I appreciate the responses. I confess that they are a bit over my head. If God the Son died, did God the Father also die? And if God the Son died but not God the Father, then doesn't that separate the Trinity? If you guys can simplify it as much as possible, I'd appreciate it.
When we say "God the Son died" we do not mean "God the Son ceased to exist" or "God the Son was annihilated". Since we don't believe those things happen(ed) to mere humans when they died, it would be strange to suddenly start believing them about the death of Christ.
When God the Son died, his spirit descended into Sheol and ransomed all of its dead, as depicted in the Anastasis icon. So just as the Son maintained his Communion with the Father when he became incarnate, so he maintained his Communion with the Father even in the pit of death itself. This Righteousness is life, and death could not hold life, nor could corruption seize it, which is why God raised Christ from the dead.
The Orthodox definition of death is the separation of the soul from the body not annihilation. God the Son did die for us on the Cross, because the human and divine natures of Christ are "without separation" as was declared at Chalcedon. We must understand that we cannot understand the mysteries of God. There is a hymn in the Eastern Orthodox Divine Liturgy, usually attributed to Justinian, but probably written by Servius of Antioch
O Only-begotten Son, who are immortal, yet didst deign for our salvation to be incarnate of the holy Theotokos and ever-virgin Mary, and without change wast made man; and wast crucified also, O Christ our God, and by thy deadth didst Death subdue; who art one of the Holy Trinity, glorified together with the Father and the Holy Spirit; save us.
This hymn clearly states that God the Son did die. The Father and the Holy Spirit did not die, but God the Son did die because the human and divine natures of Christ never separated. Whatever the human nature endured including death, the divine nature also endured for Christ is one person.
Fr. John W. Morris