All Questions
Tagged with differentiation spacetime
15
questions
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What happens if we differentiate spacetime with respect to time? [closed]
Essentially, what would differentiating space-time with respect to time provide us with? What are the constraints associated with such operations? Is it possible to obtain a useful physical quantity ...
8
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2
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How does the covariant derivative satisfy the Leibniz rule?
In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
- 103
3
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2
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165
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What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?
In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
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2
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171
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Does covariant derivative include magnitude change of a vector as well as direction change of the same vector?
Does covariant derivative include magnitude change of a vector as well as direction change of the vector? In some explanations I followed I have not noticed mentioning of magnitude change along with ...
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2
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Derivatives in the Lorentz Transformation
I am trying to better understand the Lorentz Transformation on a fundamental level and gain some intuition of it. In the Lorentz Transformation, the derivative of x' with respect to x must be a ...
- 365
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3
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680
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Solving the Euler-Lagrange equations for a complex scalar field in which the time derivatives and gradient are separate
This is found at the bottom of page 9 of David Tong's QFT lectures. The Euler-Lagrange equations for the complex scalar field:
$$\mathcal L=\frac{i}{2}(\psi^*\dot\psi-\dot{\psi^*}\psi)-\nabla\psi^*\...
- 6,963
4
votes
1
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Special relativity - Einstein's transformations
I am reading "On the electrodynamics of moving bodies" and have got to page 6 and become stuck. Is anyone able to please help explain how:
Einstein went from the first line of workings to ...
- 103
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4
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What is proper time, proper velocity and proper acceleration?
I am trying to derive the relativistic rocket equations found here [(4),(5),(6),(7),(8)] but I do not understand proper time, proper velocity and proper acceleration.
Define a point $P$ with ...
- 201
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1
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Regd. derivation of some equations in “Bertrand Spacetimes” by Pelick
We are going through "Bertrand Spacetimes" by Dr Perlick, in which he first gave the idea of a new class of spacetimes named as Bertrand spacetimes after the well-known Bertrand's Theorem in Classical ...
-3
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1
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If the space-time curve was quantified and has a mathematical function what would the derivative of the function mean? [closed]
Say I have a massive object:
This object as we know causes spacetime to bend and curve. The "maximum curve" which I would define as the line in space time that runs directly through the center of ...
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Space (distance) as a derivative [duplicate]
We are all used to the mechanical definitions of velocity, as the variation in time of the distance, the acceleration as the variation in time of the velocity, and even more other quantities like the ...
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2
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118
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Why do we use differential equations in physics instead of $h$-difference ones?
Since we don't know whether space and time are discrete or continuous wouldn't it be a better idea to use $h$-difference equations where the derivative is $$f'(x) =\frac{f(x+h)-f(x)}{h},$$ since they ...
- 159
9
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3
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2k
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Intuition behind differential operators as the basis vectors of a manifold (space-time)
I understand that in order to provide a basis for every point in space-time, the differential operators, $\partial_\mu$ (or partial derivative operator with respect to each one of the curvilinear ...
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3
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Motivation for covariant derivative axioms in the context of General Relativity
In General Relativity the idea of a covariant derivative on a manifold is quite important and it is usually defined by a set of axioms:
Let $M$ be a smooth manifold. A covariant derivative $\nabla$ ...
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4
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Do integrals of position make any sense? Do they have an application? [closed]
I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any ...
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