Undefined variable’s error when expected to have passed the content of variable

I implemented a simple class like:

class my_form {

        function __construct()
        {
            add_shortcode('my_form_custom', array($this, 'show_custom_form'));
        }

        function registration_template_form() {
        ?>              
            <div class="popup_canvas_container">            
                <form id="my_form" class="my_form" method="post" action="<?php echo esc_url($_SERVER['REQUEST_URI']); ?>">
                    <p><label for="first_field">Field1:</label>&nbsp;<input type="text" name="first_field" id="first_field" value="<?php echo(isset($_POST['first_field']) ? $_POST['first_field'] : null); ?>" required /></p>
                    [...]
                    <div class="confirmation_box">
                        <p><button name="post_form" id="post_form" type="submit"><?php _e('Submit', 'my_form_textdomain'); ?></button></p>                      
                    </div>
                </form>
                <div id="messages_box" class="messages_box">
                    <p><div id="message_label"></div></p>
                </div>              
            </div>          
            
        <?  
        }                   
                    
        function form_validation() {
            if ( strlen($this->first_field) < 4 ) { 
                $error_message = 'Too short!';
                return false;
            }
        }
                    
        function save_to_db() {         
            if ( !$this->form_validation() ) {
                ?>                   
                    <script language="javascript">$("#message_label").text("<?php echo $error_message; ?>"); $('#messages_box').dialog();</script>
                <?php
            } else {    
            }
        }
                    
        function show_custom_form() {
            ob_start():
            
            if (isset($_POST['post_form'])) {
                $this->save_to_db();
            }
            
            $this->show_template_form();
            
            return ob_get_clean();
        }
}

and , as like as I found almost everywhere on Google, to pass content’s variable to the message box div with jquery and I just did an echo so then I should show the error message displaying a dialog message.
But I keep getting in Console Log this error :

<script language="javascript">$("#message_label").text("<br />
<b>Notice</b>:  Undefined variable: error_message in <b>..../my_form.php</b> on line <b>108</b><br />
"); $('#messages_box').dialog();</script>

at this point what would be the right way to pass variable’s content to the div? Thanks in advance! Cheers, Luigi