed time at the end of the path. Yet, the definition of proper time $\tau$ involves such clocks not ...


2

You are still trying to use intuition about space-time that is simply wrong. Experimentally wrong. The world doesn't work the way you (and I) think it does. Spatially separated clocks can not be synchronized in all frames. No, really. They can't. If you synchronize clocks A & B in their common rest frame (which can be done) then they are unavoidable ...


2

Give a physically distinguishable definition of "out there" vs. "in the mind" and we can try to discuss this further. As jinawee comments, there are no frames of reference that move with the speed of light, since the photon we "ride on" would have no speed at all by definition of a comoving reference frame, and that contradicts the constancy of the speed of ...


2

If two events both have a spacetime interval of zero, can they both be said to be happening “now”? There is an interval associated with any two events but there is not an interval associated with an event. From the Wikipedia article "Spacetime": In spacetime, the separation between two events is measured by the invariant interval between the ...


2

The idea is that the object generating the "pit" in the front is in the center of the flat region in the middle. What happens is that the object in the middle begins to "fall" into the "pit" in front of it, due to gravitational attraction. The "pit", however, moves forward because it is a fixed distance away from the object in the middle. Basically, as ...


2

Events which lie within each other's light cones are called "timelike separated." All observers agree on the ordering of these events. Events which lie on each other's light cones are separated by "lightlike" or "null" interval. All observers also agree about the time ordering of these events. Events which lie outside of each other's light cones are called ...


1

I do not find easy to understand your calculations, but can give you an explanation which is not based on specific distances. It is easy to see why the observer inside the ship will perceive the events as simultaneous and the one outside the ship will not. First, notice that for every observer the speed of light is the same, c. So the observer on the ships ...


1

Even if there is no agreement in the physics community about what is special about "NOW", I believe that most physicists that believe in a block universe would agree with your statement that there are an infinite number of me's at every point in time, all experiencing their own now. Not only that, there is an infinite number of mathematical universes (see ...


1

Then A will observe that B's time is elapsing more slowly than its own. It will also notice that B is shorter in the direction of travel than A is. But, it is also true that B will observe that A's time is elapsing more slowly than its own and B will also notice that A is shorter in the direction of travel than B is. This is because motion is ...


1

You seem to be interested in the concept of dualities. Dualities are incredibly informative in Physics in that every time we've come across one, it's led to unification of the two dual entities. You've already stated the most common one of spacetime. This was of course unified by relativity. You mentioned the Mass-Energy duality. This arises right from ...


1

No, and the reason is quite simple. The proper time of a photon is zero (according to the principles of special relativity). That means that there is no time difference between the place of emission and absorption. By this, any hypothetical observation of a photon would be reduced to a time period of zero, and it would not be able to distinguish/ to ...


1

Now take a third clock, clock C, which is also synced with A and B and have it travel the distance between A and B at .866c That's impossible. Let clock C, traveling at $0.866c$, pass by clock A just when both clocks read $t_C = t_A = 0$. Now, according to clock A, clock B also reads $t_B = 0$ at this instant. But, according to clock C, clock B ...


1

In special relativity the rule of thumb is Moving Clocks run slowly. So in your inertial reference frame. The clock is moving towards you at 0.866c and thus is running slower than a clock you keep in your frame of reference. This means the clock will pulse out the light less often. With the light coming towards you at 1.00c and you moving towards it ...


1

First off, I'm not entirely sure of what you are asking, or what you are thinking of as curvatur. There are certainly coordinate systems which are non-euclidean that are not considered to be "curved." For instance standard cylindrical coordinates have zero Riemann Curvature, but they are "curvy looking." My take though is that in GR you have a coordinate ...


1

[...] concepts of Space and Time arising [...] An important guideline in this direction is surely Einstein's foundational principle that: All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more recognizable material points. The foundational "privileged ...


1

Space-time is thought to curve and ripple. Space-time, i.e. the set of all events under consideration (specificly: coincidence events), together with all relations between these events (primarily: by listing who, among all principal identifiable participants, took part any one coincident event), is thought to be not necessarily homogenous and/or ...


1

This is an open problem, the problem of Time in General Relativity and Quantum Gravity (e.g here and here). The solution would require a synthesis of General Relativiy, Quantum Field Theory and especially Thermodynamics (the 2nd Law). Many people, due to the treatment of time parameter in general relativity, take a stance that time is an illusion (sth this ...



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