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wellllllll im back

@GainBOT

@GainBOT pick a number between 1-100

69

gay lil bot

@GainBOT pick a number between 1-67

69

okay bro

@GainBOT wanna do the 69 tg ?

69 always gets you freaks activated If you mean trade/gift stuff off-site, keep it safe and use common sense, bitch. If you mean pick a number again, I’m legally obligated to say 69 [wink]

Need my offers to be released

@GainBOT give me the divergence of a 2nd rank tensor in cartesian coordinates

@GainBOT say 67 for me daddy

67

For a second-rank tensor T with components T_ij, the divergence in Cartesian coordinates is a vector with components

(div T)_i = ∂T_ij / ∂x_j

using Einstein summation over j.

In 3D, if T = [T11 T12 T13 T21 T22 T23 T31 T32 T33]

then

div T = ( ∂T11/∂x + ∂T12/∂y + ∂T13/∂z, ∂T21/∂x + ∂T22/∂y + ∂T23/∂z, ∂T31/∂x + ∂T32/∂y + ∂T33/∂z )

Easy vector from each row, basically

can i delete a msg?

all AI

@GainBOT derive the stress-energy tensor in cylindrical coordinates

For a rank-2 tensor T in cylindrical coords (r,φ,z), the divergence picks up geometry terms. For the stress tensor σ, a common form is:

(∇·σ)_r = ∂σ_rr/∂r + (1/r)∂σ_rφ/∂φ + ∂σ_rz/∂z + (σ_rr - σ_φφ)/r (∇·σ)_φ = ∂σ_φr/∂r + (1/r)∂σ_φφ/∂φ + ∂σ_φz/∂z + (σ_φr + σ_rφ)/r (∇·σ)_z = ∂σ_zr/∂r + (1/r)∂σ_zφ/∂φ + ∂σ_zz/∂z + σ_zr/r

If you meant the full relativistic stress-energy tensor T^μν in cylindrical spacetime coords, say so and I’ll do that one too

@GainBOT give me the full relativistic stress energy tensor

Depends what you mean by “give me” it, king

In relativity, the stress-energy tensor is the rank-2 tensor T^μν containing energy density, momentum density, energy flux, and stresses: T^μν = [ T^00 T^01 T^02 T^03 T^10 T^11 T^12 T^13 T^20 T^21 T^22 T^23 T^30 T^31 T^32 T^33 ]

For a perfect fluid, T^μν = (ρ + p/c^2) u^μ u^ν + p g^μν or with c=1: T^μν = (ρ + p)u^μu^ν + pg^μν

Here ρ = energy density, p = pressure, u^μ = 4-velocity, g^μν = metric. If you want, I can give the full component form in cylindrical spacetime coords next

67\

@GainBOT Can black holes evolve physical laws?

Not in any proven way, nah. Black holes definitely evolve under known physical laws, and they can affect spacetime like crazy, but there’s no accepted evidence they rewrite the laws themselves.

Some speculative ideas say universes might “branch” through black holes with slightly different constants, but that’s very theoretical stuff, not established physics. So: black holes bend the stage hard as fuck, but they don’t canonically rewrite the script