d'Alembert operator

<h2 id="the-box-symbol-and-alternate-notations">The box symbol and alternate notations</h2>
<p>There are a variety of notations for the d'Alembertian. The most common are the <i>box</i> symbol 
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
 (<a href="/facts/Unicode/DDu3MfPV">Unicode</a>: U+2610 ☐ BALLOT BOX) whose four sides represent the four dimensions of space-time and the <i>box-squared</i> symbol 
  
    
      
        
          ◻
          
            2
          
        
      
    
    {\displaystyle \Box ^{2}}
  
 which emphasizes the scalar property through the squared term (much like the <a href="/facts/Laplacian/kov9u3PT">Laplacian</a>). In keeping with the triangular notation for the <a href="/facts/Laplacian/kov9u3PT">Laplacian</a>, sometimes  
  
    
      
        
          Δ
          
            M
          
        
      
    
    {\displaystyle \Delta _{M}}
  
 is used.
</p><p>Another way to write the d'Alembertian in flat standard coordinates is  
  
    
      
        
          ∂
          
            2
          
        
      
    
    {\displaystyle \partial ^{2}}
  
. This notation is used extensively in <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>, where partial derivatives are usually indexed, so the lack of an index with the squared partial derivative signals the presence of the d'Alembertian.
</p><p>Sometimes the box symbol is used to represent the four-dimensional Levi-Civita <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivative</a>. The symbol 
  
    
      
        ∇
      
    
    {\displaystyle \nabla }
  
 is then used to represent the space derivatives, but this is <a href="/facts/Coordinate_chart/eLKXMzzR">coordinate chart</a> dependent.
</p>
<h2 id="applications">Applications</h2>
<p>The <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a> for small vibrations is of the form 
</p>

◻
          
            c
          
        
        u
        
          (
          
            x
            ,
            t
          
          )
        
        ≡
        
          u
          
            t
            t
          
        
        −
        
          c
          
            2
          
        
        
          u
          
            x
            x
          
        
        =
        0
         
        ,
      
    
    {\displaystyle \Box _{c}u\left(x,t\right)\equiv u_{tt}-c^{2}u_{xx}=0~,}

<p>where  <i>u</i>(<i>x</i>, <i>t</i>) is the displacement.
</p><p>The <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a> for the electromagnetic field in vacuum is
</p>

◻
        
          A
          
            μ
          
        
        =
        0
      
    
    {\displaystyle \Box A^{\mu }=0}

<p>where <i>Aμ</i>  is the <a href="/facts/Electromagnetic_four-potential/S8gTFNAt">electromagnetic four-potential</a> in <a href="/facts/Lorenz_gauge_condition/Uh8N3S1o">Lorenz gauge</a>.
</p><p>The <a href="/facts/Klein%E2%80%93Gordon_equation/0a5BRyy3">Klein–Gordon equation</a> has the form
</p>

(
          
            ◻
            +
            
              
                
                  
                    m
                    
                      2
                    
                  
                  
                    c
                    
                      2
                    
                  
                
                
                  ℏ
                  
                    2
                  
                
              
            
          
          )
        
        ψ
        =
        0
         
        .
      
    
    {\displaystyle \left(\Box +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\right)\psi =0~.}

<h2 id="greens-function">Green's function</h2>
<p>The <a href="/facts/Green%27s_function/rKxKKPcp">Green's function</a>, 
  
    
      
        G
        
          (
          
            
              
                
                  x
                  ~
                
              
            
            −
            
              
                
                  
                    x
                    ~
                  
                
              
              ′
            
          
          )
        
      
    
    {\displaystyle G\left({\tilde {x}}-{\tilde {x}}'\right)}
  
, for the d'Alembertian is defined by the equation 
</p>

◻
        G
        
          (
          
            
              
                
                  x
                  ~
                
              
            
            −
            
              
                
                  
                    x
                    ~
                  
                
              
              ′
            
          
          )
        
        =
        δ
        
          (
          
            
              
                
                  x
                  ~
                
              
            
            −
            
              
                
                  
                    x
                    ~
                  
                
              
              ′
            
          
          )
        
      
    
    {\displaystyle \Box G\left({\tilde {x}}-{\tilde {x}}'\right)=\delta \left({\tilde {x}}-{\tilde {x}}'\right)}

<p>where 
  
    
      
        δ
        
          (
          
            
              
                
                  x
                  ~
                
              
            
            −
            
              
                
                  
                    x
                    ~
                  
                
              
              ′
            
          
          )
        
      
    
    {\displaystyle \delta \left({\tilde {x}}-{\tilde {x}}'\right)}
  
 is the multidimensional <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> and 
  
    
      
        
          
            
              x
              ~
            
          
        
      
    
    {\displaystyle {\tilde {x}}}
  
 and  
  
    
      
        
          
            
              
                x
                ~
              
            
          
          ′
        
      
    
    {\displaystyle {\tilde {x}}'}
  
 are two points in Minkowski space.
</p><p>A special solution is given by the <i>retarded Green's function</i> which corresponds to signal <a href="/facts/Propagator/FER8tNKP">propagation</a> only forward in time<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>

G
        
          (
          
            
              
                
                  r
                  →
                
              
            
            ,
            t
          
          )
        
        =
        
          
            1
            
              4
              π
              r
            
          
        
        Θ
        (
        t
        )
        δ
        
          (
          
            t
            −
            
              
                r
                c
              
            
          
          )
        
      
    
    {\displaystyle G\left({\vec {r}},t\right)={\frac {1}{4\pi r}}\Theta (t)\delta \left(t-{\frac {r}{c}}\right)}

<p>where 
  
    
      
        Θ
      
    
    {\displaystyle \Theta }
  
 is the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Four-gradient/dkVV0lnY">Four-gradient</a></li>
<li><a href="/facts/D%27Alembert%27s_formula/H6933ImB">d'Alembert's formula</a></li>
<li><a href="/facts/Klein%E2%80%93Gordon_equation/0a5BRyy3">Klein–Gordon equation</a></li>
<li><a href="/facts/Relativistic_heat_conduction/ckn0I7yT">Relativistic heat conduction</a></li>
<li><a href="/facts/Ricci_calculus/We4yudeq">Ricci calculus</a></li>
<li><a href="/facts/Wave_equation/J2ksD2Az">Wave equation</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=D'Alembert_operator">"D'Alembert operator"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li>Poincaré, Henri (1906). <a href="https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(July)"><i>Translation:On the Dynamics of the Electron (July)</i> </a> – via <a href="/facts/Wikisource/TrMdylNf">Wikisource</a>., originally printed in <a href="/facts/Rendiconti_del_Circolo_Matematico_di_Palermo/b7fCfvG5">Rendiconti del Circolo Matematico di Palermo</a>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/dAlembertian.html">"d'Alembertian"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Bartelmann, Matthias; Feuerbacher, Björn; Krüger, Timm; Lüst, Dieter; Rebhan, Anton; Wipf, Andreas (2015). Theoretische Physik (Aufl. 2015 ed.). Berlin, Heidelberg. ISBN 978-3-642-54618-1. OCLC 899608232.{{cite book}}:  CS1 maint: location missing publisher (link) <a href="978-3-642-54618-1" target="_blank">978-3-642-54618-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>S. Siklos. "The causal Green's function for the wave equation" (PDF). Archived from the original (PDF) on 30 November 2016. Retrieved 2 January 2013. <a href="https://web.archive.org/web/20161130174612/http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/wave_equation.pdf" target="_blank">https://web.archive.org/web/20161130174612/http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/wave_equation.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

d'Alembert operator open-in-new

d'Alembert operator