Why did Jesus cry out, "My God, My God, why have you forsaken Me?"

In Matthew 27:45-46, it says, "Now from the sixth hour darkness fell upon all the land until the ninth hour. ⁴⁶And about the ninth hour Jesus cried out with a loud voice, saying, "Eli, Eli, lama sabachthani?" that is, "My God, My God, why hast Thou forsaken Me?"  If Jesus is God, why would He say this?

First of all, Jesus quoted Psalm 22:1 which begins with, "My God, My God, why hast Thou forsaken Me?".  Jesus quoted this Psalm in order to draw attention to it and the fact that He was fulfilling it there on the cross.   Consider verses 11-18 in Psalm 22:

Be not far from me, for trouble is near; For there is none to help.¹² Many bulls have surrounded me; Strong bulls of Bashan have encircled me.  ¹³ They open wide their mouth at me,  As a ravening and a roaring lion.  ¹⁴ I am poured out like water,  And all my bones are out of joint;   My heart is like wax;  It is melted within me.  ¹⁵ My strength is dried up like a potsherd, And my tongue cleaves to my jaws;  And Thou dost lay me in the dust of death.  ¹⁶ For dogs have surrounded me; A band of evildoers has encompassed me; They pierced my hands and my feet. ¹⁷ I can count all my bones. They look, they stare at me; ¹⁸ They divide my garments among them, And for my clothing they cast lots.

The term 'dogs' was used by the Jews to refer to Gentiles (cf. Matt. 15:21-28).  His heart has melted within Him (v. 14).   During the crucifixion process, the blood loss causes the heart to beat harder and harder and become extremely fatigued.  Dehydration occurs (v. 15).  Verses 16b-18 speak of piercing His hands and feet and dividing his clothing by casting lots.   This is exactly what happen as described in Matt. 27:35.

Psalm 22 was written about 1000 years before Christ was born.  At that time, crucifixion had not yet been invented.  Actually, the Phoenicians developed it and Rome borrowed the agonizing means of execution from them.   So, when Rome ruled over Israel, it became the Roman means of capital punishment imposed upon the Jews whose biblical means of execution was stoning.  Nevertheless, Jesus is pointing to the scriptures to substantiate His messianic mission.

A further comment

2 Cor. 5:21 says, "He made Him who knew no sin to be sin on our behalf, that we might become the righteousness of God in Him."  It is possible that at some moment on the cross, when Jesus became sin on our behalf, that God the Father, in a sense, turned His back upon the Son.  It says in Hab. 1:13 that God is too pure to look upon evil.  Therefore, it is possible that when Jesus bore our sins in His body on the cross (1 Pet. 2:24), that the Father, spiritually, turned away.  At that time, the Son may have cried out.

One thing is for sure.  We have no capacity to appreciate the utterly horrific experience of having the sins of the world put upon the Lord Jesus as He hung, in excruciating pain, from that cross.  The physical pain was immense.   The spiritual one must have been even greater.

That shows us clearly how much God loves us.

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Note: All Bible quotes are from the NASB Bible.