Visual walkthrough — Metric tensor — raising - lowering indices

4.10.10 · D2 · Maths › Advanced Topics (Elite Level) › Metric tensor — raising - lowering indices

Step 0 — "Component" kya hota hai? (woh cheez jo hum translate kar rahe hain)

Subtle baat yeh hai: agar basis arrows right angles par nahi hain, toh seedha recipe "kitne ${\mathbf{e}}_1$, kitne ${\mathbf{e}}_2$" ka matlab same nahi hai jaise "har arrow ke along $\mathbf{V}$ kitna pahunchta hai". Yeh do alag questions hain, aur har ek ko apni alag list of numbers milti hai. Yahi is page ki poori kahani hai.

• Red arrows basis ${\mathbf{e}}_1,{\mathbf{e}}_2$ hain — deliberately perpendicular nahi.
• Yellow arrow $\mathbf{V}$ hai.
• Dashed blue lines parallelogram rule hain: ${\mathbf{e}}_1$ ke along slide karo, phir ${\mathbf{e}}_2$ ke along, tip tak pahunchne ke liye. Unki lengths contravariant numbers $V^1,V^2$ deti hain.

Step 1 — Doosri list: recipe ki jagah projection

DOT PRODUCT "kitna along" kyun answer karta hai. Kisi bhi do arrows ke liye, $\mathbf{V}\cdot {\mathbf{e}}_{\mu }=|\mathbf{V}||{\mathbf{e}}_{\mu }|\cos \theta$, aur $|\mathbf{V}|\cos \theta$ exactly $\mathbf{V}$ ke shadow ki length hai ${\mathbf{e}}_{\mu }$ par (times $|{\mathbf{e}}_{\mu }|$). Toh dot product hi "along reach" ka measurement hai — isliye humne ise choose kiya na ki, say, koi area.

YEH KAISE DIKHTA HAI. $\mathbf{V}$ ki tip se har basis arrow par perpendicular giraao.

• Yellow: vector $\mathbf{V}$ (Step 0 jaisa hi).
• Green dashed lines: perpendicular drops. Har drop ka foot shadow mark karta hai.
• Step 0 ke blue parallelogram se compare karo: dono constructions arrows par alag jagah land karti hain. Yahi mismatch reason hai ki $V^{\mu }\ne V_{\mu }$ generally.

Step 2 — Jab dono lists agree karti hain (easy case par intuition build karo)

KYUN. Right angles ke saath, ${\mathbf{e}}_1$ ke along slide karna hi ${\mathbf{e}}_1$ par perpendicular drop hai — parallelogram rectangle ban jaata hai, aur uski side shadow hai. Unit length ke saath koi rescaling bhi nahi hoti.

Yeh high school wali ordinary Cartesian duniya hai — exactly isliye kisine tumhe kabhi nahi bataya ki components do tarah ke hote hain. Jaise hi axes tilt ya stretch hoti hain, dono lists alag ho jaati hain, aur hume ek translator chahiye. Woh translator metric hai.

Step 3 — Metric introduce karo: arrow dot-products ki ek table

YEH TABLE EXACTLY VHI KYUN HAI JO HUMHE CHAHIYE. Yeh un do cheezoon ko record karti hai jinhone Step 2 ki easy duniya todhi: arrows ki lengths (diagonal entries) aur unka tilt (off-diagonal entries). Na zyada, na kam. Yeh symmetric hai, $g_{\mu \nu }=g_{\nu \mu }$, kyunki $\mathbf{a}\cdot \mathbf{b}=\mathbf{b}\cdot \mathbf{a}$.

• Diagonal cells (blue) = har arrow ki squared lengths.
• Off-diagonal cells (yellow) = shared tilt ${\mathbf{e}}_1\cdot {\mathbf{e}}_2=|{\mathbf{e}}_1||{\mathbf{e}}_2|\cos \varphi$, jahan $\varphi$ arrows ke beech ka angle hai.
• Jab $\varphi =90^{\circ }$, $\cos \varphi =0$, yellow cells gayab ho jaate hain, aur hum Step 2 mein wapas hain. Dekho kaise table easy case ko contain karta hai.

