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62 votes
Accepted

Rotate an object about the time axis

This a great question, and leads to some interesting ideas. Firstly, the notion of a "rotation axis" is restricted to three dimensions. In more than three dimensions the rotation axis ...
mike stone's user avatar
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31 votes
Accepted

Why is clock synchronisation such a big deal in physics?

All of special relativity is based on the assumption that any observer can set up a coordinate system and then label spacetime events with their coordinates in that system. Then we can use the Lorentz ...
John Rennie's user avatar
31 votes
Accepted

Why/When can we separate spacetime into space and time?

The notion that a spacetime can be decomposed into a spacelike and timelike part is generally called a spacelike foliation, where your spacetime manifold can be decomposed into purely spatial ...
Slereah's user avatar
  • 15.2k
29 votes

What is the cause of the constancy of the speed of light in vacuum?

The invariance of the speed of light follows from the principle of relativity. This says there is no experiment that can distinguish between inertial reference frames: physical laws are the same in ...
Aiden's user avatar
  • 1,143
26 votes
Accepted

On mathematical level, what exactly is time in Newtonian mechanics?

Here is one way to address your question. "Time is defined so that motion looks simple." - Misner, Thorne, and Wheeler in Gravitation, p.23. Continue through to p. 26 where they say "...
robphy's user avatar
  • 10.7k
24 votes

What exactly would it take to show (hypothetically) that the speed of light (in vacuum) is not constant?

You are correct and in good modern treatments, people are careful to say that it's only meaningful to say that dimensionless quantities change with time. The "grown up version" of looking at ...
Andrew's user avatar
  • 42.9k
23 votes
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Is it "mathematically wrong" to ignore dual spaces, 1-forms, and covariant/contravariant indices in classical mechanics?

As long as you restrict yourself to orthonormal bases, then that's fine. The reason for this is that indices are "raised" or "lowered" via the metric, and in an orthonormal basis ...
J. Murray's user avatar
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22 votes
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Why is Noether's theorem not guaranteed by calculus?

Since the problem here appear to be coordinates, let's just stop using coordinates, and for simplicity consider the theory of a single scalar field on space(time) $M$: Our field is a function $\phi : ...
ACuriousMind's user avatar
  • 118k
19 votes

What is the relevance of the Lorentz factor in general relativity?

$γ$ is a geometrical factor that would appear in Euclidean geometry also if it were taught in a slightly different way. A transformation between two Cartesian coordinate systems with a common origin ...
benrg's user avatar
  • 22.9k
18 votes

What is the cause of the constancy of the speed of light in vacuum?

The constancy of $c$ is an experimental result, verified to great precision. When we make mathematical models that assume the constancy of $c$, we get predictions that match other experimental results ...
John Doty's user avatar
  • 15.1k
16 votes
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What does an area represent in a spacetime diagram?

In a (1+1)-Minkowski spacetime, the area of "causal diamond" of OQ (where Q is in the causal future of O) is equal to the squared-interval from O to Q. (The causal diamond of OQ is the ...
robphy's user avatar
  • 10.7k
13 votes
Accepted

Schwarzschild metric in terms of a falling observer's coordinates

Habouz asked: "What's the Schwarzschild metric in terms of a falling observer's coordinates?" The transformation rule can be found here and here: $${\rm dt = d\tau+dr \ v} / \hat g_{\rm tt} ...
Yukterez's user avatar
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12 votes
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What exactly would it take to show (hypothetically) that the speed of light (in vacuum) is not constant?

In the Minkowski metric the $dt$ is the time I measure on my clock and the $dx$, $dy$ and $dz$ are the distances I measure with my ruler. That's how those coordinates are defined. Suppose we choose ...
John Rennie's user avatar
12 votes
Accepted

How many frames of reference are there in the twins paradox of relativity?

It depends on the detailed set-up of the thought experiment. However, if you assume the travelling twin's periods of acceleration are negligibly short, then there are three inertial frames to consider,...
Marco Ocram's user avatar
  • 22.9k
12 votes

Twin Paradox (SR): How can we express the comparative length of arbitrary world-lines mathematically?

@Chris. In response to your update, find a criterion (for who ages less) based simply on what the twins observe between the moment of separation and the moment of reunion. What can they observe ? ...
robphy's user avatar
  • 10.7k
12 votes
Accepted

Why is relativity of simultaneity so special?

