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 Yearling
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Apr
20
comment How can you tell if the work done by a force is negative?
Work is positive when the object is moved in the same direction as the force, and negative when it's moved against the force.
Apr
13
comment Acceleration of particle “held in place” at $x = 1$
@user12262 Those questions didn't help. Did you invent this notation yourself or did you get it from some other source? If the latter please forward it :). "Surely a particular value of a dimensionful quantity is independent of particular choices of units; and is not just any coordinate value either." What I'm saying is, the question as posed assumes that lengths are dimensionless. The function cosh(x) doesn't make sense otherwise. So you could say the question was expressed badly. If the question were posed better it would use cosh(Ax).
Apr
12
comment Acceleration of particle “held in place” at $x = 1$
@user12262 I'm not familiar with the notation you're using in the first comment. "Well, that's not quite the same as applying $d/d \tau$ outright in any case." You're right, it's not. That definition of acceleration is chosen because the operator $d/d \tau$ does not map tensors onto tensors while the operator $\mathbf v \cdot \nabla$ does. "But, notably, re-introducing "c"... So what's missing?" The function cosh(x) implies dimensionless x. If you're working in more common units the function would need to be cosh(Ax) for constant A with dim. $[L]^{-1}$.
Apr
8
comment Acceleration of particle “held in place” at $x = 1$
@user12262 In general: $$\mathbf a = (\mathbf v \cdot \nabla) \mathbf v$$ where $\nabla$ is the covariant derivative. When the metric is Minkowskian the operator $\mathbf v \cdot \nabla$ reduces to $d/d \tau$, however this is not true in general. The quantities appear dimensionless because in GR we tend to work in geometric units where c=G=1.
Apr
8
reviewed Approve Why do shiny surfaces reflect more light than other surfaces?
Apr
7
comment Acceleration of particle “held in place” at $x = 1$
@user12262 The components $u^\mu = \frac{d x^\mu}{d \tau}$ are certainly coordinate dependent. The quantities $dx_\mu dx^\mu$ and $u_\mu u^\mu$ are coordinate invariant. On curvilinear/non-inertial coordinates the definition of the acceleration four-vector makes use of the connection via a covariant derivative: $$a^{\mu} := u^{\nu} \nabla_\nu u^{\mu}.$$ This is done to ensure that $a_\mu a^\mu$ is also invariant, and corresponds to the square of the magnitude of the acceleration the particle would actually experience. Thanks for the +1 :)
Apr
7
comment Acceleration of particle “held in place” at $x = 1$
@user12262 I derived both the x-coordinate acceleration AND the physical (proper) acceleration (magnitude of 4-accel). They just happened to be equal due to the form of the metric ($g_{11}=1$).
Apr
7
comment Acceleration of particle “held in place” at $x = 1$
@garyp Apologies. I've removed the final answer as well as the expression for four-acceleration.
Apr
7
revised Acceleration of particle “held in place” at $x = 1$
Removed final ans to leave some work up to OP
Apr
7
comment Derivatives involving four vectors
I'm not clear on exactly what you're trying to do. Are you trying to "break up" the derivative into its temporal and spatial components? If so why? Could you give an example of where this would be useful?
Apr
6
answered Acceleration of particle “held in place” at $x = 1$
Apr
6
comment The Theoretical Minimum: Confusion Over Susskind's Reasoning for mutually orthogonal states
@MonaLisaOverdrive Have a look at the answer I posted last night. Since the graphical/Cartesian representations were confusing you I did the problem explicitly using only bra-ket notation. Let me know if this helps or if you're still confused.
Apr
6
answered The Theoretical Minimum: Confusion Over Susskind's Reasoning for mutually orthogonal states
Feb
19
reviewed Approve eigenvalues and eigenvectors for a generalized Pauli matrix in spherical coordinates
Feb
19
reviewed Approve How to test the brightness of the light bulb?
Feb
10
awarded  Yearling
Feb
1
reviewed Approve Why is Iron the most stable element?
Jan
20
reviewed Approve Why are four-legged chairs so common?
Jan
15
reviewed Approve Light reflecting off a sphere
Jan
13
reviewed Approve Partition function of primon bosonic gas