Foundations — Metric tensor — raising - lowering indices

4.10.10 · D1 · Maths › Advanced Topics (Elite Level) › Metric tensor — raising - lowering indices

Is page par koi assumption nahi hai. Agar parent note mein $V^{\mu }$, $g_{\mu \nu }$, ya ${\delta }^{\mu }{}_{\nu }$ jaisa koi symbol tha aur aap sure nahi the ki har chhota mark kya matlab rakhta hai, to yahan hum unhe ek-ek karke build karte hain — aisa order mein jahan har piece sirf usse pehle waale piece par tika ho. Upar se neeche padho.

0. "Vector" aur "basis" actually kya hote hain

Arrow ko numbers mein badalne ke liye hume reference arrows chahiye honge jisse measure kar sakein. Woh reference arrows hi basis hain.

Koi bhi arrow $\mathbf{V}$ ko ${\mathbf{e}}_1$ ki kuch matra aur ${\mathbf{e}}_2$ ki kuch matra jodd kar banaya ja sakta hai. Woh "matraayen" hi woh numbers hain jise hum components kehte hain. Lekin — is poore topic ke liye yeh crucial hai — reference arrows ka right angles par hona zaroori nahi hai, aur na hi unki length 1 honi zaroori hai. Yahi freedom hai jis ki wajah se components ke do flavours aate hain.

1. Greek index $\mu$ aur dummy index — bas ek counter, kuch nahi

Parent page $\mu$ (mu), $\nu$ (nu), $\alpha$, $\beta$ jaisi letters se bhari hai. Yeh scary lagte hain. Hain nahi.

Yeh parent page padhne ki sabse badi skill hai. Jab aap $g_{\mu \nu }{V}^{\nu }$ dekhte hain, to $\nu$ repeat ho raha hai ($g$ mein neeche, $V$ mein upar), to yeh secretly ${\sum }_{\nu }{g}_{\mu \nu }{V}^{\nu }$ hai, aur $\mu$ woh ek free label hai jo bacha hua hai.

2. Upper vs lower index: "amount" ke do flavours

Ab topic ki star confusion: $V^{\mu }$ (index upar) versus $V_{\mu }$ (index neeche).

"Shadow" flavour ki poori detail Dual space and covectors mein hai; "recipe" flavour standard vector building hai.

3. Dot product — woh machine jo measure karti hai

Metric define karne se pehle hume woh ek operation chahiye jis par sab kuch bana hua hai.

Do facts jinhe hum bahut use karenge:

• Dot product se length: $\mathbf{A}\cdot \mathbf{A}=|\mathbf{A}{|}^2\cos 0=|\mathbf{A}{|}^2$. To ek arrow ka khud se dot product uski length squared hai. Yahi wajah hai ki metric lengths ko control karta hai.
• Bilinearity: dot product "har slot mein linear" hai — numbers bahar nikal sakte ho aur sums split kar sakte ho: $(a\mathbf{A})\cdot (b\mathbf{B})=ab(\mathbf{A}\cdot \mathbf{B}),(\mathbf{A}+\mathbf{B})\cdot \mathbf{C}=\mathbf{A}\cdot \mathbf{C}+\mathbf{B}\cdot \mathbf{C}.$

Parent ki length derivation ka har "$=$" inhi do facts mein se ek hai, kuch nahi. Gehri background: Inner product spaces.

4. Metric $g_{\mu \nu }$ ko un pieces se build karna jo ab tumhare paas hain

Ab sab kuch ek saath click karta hai. Metric ko reference arrows ke saare dot products ki table ke roop mein define kiya jaata hai.

Kyunki dot product ek number hai, $g_{\mu \nu }$ ek numbers ka grid hai — ek matrix. 2D mein: $g=\left(\begin{array}{cc}{\mathbf{e}}_1\cdot {\mathbf{e}}_1& {\mathbf{e}}_1\cdot {\mathbf{e}}_2\\ {\mathbf{e}}_2\cdot {\mathbf{e}}_1& {\mathbf{e}}_2\cdot {\mathbf{e}}_2\end{array}\right).$

• Diagonal entries $g_{11},g_{22}$ har reference arrow ki lengths-squared hain (ek arrow khud se dotted).
• Off-diagonal entries $g_{12}=g_{21}$ measure karte hain ki do axes kitni tilted hain ($\cos \vartheta$ times lengths). Agar axes perpendicular hain, $\cos 90^{\circ }=0$, to off-diagonals zero ho jaate hain.

Yeh "pairwise dot products ka grid" bilkul wahi hai jo Line element and ds^2 aur Minkowski spacetime pages encode karte hain — $d{s}^2$ formula hi metric hai likha hua.

5. Kronecker delta ${\delta }^{\mu }{}_{\nu }$ aur inverse metric $g^{\mu \nu }$

Do aur symbols jo parent bina ceremony ke use karta hai.

6. Reading skills ko ek saath jodna

Parent ki key line lo aur ise upar ki saari cheezein saath padho: $V_{\mu }=g_{\mu \nu }{V}^{\nu }.$

• $V_{\mu }$ — slot $\mu$ mein shadow number (free index, to yeh har $\mu$ ke liye hold karta hai).
• $g_{\mu \nu }$ — measuring grid, row $\mu$, column $\nu$.
• $V^{\nu }$ — recipe numbers.
• Repeated $\nu$ ($g$ mein neeche, $V$ mein upar) ⇒ $\nu$ par sum karo.

To $V_1=g_{11}{V}^1+g_{12}{V}^2$: grid ki row 1 lo, recipe list se dot karo. Yeh exactly matrix-times-vector hai. Poora topic hai "apni number list ko sahi grid se multiply karo."

Higher rank ke tensors (do ya zyada indices) ek index ek waqt mein same reading rules follow karte hain — dekho Tensors — definition and transformation laws aur, metric ki derivatives se aage ki machinery kaise banti hai uske liye, Christoffel symbols.

Equipment checklist

Aap Metric tensor — raising - lowering indices ke liye ready hain jab aap bina dekhe in mein se har ek ka jawab de sako:

Prerequisite Map

Connections

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