The crucial mistake you are making is to focus on the reception of the signals, not the events that generated them. Clearly Sally receives the blue and red signals at different times, and that would ...
Marco Ocram's user avatar
  • 22.9k
11 votes

Why do we need a unit vector?

You'll find plenty of examples from classical, introductory mechanics and physics. For instance, you will often need to extract the components of a vector, such as a force vector or a velocity vector. ...
Steeven's user avatar
  • 49.5k
11 votes

Why is Noether's theorem not guaranteed by calculus?

It might be easier to see what's going on by making a few simplifications: First, we can work in $0+1$ dimensions -- in other words, we can work with ordinary particle mechanics, where the action is ...
Andrew's user avatar
  • 42.9k
10 votes

Proper conceptualization & notation for vectors, $n$-tuples, and matrices in physical space

I think the core of your question is a very commonly-misunderstood subtlety, so I'll begin with a seemingly abstract example. Consider the vector space $V$ which consists of formal polynomials of ...
J. Murray's user avatar
  • 63.3k
10 votes

Twin Paradox (SR): How can we express the comparative length of arbitrary world-lines mathematically?

The twin who ages less is the twin for whom the following mathematical formula $$ \tau = \int \sqrt{1-\frac{v(t)^2}{c^2}}dt$$ is smaller than the same mathematical formula for his twin. In this ...
Dale's user avatar
  • 86k
10 votes

Which comes first, basis vectors or coordinates?

You can do either one. Any choice of coordinates $x^i$ on a neighborhood $U\subseteq M$ induces a basis $\frac{\partial}{\partial x^i}$ for the tangent space $T_pM$ at each point $p\in U$. Given a ...
J. Murray's user avatar
  • 63.3k
9 votes

What is the cause of the constancy of the speed of light in vacuum?

This is not a question of cause and effect, but a question of definition. A local inertial frame is defined as one where locally you measure the speed of light to be $c$ (among other properties). So ...
Dale's user avatar
  • 86k
9 votes

Why is relativity of simultaneity so special?

For one thing, in the sound example the pressure waves are moving through the air or some fluid medium. You could choose a reference frame where the bulk motion of the fluid with respect to the ...
RC_23's user avatar
  • 6,332
8 votes
Accepted

What is the difference between the metric (tensor), $g_{\mu\nu}$, and the invariant interval, ${ds}^2$?

If you want to be super systematic about language and not overloading the terminology, you can say the following. Fix a smooth $n$-dimensional manifold $M$. A pseudo-Riemannian metric tensor field on ...
peek-a-boo's user avatar
  • 4,420
8 votes

Is there are relationship between the Ricci scalar and the determinant?

The determinant of the metric tensor $\det g $ is a scalar density of weight $+2$, and thus, when expressed in terms of local coordinates, transforms with the square of the Jacobian of the mapping. It ...
K.T.'s user avatar
  • 1,158
8 votes
Accepted

Can the metric tensor always be diagonalized?

It is always possible to find coordinates such that the metric is diagonal at some chosen event (indeed, it can even be Minkowski at a chosen event). In more than 3 dimensions it is not always ...
Andrew Steane's user avatar
8 votes
Accepted

If momentum is a covector, how does $p=mv$?

Yes, OP points to the fact that classical mechanics typically relies on the existence of a fiducial/distinguished/background metric structure on the configuration manifold so that we can apply the ...
Qmechanic's user avatar
  • 188k
8 votes
Accepted

What is the difference between Poincare symmetry and coordinate independence in field theory?

First, there is something important to clarify here, QFT has nothing to do with this. This is a statement that arises in classical field theory before quantization ever comes into the picture. ...
Wintermute's user avatar
8 votes

Uniqueness of the diagonal form of metric

The diagonal form of the metric is highly non-unique and we can illustrate that with a much simpler example than Schwarzschild. The $\mathbb{R}^3$ metric is diagonal in Cartesian coordinates: $$ds^2=...
Gold's user avatar
  • 34.2k
8 votes

Why does it matter that a passive transformation can be "interpreted" as an active transformation?

The subtlety arises from the fact the association between matrices and linear maps is non-canonical and requires a choice of a preferred basis of the vector space. An active transformation on $V$ is a ...
Fraxinian's user avatar
  • 126

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