Step 4 — Lowering derive karo: recipe ko shadows mein badlo

KYA / KYUN, line by line. Shadow ki definition se shuru karo (Step 1), phir recipe substitute karo (Step 0):

Term by term:

• $\mathbf{V}\cdot {\mathbf{e}}_{\mu }$ — woh shadow question jiska hume chahiye jawab.
• $V^{\nu }{\mathbf{e}}_{\nu }$ — $\mathbf{V}$ ko uski recipe se replace karo (woh numbers jo hamare paas hain).
• scalar $V^{\nu }$ ko bahar nikalo — dot product bilinear hai, numbers slide out ho jaate hain.
• ${\mathbf{e}}_{\nu }\cdot {\mathbf{e}}_{\mu }$ by definition metric table ki ek entry hai.

Toh metric woh machine hai jo tumhare paas wali list (recipe, upper) ko tumhari chahiye wali list (shadow, lower) mein convert karti hai.

Picture $V_1$ ka arithmetic dikhati hai: har contravariant number $V^{\nu }$ ko table ki row $1$ mein store tilt/length se multiply kiya jaata hai, aur results sum kiye jaate hain. Jab metric off-diagonal hota hai tab $V^2$ term $V_1$ mein "leak" karta hai — woh leak tilt hai.

Step 5 — Raising derive karo: tumhe inverse table use karni padegi

INVERSE KYUN, NA KI $g_{\mu \nu }$ PHIR SE. Lowering ne $g_{\mu \nu }$ se multiply kiya. Wapas jaane ke liye us multiplication ko cancel karna padega — aur jo cheez matrix ko cancel karti hai woh uska inverse hai. Lowering formula ko $g^{\alpha \mu }$ se multiply karo aur sum karo:

Loop $V^{\mu }\stackrel{g_{\mu \nu }}{\to }{V}_{\mu }\stackrel{g^{\mu \nu }}{\to }{V}^{\mu }$ tumhe ghar wapas laata hai — kyunki $g^{\mu \alpha }{g}_{\alpha \nu }={\delta }^{\mu }{}_{\nu }$ exactly "table phir undo-table = kuch nahi karna" hai.

Step 6 — Har case, ek plate par (degenerate & signed metrics)

Hume koi bhi scenario undrawn nahi chhodna. Teen regimes sab cover kar lete hain:

• Left panel (green): identity — dono lists ek doosre ke upar baith jaati hain.
• Middle (blue): diagonal stretch — perpendicular lekin rescaled, arrows direction mein line up rehte hain, numbers rescale hote hain.
• Right (red): ek negative sign covariant version ko doosri side par flip kar deta hai — yahi indefinite metric ki pehchaan hai.

Degenerate warning: agar kisi $g_{\mu \nu }$ ka determinant $0$ hota (ek squashed basis jahan ek direction collapse ho jaaye), toh inverse $g^{\mu \nu }$ exist nahi karta aur raising impossible ho jaata. Ek usable metric non-degenerate honi chahiye — yahi two-way translation ki cost hai.

Ek-picture summary

Ek diagram, poora walkthrough: contravariant $V^{\mu }$ (recipe / parallelogram) left par, covariant $V_{\mu }$ (shadows / perpendicular drops) right par, aur metric table $g_{\mu \nu }$ (down = lower) aur uska inverse $g^{\mu \nu }$ (up = raise) dono ke beech cross karte arrows ke roop mein.

Connections

• Metric tensor — raising - lowering indices (parent)
• Einstein summation convention
• Inner product spaces
• Dual space and covectors
• Line element and ds^2
• Minkowski spacetime
• Tensors — definition and transformation laws
• Christoffel symbols