4.10.10
Ek vector kisi space mein rehta hai. Geometry karne ke liye (lengths, angles, dot products) tumhe ek ruler chahiye jo bataye ki measure kaise karna hai. Woh ruler hai metric tensor g μν
KYA karta hai: usi vector ko describe karne ke do "dialects" ke beech convert karta hai — contravariant components V μ covariant components V μ KYUN chahiye: non-Cartesian / curved / Minkowski spaces mein, V μ V μ alag numbers ki lists hoti hain. Dot product A ⋅ B ∑ A μ B μ g μν A μ B ν KAISE : g μν lower karo, aur iske inverse g μν raise karo.
Definition Contravariant vs covariantTangent space ke liye ek basis { e μ }
Contravariant components V μ V = V μ e μ opposite transform hote hain — basis ko chhota karo, components bade ho jaate hain.Covariant components V μ project karke milte hain. Yeh basis ke saath transform hote hain.
Plain Cartesian coordinates mein orthonormal axes ke saath dono same hote hain, isliye school mein kabhi mention nahi hote.
Intuition "Opposite" naming kyun?Units metres se centimetres mein badlo. Ek basis vector e x chhota ho jaata hai (1 cm < 1 m). Lekin ek fixed physical displacement mein unki ginti badi ho jaati hai (300 cm vs 3 m). Components basis ke contra (ulte) vary karte hain — isliye contra variant, upper index.
g μν ≡ e μ ⋅ e ν g μν = g ν μ har pairwise dot product store karta hai. Inverse metric g μν g μα g α ν = δ μ ν ( Kronecker delta ) .
KYUN yeh sahi object hai. V = V μ e μ ∥ V ∥ 2 = V ⋅ V = ( V μ e μ ) ⋅ ( V ν e ν ) = V μ V ν ( e μ ⋅ e ν ) = g μν V μ V ν . g μν forced ho jaata hai jab hum components se lengths chahte hain.
Derivation.
V μ = V ⋅ e μ = ( V ν e ν ) ⋅ e μ = V ν ( e ν ⋅ e μ ) = g μν V ν .
Raising. Dono sides ko g α μ g α μ g μν = δ α ν g α μ V μ = g α μ g μν V ν = δ α ν V ν = V α .
Mnemonic "g matching index khaata hai, naya wala deta hai"Metric Down, Lower Down — down indices wala g lower karta hai. Up indices wala g g μν raise karta hai. Repeated (summed) index hamesha ek baar up, ek baar down aata hai.
Ek baar mein ek index raise/lower karo, g T μ ν = g ν α T μα , T μν = g ν β T μ β , R μν = g α β R μα ν β . KYUN ek ek karke: g ek free index ko dummy ke saath pair karti hai, exactly usi slot ko convert karti hai.
Ek neat consistency check: lower karo phir raise karo to original wapas milna chahiye.
g μ β ( g β ν V ν ) = ( g μ β g β ν ) V ν = δ μ ν V ν = V μ . ✓
Worked example (a) Minkowski spacetime, signature( − , + , + , + ) g μν = diag ( − 1 , 1 , 1 , 1 ) U μ = ( 2 , 1 , 0 , 0 )
U 0 = g 0 ν U ν = g 00 U 0 = ( − 1 ) ( 2 ) = − 2 Kyun? Diagonal metric ⇒ sirf ν = 0 U 1 = g 11 U 1 = ( 1 ) ( 1 ) = 1 Toh U μ = ( − 2 , 1 , 0 , 0 ) yahi indefinite metric ka poora point hai.
Norm: U μ U μ = ( − 2 ) ( 2 ) + ( 1 ) ( 1 ) + 0 + 0 = − 3.
Worked example (b) 2D polar coordinatesLine element d s 2 = d r 2 + r 2 d θ 2 ⇒ g μν = ( 1 0 0 r 2 ) g μν = ( 1 0 0 1/ r 2 ) V μ = ( V r , V θ )
V r = g r r V r = V r V θ = g θ θ V θ = V θ / r 2 1/ r 2 θ e θ r
Worked example (c) Off-diagonal metricg μν = ( 1 1 1 2 ) V μ = ( 3 , 1 )
V 1 = g 1 ν V ν = g 11 V 1 + g 12 V 2 = ( 1 ) ( 3 ) + ( 1 ) ( 1 ) = 4. V 2 = g 21 V 1 + g 22 V 2 = ( 1 ) ( 3 ) + ( 2 ) ( 1 ) = 5. Dono terms kyun include kiye? Off-diagonal g 12 = 0 ignore nahi kar sakte .
V μ V μ Kyun sahi lagta hai: Cartesian orthonormal coordinates mein g μν = δ μν hain equal — yahi sabka pehla experience hota hai.
Fix: woh equal hote hain sirf tab jab g = 1
Common mistake Dot product= ∑ μ A μ B μ Kyun sahi lagta hai: jaana-pahchana Euclidean formula.
Fix: invariant hai A μ B μ = g μν A μ B ν ek up, ek down hona chahiye. Do ups (ya do downs) valid contraction nahi hai.
g μν raise karna.
Kyun sahi lagta hai: "metric index gymnastics karta hai, toh sab ke liye use karo."
Fix: down-indexed g μν lower karta hai; raise karne ke liye tumhe inverse g μν g
Recall Feynman: 12-saal ke bache ko samjhaoSocho ek treasure map stretchy rubber pe bana hai. Treasure kahan hai yeh batane ke liye tum ya toh steps ginoge ("3 east, 2 north") ya describe karoge ki yeh har direction ke saath kitna align karta hai. Flat un-stretched map pe dono descriptions same numbers dete hain. Lekin agar rubber ek direction mein zyada stretched hai, toh dono descriptions alag hogi — aur tumhe ek choti "stretch table" (the metric) chahiye ek ko doosre mein translate karne ke liye. Index lower karna = stretch table use karo; raise karna = undo table use karo.
Recall Active recall checkpoint
g μν V μ = g μν V ν Inverse metric kyun indices raise karta hai?
diag ( − 1 , 1 , 1 , 1 ) U μ = ( 2 , 1 , 0 , 0 ) U μ
Metric tensor g μν Basis vectors ka dot product, g μν = e μ ⋅ e ν
Index lower karne ka formula? V μ = g μν V ν
Index raise karne ka formula? V μ = g μν V ν g μν
g μν g μν g μα g α ν = δ μ ν
V μ V μ Woh equal hote hain sirf jab g μν = δ μν
A B A μ B μ = g μν A μ B ν
Minkowski ( − , + , + , + ) Sign flip ho jaata hai: V 0 = − V 0
Polar metric aur V θ V θ g θ θ = r 2 g θ θ = 1/ r 2 V θ = V θ / r 2
Dot product A.B = g_mu_nu A^mu B^nu
